Risk Management Value at Risk by als32300

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Risk Management
1     Foundations
1.1     Terms and Definitions
In 1973 the stable exchange rates disappeared. Risk mangement became a well-used term.
In a perfect market one can evaluate projects seperately ⇒ Modigliani&Miller Theorem
which states that value of a firm can be evaluated without ist capital structure holds ⇒
one has to be precise when talking about risk management.
Broad definition of risk management: One is always interested in an optimal combination
of risk and return.
Narrow definition of risk management: A job of the finance department in a corporation.
An upper limit for the overall risk is set by the finance department. Then the risk is divided
to different sections. Afterwards a Portfolio is selected to optimize risk management.
People often define risk management as risk minimization. This is not true because this
might not be the optimal combination. No risk ⇒ no return. One must take some risk
in order to earn a risk premia. One optimizes over risk and risk premia.

1.2     Reasons and jobs for risk management
Modigliani&Miller states that in a perfect market risk management is irrelevant. Equal
access: Everything done by a compnay can be also done by an individual. This is a reason
for the M&M arguement. The financing policy has no impact. The investment policy is
then also independent of the financing policy.
Investment: Series of random cash flows. If one knows the market value of each cash flow,
then the investment policy becomes very simple. M(˜1 , e2 , . . . , eT ) ⇒ No risk management
                                                      e ˜           ˜
is needed.
Fisher seperation: One can decide of each project seperately by looking at the market
value of the series of cash flows.
Example: Exporting firm to Japan:
py w1 − k = profit/loss where py : Price of Yen, w1 : Exchange rate in 1 year, k: Cost.
   ˜
In a perfect capital market: py f1 − k with f1 : Forward rate ⇒ Only deterministic figures.

Imperfections in capital markets:

    1. Insolvence cost: Costly in imperfect markets. Irrelevance arguement does not hold
       anymore. Without risk management one rejects some projects which might be
       valueable. For example reducing bancrupcy cost with risk management.

    2. Taxation: Government can be seen as a silent stock holder. The governmennt has
       a call option on profits:




                                             1
      If the volatility is high, the market value is high. If risk management reduces the
      volatility, then the price of the call option is reduced.

  3. Lack of information: An individual has not all information to do risk management
     on his own.

  4. Transaction costs are very high when individuals need to do risk management.

  5. Agency problems: Start with the idea of asymmetric information. Stockholders want
     to hedge risk so that there are no excuse for losses. They want risk management to
     evaluate the managers performance.

Not everybody has a unique view on risk management:
Bondholders ⇐⇒ Stockholders
Employees ⇐⇒ Owners, etc.
Legal side: HGB 317II and AG 91

1.3    Types of risk
  1. Functional classification: Sales department, Financing, Purchasing and Production
     ⇒ each one has a risk

  2. Origin classification: Nature risk, behavioral risk, legal risk, political risk, supply-
     and demand risk, risk in research and devolopment, trade vs. non- trade risk

       a) foreign exchange risk
       b) storm, fire vs. demand risk

  3. Financial intermidiar risk:

       a) default risk
       b) price risk
          ba)   interest rate risk
          bb)   stock market risk
          bc)   exchange rate risk
          bd)   liquidity risk (quick selling risk, refinancing risk)
       c) operational risk (computer breakdown, moral hazard ⇒ Basel II)
       d) economic risk

1.4    Instruments of risk management
  1. Selection of investment projects

  2. Measures about reducing potential damages

  3. Constractual arrangements to reduce our risk




                                              2
ad 1. High risk stocks: New economy
      Low risk stocks: Power utility companies
      May companies nowadays concentrate on corps business. It is risky but it is easier
      to run a competitive company. It costs diversification.

ad 2. Many damages can be offset by setting up a set of securities. Many companies are
      forced to do so by law. Insurance companies also insist companies to do so.

ad 3. The more risk is shifted to another person the higher the price will be.
      Example: House owners vs. rent, leasing contracts, limited liability.
      Derivatives hedge price risk. Hedging a foreign exchange rate does not cost a risk
      premium because if one side earns a positive risk premium the other side earns a
      negative one.
      So in talking about hedging, one has to be aware of risk premium.
      Insurance copanies have the problem of adverse selection (some insurance compa-
      nies sell life insurance to sick people and some do not) and moral hazard ( less
      carefulness, fraughtful behavior)




                                           3
1.5    Different objective variables of risk management
                  e ˜               ˜
Cash flow series: (˜1 , e2 , . . . , e2 ) Measurement of economic risk.
One could manage the risk of each period seperately. But it is a very dangerous policy
because there might be a correlation between these cash flows. If the correlation is negative
everyone implying this strategy will run into bancrupcy. (Looking at the sum of the cash
flows)
              ˜
              e2        ˜
                        e3
M V1 = e1 + (1+k) + (1+k)2
       ˜
M V1 |I1 = e1 |I1 + E(˜2 |I1 ) + E(˜3 |I1 )
                      e
                     (1+k)
                                   e
                                 (1+k)2
k= risk- adjusted discount rate= risk- free rate + some risk premium
The model is simplified because a constant k is assumed.
Adjusting for that:
                    E(˜2 |I      E(˜3 |I
M V1 |I1 = e1 |I1 + (1+k|I1 ) + (1+k|I11))2
                       e
                            1)
                                    e

If the risk- free rate rises for example people are more likely to invest in bonds. So stock
prices might fall or remain the same. So this model becomes very complex because it is
hard to estimate even the standard derivation.
Example:
100 for 5 years:

 (1) Buy a zero bond with yield 6% which matures after 5 years.

 (2) Buy a coupon bond at par value which pays 6% annually.

 (3) Buy a floating rate note at par value which pays LIBOR (One knows only the first
     12 month forward rate) annualy.

Cash flow series:

 (1) Zero bond after five years: 100 ∗ 1, 065 = 133, 82. No cash flow risk.

 (2) Coupon bond: No cash flow risk if one consumes the 6 immediately. Only cash flow
     risk if the money is reinvested.

 (3) FRN: cash flow risk

Market value at date 1 ex coupon:

 (1) Zero bond: Highest risk

 (2) Coupon: Smaller risk

 (3) FRN: No risk

Assumption: No default risk
V1 changes as the interest rate changes in the coupon bond. The level of change is
measured by the interest rate.




                                             4
     Assumption: Term structure of interest rate is flat.
It changes only by parallel shifts. This model is a problem because the market is not
arbitrage free.
r1 : Level of yield at time 1.
E0 (r1 ). dr1 is close to the expectation. Only small deviation from the expected value.
          ∂V1
dV1 = ∂(1+r1 ) dr1




                                  ∂V1
     Slope of the tangent: ∂(1+r1 )
Since we are only considering small changes in the yield, the above model can be implied.
                                                     ∂V1
Risk measured in standard deviation: σ(dV1 ) = ∂(1+r1 ) σ(dr1 )
                          ˜
dV1 + V1 (E0 (r1 )) = V1 (random)
σ(dV1 ) = σ(V1 )˜
               r
σ(dr1 ) = σ(˜1 )
Result: σ(V  ˜1 ) = ∂V1 σ(˜1 ) Only holds for small changes.
                               r
                      ∂(1+r1 )
Critical: Slope of the tangent.
   ∂V1       1
∂(1+r1 )
         ∗ 10000 = price value of a basis point
V1 = T (1+rt )−1 (Value at year 1)
         t=1
                  e
                    1
   ∂V1
∂(1+r1 )
         = T −(t − 1) (1+r1 )t
               t=1
                               et

ei : Future cash flow at year i.
Duration:
                                                −t+1
                                         e
D1 = − ∂(1+r1 ) ∗ 1+r1 = T (t − 1) T t (1+r1 ) −t+1 := T (t − 1)gt
            ∂V1
                       V1      t=1                       t=1
                                            e (1+r1 )
                                         t=1 t
  T
  t=1 (t   − 1)gt = 1 used as weights.

Returning to the evaluation of the market value:
Duration:

 (1) Zero coupon bond: 4 years.

 (2) Coupon bond: Assumption: E0 (r1 ) = 6%
                 6           6           6           6             6            1
     D1 = [0 ∗ 1,060 + 1 ∗ 1,061 + 2 ∗ 1,062 + 3 ∗ 1,063 + 4 ∗   1,064
                                                                       ]   ∗   106
                                                                                     = 3, 47

 (3) FRN: D1 = 0 (shown later)


                                                 5
 The larger the time to the next payment the more sensitive the price is to a change in
 interest rates.

 Modified duration:
   m       D1   −∂V1   1
 D1 = 1+r1 = ∂(1+r1 ∗ V1 (relative price change)
 For the coupon bond:
 D1 = 3,47 = 3, 27 years
   m
          1,06
   ∂V1         m
 ∂(1+r1 )
          = −D1 V1
 For the coupon bond:
 -3,27*106% ⇒ 6% → 7%: ∆V1 = −3, 27 ∗ 106% = −3, 47%
 For the zero coupon bond the relative price risk is higher because it has a higher duration.
 ⇒ highest price risk.
 Considering the FRN: It is paid the EURIBOR
 e1 = 100r01
 e2 = 100r12 (Not known today; only at date 1)
 e3 = 100r23
 e4 = 100r34
 e5 = 100r45 + 100
 t=4 ex coupon: V4 = 100∗(1+r) ) = 100
                         (1+r45
                                45


 t=3 ex coupon: V3 = 100∗(1+r) ) = 100
                         (1+r34
                                34


 This can only be true if D is 0.
  ∂V1             V1
 ∂(1+r1
        = −D1 ∗ (1+r1 ) = 0 ⇒ D1 = 0.
 The market value is not always 100:
 V1 = 100 but VM ay26th = 100∗r01 +10011 (Not the same)
                           (1+rM ay26th ) 12
 ⇒ There exists some risk. ⇒ It is like a zero coupon bond now. ⇒ 11 month duration.

 So the price risk for the market value is just opposite of the cash flow risk. At date
 t=5 there is no market value risk at the zero coupon bond nor on the coupon bond. So
 the market value risk is not independent of the reference date.
 Transaction risk
 Competitors behavior has a hugh impact on cash flow risk (car market for example).
 Thus, economic risk is very hard to apply. Idea of transaction risk: Only considering risk
 that has been already contracted.
 Example: Foreign exchange risk ⇒ US-Dollar much weaker now than two years ago. ⇒
 short term hedging cannot be very effective.
 Profit
 Two figures are given to the manager to reach a good balance sheet:
   (1) Legal earnings manipulation
   (2) Risk management

Ad (1) European accounting rules are changing ⇒ International financial report system
Ad (2) Risk management has to be based on accountig rules. ⇒ It differs from other risk
       management.
       Example: Hedge accounting ⇒ value changing in foreign exchange rates ⇒ net this
       position or not

                                               6
Anglo-saxon companies emphazise more on market value risk whereas European compa-
nies concentrate more on profit risk and cash flow risk.

1.6    Time dimension of risk management
One possibility is concentrating risk management at the end of a year. Another possibility
would be concentrating risk management at maturity date or refinancing date.
There might be a chance on doing risk management over a whole period. (Example:
Setting a floor for a colleteral)
Static risk management vs. Dynamic risk management
Dynamic risk management is much more complicated than static risk management.

1.7    Risk measures
There is no general accepted concept of risk. In economics three different kinds of risk
are used:

  1. Standard deviation:

       a.) The single security returns are needed.
      b.) The correlation between the different returns are needed.
       c.) If the variables are normally distributed it is a perfect risk measure.




                                             7
        Most people prefer positive skewness




        C1 = max(S − K, 0) ⇒ Standard deviation does not make sense.

2. Most people have loss aversion. A thumb rule states that the disutility of a loss is
   ten times as high as the utility of a profit.
   Practitionar:




  risk perception: E[g(y − y )]
                            ¯
  Practitionar does not like the variance because it measures both sides equal.




                                         8
3. Value at risk:
   It starts with the probability of the default of a company.




  The board of a firm sets up a percentage line under which the probability of default
  has to remain. This policy is called risk budgeting.
  Example: Bank with equity capital of 100”Euro
                                  retail bank     20”Euro
                                corporate bank    44”Euro
                                   brokerage      10”Euro
                                    reserve       26”Euro




  z = y−E(y) Normailzed variable.
        σy
  With normalized variables on is able to calculate fractiles: z0,01 = −2, 33 for α = 1%

                                         9
yα = zα σy + E[y] = −2, 33σy + E[y]
|yα | = −zα σy − E[y]
So one has to have a normal distribution. risk budget≥ |yα |
The value at risk is very easy to communicate. Criticism:

  1. Planly looking at value at risk for returns does not make much sense. But it
     makes sense for looking at defaults.
     Example:




  2. It depends on the fractile which project is riskier.
     Considering a call: Exercising probability equals 0,8%; value at risk 1%. Sell
     the call ⇒ with a probability of 99% one sacks in the call price.




     The value at risk is negative: VaR=−C0
  3. Considering a bank:
     (1) Give one loan of 3mio Euros; probability of default=0,8%
     (2) Give three loans of 1mio Euros each; probability of default= 0,8%; defaults
         are independent.


                                     10
          VaR 1% fractile
          Normally one would guess that the first choice is more risky because it neglects
          the law of diversification.
          (1) VaR=0
          (2) P(0 defaults=0, 9923 = 97, 62%
              P(1 default)=3 ∗ 0, 9922 ∗ 0, 088 = 2, 36%
              P(2 dafaults)=0,002%
              P(3 defaults)≈ 0




              The VaR=1mio Euro.
       4. This leads to the definition of the fractiles. Consider VaR=50mio Euros:
          It makes no difference whether
                                          L=50”   1%
                                          L=50” 0,1%
                                          L=100” 0,5%
                                          L=500” 0,4%
          ⇒ The greater probability of higher losses is not taking into calculations.

     Coherent risk measures try to avoid the pitfalls of VaR ⇒ more complicated, so
     practitions do not like it. For a portfolio manager VaR is not a reasonable concept.

Derivation of efficient portfolios: Normally we are interested in an optimal combination
of risk and return.
Hedging purpose: Reduce the risk of a given distribution.
Speculating purpose: One is interested in a high risk premium.
In talking about a portfolio one cannot distinguish between these two. One exeption is
an insurance contract (Hedging contract).
Sales department often accepts offers and the the finance department has to do the risk
management. But not even in this worse scenario blind risk minimizing is always the best
solution.




                                           11
2     Static risk management
2.1    Some definitions
Static risk management: Build up a position today for risk management for a certain time
period and do not revise this position during that period.
Dynamic risk management: Build up a position today for risk management but revise it
everytime we get a new information or the passus of time forces us to revise this position.
Perfect hedge: Hedge which completely eliminates the risk of some risk factors.
Imperfect hedge: Hedge which only partially eliminates the risk of some risk factor.
y = y(R1 , . . . , Rn , ) (y=Objective variable, Ri =Systematic risk factors, = Unsystematic
risk factors)
Systematic risk factors effect many variables. Unsystematic risk factors only effect one
objective variable of the company. This destinction is important because one cannot
hedge the unsystematic risk factor.
Example: Bond
Most important systematic risk factor: interest rate.
Default risk: Unsystematic and systematic:
    - macro factors (boom and recession)
    - micro factors (unsystematic)

2.2    Management of a linear risk with forward contracts
Consider a two date model with linear risk: y = a + bR +




Linearity conditions (mathematically):
Least squares E( ) = 0 or E( |R) = 0∀R
Assumptions:

                                            12
    - perfect capital market

    - no taxation

There exists a forward contract on the systematic risk:
f=forward price
payoff forward contract: x(R-f)
⇒ We can come up with a perfect hedge:
yg =(overall risk)y + x(f − R) = a + bR + + x(f − R) = a + (b − x)R + + xf
⇒ if x=b the systematic risk is completely hedged.
We want an optimal policy between risk and return:
M inσ 2 (yg ) − λE[yg ] (λ 0) (λ is the risk aversion coefficient)
If someone is extreme risk averse, λ approaches 0. If someone is not risk averse, λ is
highly positve ⇒ σ 2 has no impact anymore.
                                          ¯                                             ¯
(b − x)2 σ 2 (R) + σ 2 ( ) − λ[a + (b − x)R + xf ] = (b − x)2 σ 2 (R) + σ 2 − λ[(b − x)(R − f ) + bf ]
(Constants as bf have no impact on optimization)
Differentiating with respect to b-x:
                            ¯
⇒ 2(b − x)σ 2 (R) = λ(R − f ) = λπ (π= risk premium)
The sign of the risk premium plays a key role in hedging:
b − x∗ = λ σ2π ⇒
            2   (R)
     
      > 0: bachwardation
     
π =  =0: fair market
     
              < 0: contango
         
          >0 if backwardation ⇐⇒ underhedge(2)
         
b − x∗ =         =0 if fair market ⇐⇒ full hedge(1)
                  <0 if cantango ⇐⇒ overhedga(3)
         
         

 (1) Full hedge: Exposure to risk factor R is completely eliminated. (Example: Selling
     forward foreign exchange rate). If no risk premium is earned one tries to get out of
     this risk factor.

 (2) If there is a positive risk premium one has to take into calculation that something
     might be earned by undergoing this risk. Open position (b-x) increases with the
     risk premium but declines with higher risk measured by the variance.

 (3) Negative exposure to risk times a negative risk premium gives one a positive profit.

Speculation motive: Earn a positive rsik premium.

2.3     Management of several linear risks with forward contracts
Now: Several linear systematic risk factors:
y = a + n b i Ri +
             i=1
                         ¯
yg = y + n xi (fi − Ri ) = a + n [(bi − xi )(Ri − fi ) + bi fi ] +
              i=1                    i=1
M inσ 2 (yg ) − λE[yg ] = ni=1
                                n                              2
                                j=1 (bi − xi )(bj − xj )σij + σ − λ[a +
                                                                             n
                                                                             i=1 ((bi   − xi )πi + bi fi )]
(σij = Cov(Ri , Rj ), constants as bf and σ are eliminated)
⇒ Same solution as for the Markowitz problem
= (b − x) S(b − x) − λπ (b − x)
First order condition:

                                                 13
2S(b − x) = λπ
Unique solution if the risk factors are linear independent:
b − x = λ S −1 π
         2
If π = 0 ⇒ b − x = 0 ⇒ each single risk factor is fully hedged.
2 risk factors R1 , R2 :
π1 = 0, π2 = 0 ⇒ does not even imply a full hedge of the first factor. It gets even more
complicated since the factors might be correlated.
Additional remarks:

  1. The foreign exchange market has a risk premium of almost zero. If transaction costs
     are included only a partial hedge can be done. There might exist different beliefs
     on the risk premium. In this case there exist also only a partial hedge.

  2. Estimating the regression: Significance of risk factors. These might not be stable
     over time.

If companies are concerned with insolvency VaR is a good risk measure. Since now we
have only applied the variance as a measure of risk in the above models. So now we try
to to combine these two risk measures. This is quite simply if our objectve variable is
normally distributed:
Risk budget: y ˆ
Prob(yg ≤ y ) ≤ α (→ zα (α = 1% ⇒ zα = 2, 33))
            ˆ
VaR constraint: −E(yg ) + |zα |σ(yg ) ≤ |ˆ|
                                         y
Example: α = 1%, y = 10 Euro ⇒ −E(yg ) + 2, 33σ(yg ) ≤ 10
                     ˆ
e(yg ) ≥ −|ˆ| + |zα |σ(yg )
           y




If the two lines do not overlap we have no feasible solution.
Final remark:
M inV aR − λE(yg ). This is nonsense. Stock-holders want variance as risk measure.

2.4    Cross-hedging of linear risk
In reality there may not exist a forward contract on any systematic risk factor. But there
might exist a forward contract on a related risk.

                                           14
y = a + bR1 + ; E( ) = 0
There exist no forward contract on R1 . But a forward contract exists on a related factor
R2 .
R1 = c + dR2 + η; E(η) = 0; E(η|R2 ) = 0.
Example: Holding Bayer stocks which have no forward on systematic risk. One could
hold a forward contract on the DAX.




RBayer = c + βRDAX + η (unsystematic risk of Bayer)
yg = a + bR1 + + x(f2 − R2 ) = a + b(c + dR1 + η) + + x(f2 − R2 ) = a + (bd − x)R2 +
bc + bη + + xf2 = a + (bd − x)(R2 − f2 ) + bc + bη + + bdf2
                                    2
σ 2 (yg ) = (bd − x)σ 2 (R2 ) + b2 ση + b2 σ 2
Assumption: Cov[R2 , ] = Cov[R2 , η] = 0
E[yg ] = a + (bd − x)π2 + bc + bdf2
M inσ 2 (yg ) − λE(yg )
First order conditions:
2(bd − x)σ 2 (R2 ) − λπ2 = 0 ⇒
bd − x = 2σλπ2 2 )
              2 (R

Two changes in comparision of our earlier models:

   - product bd
                        π2
   - risk premium:   σ 2 (R2 )

Implication:
New type of risk η:
                ,R          σ(R
bd = b Cov(R12 ) 2 ) = bρ12 σ(R1 ) =b*Hedgeratio.
         σ 2 (R                 2)
The greater the noise risk η teh lower is the correlation coefficient. So η effects the
hedgeration. If the hedgeratio delines the hedge is not as good anymore. If d declines, bd
declines and so the hedge position falls down.

Different approach:
                 ˆ
R2 = e + f R 1 + η
If both variables R1 and R2 are bivariate normally distributed it makes no difference if
one is looking at the regression of R1 or R2 .

                                           15
     ¯
Let f denote the forward price:
                       ¯                        ¯
yg = a+bR1 + +x(f2 −R2 ) = a+bR1 + +x(f2 −e−f R1 −ˆ) = a+(b−f x)R1 + +x(f2 −e−ˆ)
                                                               η                       ¯       η
  2                   2 2          2   2 2
σ (yg ) = (b − f x) σ (R1 ) + σ + x σ (ˆ)   η
                                     ˆ
E(yg ) = a + (b − f x)E(R1 ) + x(f2 − e)
First order condition:
                                                     ¯
−2f (b − f x)σ 2 (R1 ) + 2xσ 2 (ˆ) − λ(−f E[R1 ] + f2 ) = 0 ⇒ f (b − f x)σ 2 (R1 ) + 2xσ 2 (ˆ) =
                                 η                                                          η
λ ¯
  (f − f E[R1 ])
2 2
Assumption: λ = 0
                                                      2 (ˆ)
                                                         η 1
f (b − f x)σ 2 (R1 ) − xσ 2 (ˆ) = 0 ⇒ (b − f x) = x σσ (R1 ) f
                             η                       2

b-fx would be the risk minimization form older results if there were no noise.
           1 2 (ˆ)
                 η
x[−f x − f σσ (R1 ) ] = −b ⇒ x = 1 bσ2 (η)
               2                           ˆ
                                      f+ f
                                             σ 2 (R1 )
Buying the forward contract means buying an additional noise risk. So one has to be
careful in hedging.

Question: Are these hedging policies the same? They are the same under a bivariate
normal distribution.
What is the highest possible hedging effectiveness? Hedging effectiveness is measured by
the highest possible reduction in risk.
Under the assumption λ = 0 and using the first model:
x=bd risk minimization policy
σ 2 (yg ) = b2 ση + σ 2
                2

R1 = c + dR2 + η
σ 2 (R1 ) = d2 σ 2 (R2 ) + ση  2
                       2 2 (R
ση = σ 2 (R1 )[1 − dσσ(R1 ) ) ]
  2
                        2
                              2

     σ(R2 )                          ,R
d sigma(R1 ) = ρ12 (d = Cov(R12 ) 2 ) ) ⇒ = σ 2 (R1 )[1 − ρ2 ] ⇒
                              σ 2 (R                       12
σ 2 (yg ) = b2 σ 2 (R1 )[1 − ρ2 ] + σ 2
                              12
The impact of hedging is the term [1 − ρ2 ]. This is the hedging effectiveness. So the
                                                  12
scaling has no effect because the correlation is not effected by scaling. Thus, one uses the
hedge instrument which ahs the highest ρ2 . This implies the highest hedging effective-
                                                  12
ness. Practitionist say that they only consider hedging instruments with |ρ12 | ≥ 0, 7.

Returning to the example:
βDAXspot vs. βDAXf orward
            Cov(RBayer ,RDAXspot )       Cov(PBayer1 ,PDaxspot1 )∗PDaxspot0
βDAXspot =      σ 2 (RDAXspot )
                                   =         σ 2 (PDAXspot1 )∗PBayer1
                                                                                (R is return now; RBayer =
PBayer1
PBayer0
        )
                 Cov(RBayer ,R
                             DAXf orward )         Cov(P        ,P
                                                  Bayer1 Daxf orward1       )∗P
                                                                          Daxf orward0
βDaxf orward =      σ 2 (RDAXf orward )
                                          =        σ 2 (PDAXf orward1 )∗PBayer1
(1 + f orward)PDAXspot0 = PDAXf orward0
PDAXspot1 = PDAXf orward1 (Maturity date)
                β
So βDAXspot = DAXf orward
                     1+r
Assumption: βDAXf orward =d=0,8=Hedgeratio
100000 Bayer stocks; PBayer0 =20; value of Bayer stocks=2000000
Exposure in DAX forwards: 2”*0,8=1,6”
                                         π2
x = bd − λ σ2π2 1 ) = 1, 6 − λ σ2 (RDAXf orward )
            2  (R                 2
                                                                        λ         0,04              λ
Assumption: π2 =0,04; σ(RDAXf orward )=0,2 ⇒ x = bd −                   2
                                                                            ∗     0,04
                                                                                         = 1, 6 −   2



                                                           16
x = P riceofM V of P F
            hedgeinstrument
                            ∗ Hedgeratio
This is a static hedge. In cross hedge one can loose on the stock and on the hedge instru-
ment. So it has to be observed continously.

Basis risk:
Example: Dollar claim for first of April 2005.
1. Probabililty: Undergo a forward contract. The problem is that it might be not possible
to undergo this contract due to the first of April 2005.
2. Probabiltiy: Reinvesting Dollars on first of April until third Friday of June. Then
undergo a forward contract with the bank to the thrid Friday in June. Then the risk is
with the bank.
3. Probability: Basis risk.
The forward contract trades today at f0 . No arbitrage relationship:
 f0               f1.4.           fT = sT
                   ft
  - 1 + rEuro = st (1 + rDollar )    -
                  1+rEuro (T −t)
  -      ft = st 1+rDollar (T −t)    -
                              20
              1+r          ) 365
f1.4. = s1.4 (1+r Euro1.4.      80
                  Dollar1.4. ) 365

Basis:=ft −                           −rDollar1.4. )(T
                st = st (rEuro1.4.Dollar1.4. (T −t) −t)
                                 (1+r
Basis risk=σ(ft − st )
What determines the basis risk: st , difference in the money market rate, rDollar1.4
Arbitrage relationship:
                       −rDollar1.4. )(T
ft − st = st (rEuro1.4.Dollar1.4. (T −t) −t) + t
                 (1+r




For a short intervall the basis risk in low. Then it gets higher. But at maturity date the
basis risk is zero.
s1.4. + y(f0 − f1.4. ) = s1.4. + y[(f0 − s1.4. ) + (s1.4. − f1.4. )] y=number of contract sold;
s1.4. − f1.4. = negative basis
Profit/Loss= change in forward price
Hedgeratio:
                                      σ(s
d = ρ12 σ(R1 ) = ρ12 (s1.4. , f1.4. ) σ(f1.4. )
         σ(R2 )                           1.4. )
The correlation coefficient declines because of basis risk ⇒ Hedgeratio is declining ⇒
smaller hedge position.

                                                          17
Exchange traded fund (certificate) is a security which perfectly replicates the price of
a stock index.
Advantage:
    - very low bid- asked spreads

    - very well diversified

    - liquid market
Disadvantage:
    - Difference between price index (whenever a dividend is paid the stock looses in
      value: DOW) and performance index (dividend protected: DAX)

    - most indices are price indices

    - one only wants to be invested in performance indices → otherwise one could loose
      money
Organisation: Banks sell this exchange traded funds. As a trader which has the risk the
banker has to hedge the risk. One cannot hedge it just by buying all 30 DAX stocks
because of the different bid- asked spreads. So one has to follow a so called tracking
strategy which has the goal to minimize basis risk and at the same time safing transaction
costs.
Rp = tracking return
 ˆ
R= DAX return
        ˆ
(Rp − R)= tracking error
Strategy: Just buy the 20 biggest stocks in the DAX to safe transaction costs.
ˆ
xi : weight of stock i in the index
xi : weight of stock i in the tracking portfolio
i ∈ I, J ⊂ I with xj = 0
                                                                                ˆ
So we are interested in minimizing the variance of the tracking error σ 2 (Rp − R)
              ˆ
Ri = αi + βi R + γi ∆Rsi with Rs = industry return.
                          ˆ
1. Stage: Rs = αs + βs R + s (Industry return on DAX) and s = ∆Rs
So the problem of multicorrelarity is avoided.
                         ˆ
2. Stage: Ri = αi + βi R + γ∆Rsi + i
The tracking protfolio should exist of
    - a good representation of each industry of the index

    - a good representation of the industry structure
                          ˆ                 ˆ            ¯ˆ
M inxi ,i∈I−J σ 2 (Rp − R) = E[((Rp − R) − (¯ Rp − hR))2 ] = E[( i (xi − xi )Ri − i (xi −
                                                   h                                ˆ
                                                                                            ˆ
 ˆ i )¯ Ri )2 ] = E[( i (xi − xi )(Ri − hRi ))2 ] = i j σij (xi − xi )(xj − xj )(βi βj σ 2 (R) + γi γj ∗
X h                           ˆ         ¯                         ˆ         ˆ
Cov(∆Rsi , ∆Rsj ) + Cov( i , j ))
One has to watch the index closely in order to adjust the portfolio properly. The exchange
traded funds are related to passive portfolio management.

Dangers of cross- hedging:

                                                  18
   - Ex ante risk is reduced by [1 − ρ12 ]2 but you want to be protected ex post. You
     could loose on both siedes. So it is an imperfect hedge. The hegde has to currently
     observered. (DAX ↑, PF ↓)

   - The loss may exceed the VaR border
                            marketvalueatyourportf olio
   - Hedge par=Hedgeratio* marketvalueof hedgeinstrument

⇒ currently adjusting hedge position

2.5    Management of non- linear risk
Since now: y = a + bR + (linear exposure to a). But often we have no linearity:
Example:




   .




   .




   .




   .
Exports are atrractive if the exchange rate is high:




   .




   .




                                            19
   .
An option might give you such a hedge position. If there is no such option available you
could create a portfolio of options.
Earlier objective function: σ 2 (yg ) − λE[yg ]
Problem:




   .




   .




    .
Variance might not be a good risk measure anymore if strong deviation from normality
exists. The diagramm 1 is called delta- gamma- hedging:
∆: First differential of the price to the risk factor.
γ: Second differential of the price with respect to the risk factor.
If you are lucky, γ is stable. Then look for a hedging instrument that has a stable γ and
then hedge away convexity of the position. If this does not hedge ∆ then you can add a
forward contract ⇒ hedges away the slope.

2.6    Risk management given non- hedgeable risk
The noise term is non- hedgeable. This is equal to not tradeable. But there also exists
risk which is non- hedgeable. (for example: job risk at large parts). We saw already that
the noise term reduces the hedgeratio.
Example: Construction company ⇒ International competition for building a damb for 5”
Dollars. The submission date is the end of September. You have exchange rate risk.
Question: Should you hedge it even though you do not know whether you get the job. So
you have an unhedgeable submission risk.
Two stages hedge procedure:




   .




                                           20
   .




   .
1. Strategy: Hedge 5” Dollars from 0 to Christmas by a forward
2. Strategy: Hedge 5” Dollars from 0 to Christmas by an option:




   .




   .




     .
f0 ; Forward price today expiring at Christmas.




   .




   .




  .
One would prefer the put option. With a forward you have the double bad luck problem:

  1. Loose submission

  2. Forward price might be very high at Christmas ⇒ Unlimited loss possible.

With the option you do not have unlimited loss. In case of a loss you have to pay the
compounded put price.


                                           21
3     Organization of risk management
3.1    Macro- vs. microhedge
Question: Who is responsible for risk management?

    - Chief risk officer

    - Chief risk commitee

         - Deciding about the risk budgets for all devisions.
         - Risk controlling: How to measure risk; profits and losses of traders
         - Establishing a system of monitoring all devisions ⇒ Did they cross the budgets
         - Risk reporting
         - Specific model risk: Does it match reality ⇒ Follow many risk models at the
           same time.

Micro hedge: A hedge policy where we hadge every single transaction seperately.
Example: Bank selling a swap ⇒ Hedging interest swap. But it does not make sense to
hedge every single swap.
Macro hedge: A hedge policy where we hedge a portfolio of single transaction.
Example: Portfolio of swaps

You could also hedge the risk for the bank as a whole. Of course only the systematic
risk is hedges. Under the assumption of liquid markets you could do that. Then the price
of the risk might be fix.
In reality we face many problems: A bank has a loan department which are non- hedge-
able. This is completely different of treating stock price risk. So nobody can be an expert
in all this devisions.
So the above procedure is not often be done. Risks are split in independent devisions.
We end up with an mixture of micro- and macrohedge: Within a devision the risk is put
together but the whole risk is split.
Further pitfall:

    - Lack of information in an international company

    - Motivation problem: You have to motivate traders to make much money for the
      company. So they offer traders much freedom. Macrohedge would contradict each
      of the traders trading seperately.

Devisions of risk management:

    - asset liabilty management risk (interest rate risk)

    - stock price risk

    - foreign exchange rate risk

    - default risk


                                            22
   - operational risk
   - settlement risk: agreements on trading ⇒ computer problem ⇒ you are not paid
If these devisions are treated separately the correlation is ignored. By having separate
devisions some might use their risk budget and some do not ⇒ problem of the best use
of equity.
International accounting rule 39 only allows microhedging so far.

3.2    Risk budget for devisions
Risk budget definition as in Value at Risk. You cannot exceed a certain fractial of the
P/L- distribution.
Question: What is the horizon of the P/L- distribution? (one day, one month, one year)
   - Basel II suggests a 10 day horizon for trading books (some banks do one day)
     (trading devision)
   - for loans three month and mortage loans even 10 years (loan devision)
How do you connect these two?
Another question is whether to exceed the risk budget in case of a profit and cut back
risk budget in case of a loss. Banks do not exceed the risk budget but cut it back in case
of substantial losses.

3.3    Risk budget for traders
Additive proposal: Allocate risk budgets among traders such that they add up to the
total amount.
Superadditive proposal: Most traders do not use their risk budgets as a whole. So the
risk budget is grossed up by a certain number:
                       1
   i xi = 50 ⇒    i ( 1,7 xi ) if only 70% are used.
But now exists a problem: The risk budget might be overused.
Thus, a supervisior is in charge to see that the budget is not overused. So the senior
manager might have to take counteraction.
How do you come up with risk budgets for a single trader?
The computer might have a software to display the used risk budget. The inputs are
given by the back office.
In addition banks might come up with limits to single risk factors.
Example: Derivative trading:
   - index risk
   - stock prices risk (big companies)
   - interest rate risk
   - volantility of the index
You come up with limits for the sensitivities for the risk factors:
 ∂M
∂RFi
     ≤ αi ∀i
Most banks come up with other limits:

                                            23
   - country limits
   - currency limits
   - company limits

3.4    Risk controlling




Bank A looses ∆ ⇒ The trader brings it to a bank C where he owns an account.
Risk management should try to avoid such fraughtful behavior.
A bank has to have a strict seperation between front office (traders) and back office. It
is a pitfall to neglect the back office. It has to check the price, legal action and all other
actions. ⇒ Monitoring process
Monitoring:
   - all telephone calls are taped
   - communication between different back offices
   - observing limits
   - P/L diagramm for each trader every day ⇒ sent to chief risk officer

3.5    Risk repeating
Requires a lot of transparancy.
Example: Deutsche Bank talks 40 pages only about risk management in the internet.
Basel II:
   - equity capital requirement
   - supervisory review process/ interaction between bank and bank regulators
   - market discipline: philosophy that all banks match each other and honor good
     behavior
⇒ Basil II emphasizes on transparency

                                            24
4     Dynamic Risk Management
4.1     Reasons for dynamic risk management
It is very expensive in terms of transaction costs. So you should think about the neces-
sarity in comparison to static risk management.
1. Reason: Static risk management:

     - model

     - parameter estimates

⇒ Problems:

    1. Model risk:

          - We are not sure which model is the best one
          - Estimation risk in Econometric models

    2. The result is subject to estimation error

Result: Do not be to confident about static risk management

2. Reason:
You sometimes have borders for market values you have to stay within.

3. Reason:
Exporting companies for example have to hedge daily.

4. Reason:
Cross- hedging might be a problem: ⇒ other strategy: Constantly hedging the spot rate.
Particulary important for long term hedge.
⇒ Forward and options are only good for short term hedges.

Example of dynamic hedges:

     - Synthetic risk management

     - Roll- over hedge

4.2     Synthetic risk management
You have a reasonable spot market but no liquid derivative
Statically complete market: You can buy today any random cash flow payable at some
date T.




                                             25
Any claim you can think of is buyable today. Not very realistic because the number of
derivatives is limited.
Dynamically complete market: You can design a trading strategy which you adjust quite
often over time so that you can reach any random cash flow y.
Difference: In a statically complete market you do not revise your position whereas in a
dynamically complete market you revise it any time.

Example of a dynamically complete market: Black- scholes - Model




BSM:
dS
   = µdt + σdzt (µ= Drift, dt= Length of the intervall, dzt = Wiener process)
S         √
σ(dzt ) = dt
V0 =1”Euro
VT ≥1”Euro
        ˆ          ˆ
Floor: VT =1”Euro, VT ≤1”*exp(rT)




                                      VT
In order to come up with a floor     exp(rT )
                                               this fraction of the initial endowment has to be
invested in a ten year zero bond.
Constant proportion strategy:

                                                 26
Put a constant proportion in the market portfolio and the rest in a risk free asset.
Example:
          VT
V0 − exp(rT ) =400’
If x=0,7 ⇒ 0,7*400’ is invested in the market portfolio and 0,3*400’ in the risk free asset.
On a bad day the stock proportion is mismatched ⇒ the proportion is lower ⇒ a self-
financing strategy has to be followed to reestablish the stock proportion ⇒ this has to be
done very frequently.
What do you get after say ten years:
dVt = Vt [(1 − x)rdt + x dStt ] ∀t ∈ [0, T ] (1)
                               S
Itos Lemma:
                                               2 g(V
dg(Vt , t) = ∂g(Vt ,t) dt + ∂g(Vtt,t) dVt + 1 ∂ ∂V 2 ,t) σ 2 (Vt )dt
                   ∂t          ∂V           2
                                                     t
                                                   t
In our case:
g(Vt , t) = lnVt
dlnVt = 0 + Vt dVt − 1 v12 σ 2 (dVt )dt
                  1
                           2 t
⇒ plugging into (1)
                                 1
⇒ = (1 − x)rdt + x dStt − 2 x2 σ 2 dt
                           S
Assumption: x=1 ⇒ Vt = St
dlnSt = dStt − 1 σ 2 dt
             S      2
                                         1            1                        1
dlnVt = (1 − x)rdt + x[dlnSt + 2 σ 2 dt] − 2 x2 σ 2 dt = [(1 − x)r + x 1 σ 2 − 2 x2 σ 2 ]dt + xdlnSt
                                                                       2
We integrate in [0,T]:
                                   1
lnVT − lnV0 = [(1 − x)r + 2 σ 2 x(1 − x)]T + x[lnST − lnS0 ]
ln( VT ) = . . . + xln( ST )
    V0                    S0
VT
 V0
    = exp([(1 − x)r + 1 σ 2 x(1 − x)]T )( ST )x
                             2                   S0
Gross return on the protfolio:
VT
 V0
    = c( ST )x
           S0
If x=0 ⇒ money is just invested in the risk free asset: exp(rT)
If x=1 ⇒ VT = ST
               V0     S0
If x > 1 ⇒ convexe relationship
If x < 1 ⇒ concave realtionship

In this strategy there exists no risk of bancrupcy because if the market declines and
you have borrowed money you also have to cut back debt in order to maintain a constant
proportion x.
Crucial assumption: You have enough time to adjust your position: This is due to the
diffusion process ⇒ no jumps in price changes.
In the constant proportion strategy no derivative is needed. In the Black- Scholes world
we will follow a constant proportion strategy if we have constant RRA:
dS
 S
   = µdt + σdzt
We saw already that this process is only achieved with a constant RRA by a representa-
tive investor:
u(VT ) = 1−γ VT1−γ
           1

Assumption: Statically complete market
M axE[u(VT )] s.t. V0 = E0 (VT π0T ) (Budget constraint)




                                                27
                                   0
Complete market: V0 = z VT z πz
             0
            πz
π0T z = P rob(z)
First order conditions:
P orb(z)u (VT z ) = λP rob(z)π0T z
VT−γ = λπ0T z , ∀z
   z
No representative investor:
                1
uj (VT j ) = 1−γj VT j
               γj
⇒ FOC: VT jz = λj π0T z , ∀z
          −γ
         VT z
= λj      λ
              ,     ∀z
  −γ
VT jzj
  −γj    =
V0j
              −γj      −γ
 λj       V          V0
  −γj (      )              =
         Tz
V0j     V −γ  0
                      λ
            −γ
λj f racVT z j V0−γ )( V0j )−γj =
¯                     VT jz

¯ T
λj ( VV0z )−γ =
¯
λj ( ST0z )−γ
      S
                       1
                     −λ         γ
      VT zj         ¯
                j ST z γj
⇒      V0z
                 ( S0 )
                  = λj
           γ
⇒ x j = γj
If our investor has the same RRA as the representative investor ⇒ xj = 1 ⇒ Our investor
still follows the constant proportion strategy.
Under the assumption of perfect markets you can always assume statically complete mar-
kets. In reality the constant proportion strategy is just a mean to come close to reality.
Adjusting the portfolio all the time is too expansive. Transaction costs will kill you. The
portfolio is only revised if the difference between the theoretical portfolio and the actual
portfolio exceeds a certain margin. The higher the transaction costs the higher the mar-
gin.
Another problem: Tracking error:
Problem to implement this strategy because you only revise your portfolio if a certain
margin is exceeded.

Generalization of the strategy if somebody is not happy with the constant proportion
strategy:




                                            28
The strategy is similiar to option pricing: Binomial tree:




We are still in the Black- Scholes world ⇒ we can derive the Arrow- Debreau prices. Each
note at S defines a certain return between t and T. We need to derive a dynamic portfolio
option which gives us this payoff.
What is the duty of the portfolio manager?




         0       0
VA = πAa Va + πAb Vb
         0       0
VB = πBb Vb + πBc Vc
         0        0
Vα = παA VA + παB VB
  0     1 RF −d
πu = RF u−d
πd = R1F u−RF
  0
           u−d
u,d = gross return in different states
Final step:
V1 = V0 [(1 − x0 )RF + x0 RM 1 ]
V2 = V0 [(1 − x0 )RF + x0 RM 2 ]
V1 − V2 = V0 x0 (RM 1 − RM 2 ) ⇒ x0 = (VM−V 2/V2
                                        1
                                      R 1 −RM
                                               0


It is like the hedge ratio. We know RM 1 − RM 2 because we have a geometric Brownian
motion. If the binomial tree has many periods we aprroach a log- normal distribution:

                                            29
u = u(µ, σ, n)
d = d(µ, σ, n)
⇒ Cox- Ross- Rubenstein Model

Exactly the same procedure is done in banks when having sold a strange option claim to
a costumer. It has to hedge the risk in creating a long position which has exactly the
same payoff as the short position.
Looklack option: Payoff depends on what has happened between the contract and the
time of maturity. The same strategy is applied.
Synthetic risk management: We always adjust our stock position.

4.3    Roll- over Hedge
Traders do not hold a short position for say two years and in order to hedge this position
hold a long position for two years. Traders prefer roll- over strategies: Short term contracts
are rolled- over for two yeras: Short- term contracts are more liquid ⇒ have lower bid-
asker spreads. The corresponding long term contracts may not even exist. Transaction
costs are higher when doing a short- term strategy (16 times as high)
Reasons:

    - Traders may have the feeling that the short- term contracts allow them to do better
      than the market by optimal timing ⇒ questionable

    - The market value might not change as much in short- term contracts ⇒ questionable

    - Impressing the boss

So an existing long- term contact is probably the better strategy.
Example: Metallgesellschaft:
It sold long- term contracts in over- the- counter- markets hedged by short- term contracts.
Thus, it used a hedge ratio of one.
The treatment of commodities differs from the treatment of money.
Commodities:
If you have oil in a tank and the price rises you would sell it. It gives you an option. This
is called convenience yield: real option value- storage cost ⇒ very volantile
                       1
S0 (1 + r0 − c0 ) = f0 with:
r0 = money market rate for 1 year
c0 = convenient yield
  1
f0 = forward rate in 1 year quoted today.
We cannot use it as a standard no arbitrage relationship because c0 is not known ⇒ any
roll- over strategy runs into problems.
Relationsship for any date t:
st (1 + rt − ct ) = ftt+1
Assumption:
If the market retrun and the convenience yield are deterministic we can come up with a
perfect roll- over hedge.
Proof:
Assumption: no margin requirements


                                             30
               t          payments                                     payments
               0             X*0                                             -
                                   1
               1         X0 (β1 − f0 )                          X0 (s1 − s0 (1 + r0 − c0 )
                                   2
               2         X1 (β2 − f1 )                          X1 (s2 − s1 (1 + r1 − c0 )
               .
               .               .
                               .                                             .
                                                                             .
               .               .                                             .
                              T −1
          T-1 XT −2 (sT −1 − fT −2 )    XT −2 (ST −1 − sT −2 (1 + rT −2 − cT −2 )
           T        PT − s T         PT − sT + XT −1 (sT − sT −1 (1 + rT −1 − cT −1 )
Payments:
                 cT
T-1:−sT −1 (1 + 1+r−1 + XT −2 (sT −1 − sT −2 )
                    t−1
T:PT − sT −1 (1 + rT −1 − cT −1 )
You gain T-1 from T by discounting by (1 + rT −2 − cT −2 )
Risk minimizing strategy: roll- over optimization.
This payment is not randomly as of date T-2:
T-1:XT −1 = 1 Hedgeratio
                    CT −1
T-2:XT −2 = (1 − 1+rT −1 )XT −1
Discounted payment: c
        T −1
−XT −2 fT −2       −(1− 1+r −1 )(1+rT −2 −cT −2 )sT −2
                          T
                                                                      cT −1             cT −2
 1+rT −2
               =           T −1
                                  1+rT −2
                                                         = −(1 −     1+rT −1
                                                                             )(1   −          )s
                                                                                       1+rT −2 T −2
                ct+1
Xt = (1 −      1+rt+1
                      )Xt+1 ,     t = 0, . . . , T − 2

                                         t             payment
                                         0                -
                                                    1
                                         1     −X0 f0 = −X0 (1 + r0 − c0 )
                                         2                -
                                         .
                                         .                .
                                                          .
                                         .                .
                                       T-1                       -
                                        T                       PT
             c1          c2                 cT
X0 = (1 − 1+r1 )(1 − 1+r2 ) . . . (1 − 1+r−1 )XT −1
                                               T −1
In an arbitrage free market the profit of this strategy should be zero:
       1
−X0 f0 (1 + r1 )(1 + r2 ) ∗ . . . (1 + rT −1 ) + PT = 0
          c1       c2                 cT
−(1 − 1+r1 )(1 − 1+r2 ) . . . (1 − 1+r−1 )s0 (1 + r0 − c0 )(1 + r1 ) . . . (1 + rT −1 ) + PT = 0
                                         T −1
PT = s0 (1 + r0 − c0 )(1 + r1 − c1 ) . . . (1 + rT −1 − cT −1 )
The difference between the money market you pay and the convenience yield you earn is
called cost of carry.
         1
PT = f0 long term forward rate as of date 0.
This is only theoretically since there is no long forward rate for oil in reality. There also
exist transaction cost, taxes, ...
Metallgesellschaft followed a hedge- ratio of 1:
Assumption: ci > 0 ⇒ XT −2 < 1




                                                           31
Assumption: ci < 0




Even in our simple model the hedgeratio does not equal 1 all the time. This would only
be the case if ci = 0.

We now again consider the risk minimization policy. Assumption now: ct and rt are
random.
T-1:XT −1 = 1 same strategy
                      cT
Payment: −sT −1 (1 − 1+r−1 ) + XT −2 (sT −1 − st−2 (1 + rT −2 − cT −2 ))
                         T −1
     ˜       ˜
T-2: rT −1 , cT −1




                                 Cov(·)
hedge ratio= slope = σ2 (payof f hedgeinstrument)
                       T −1                   c
           Cov(sT −1 −fT −2 ,sT −1 (1− 1+r −1 ))
                                         T
                                                  T −1
XT −2 =                             T −1
                      σ 2 (sT −1 −fT −2 )
                                 T −1
(XT −2 =Hedge ratio, fT −2 has                    no impact on the variance)
                         cT
  Cov(sT −1 ,sT −1 (1− 1+r −1 ))       c               cT
                           T −1
=           σ 2 (sT −1 )
                                    ( 1+r−1 = ET −2 ( 1+r−1 )
                                        T
                                          T −1            T −1
                                                                   +   T −1 )
                                   cT
        Cov(sT −1 ,sT −1 (ET −2 ( 1+r −1 )+ T −1 ))
=1 −                    σ 2 (sT −1 )
                                       T −1




                                                              32
=1 − ET −2 ( 1+r−1 ) − Cov(sT −1 ,sT −1 T −1)
                   cT
                      T −1          σ 2 (sT −2 )
 T −1 is pretty small ⇒ main driver of the covariance is the change in the spot rate.
Cov(sT −1 ,sT −1 T −1)
      σ 2 (sT −2 )
                         = ρ(sT −1 , st−1 T −1 )
σ(sT −1 t−1 )
          (ρ will be pretty high and the last term between 0,1 and 0,2)
  σ(sT −1 )
So the - risk is like a basis risk which reduces the hedgeratio.




If you follow a strategy of a hedge ratio of 1 it may be grossly misleading. Minimizing
the variance period by period is called a miopic strategy.
Is this a reasonable policy?
It seems to be reasonable if changes in spot prices follow an additive random walk.
Additive random walk:
st+1 = st + ηt+1 with ηt+1 iid. Miopic or local variance minimization policy.
Is this strategy reasonable for Metallgesellschaft?
The OPEC wants to keep the oil price between 22 and 28 Dollars.
This is a strong deversion from the additive random walk. It depends conversely on future
prices (Mean- reversion piolicy). In having the mean- reversion neighborhood you should
do nothing for the first years and shortly before maturity undergo a dynamic hedging
policy. Another strategy would be to do the hedge with an option. But then you face the
same problems of modelling and hedging.
Metallgesellschaft defaulted because all forwards had turned badly on them. So the
COMEX asked for hugh margin requirements.
Profit risk: There might be accounting rules with drive you into bancrupcy.




                                            33
5     Managing interest rate risk
Excursus: Foundations of term structure models
Modelling the term structure of interest rates: requirements




Internal rate of return for different coupon bonds which have the same maturities is not
the same because of duration. So today we talk about zero bonds and calculate the yield
for different maturity dates. A coupon bond is a portfolio of different zero bonds.
                                     0    1   2
                                     P0   c 100+c

P0 = Rc + 100+c
        01
               2
             R02
We know R01 , c and observe P0 in the market ⇒ so we can derive the whole equation:
100+c
   2
 R02
       = P0 − Rc 01
It is almost a zero bond when Rc is invested at t=0. Synthetic zero bond.
                               01
The term structure changes. So you have to come up with a model which describes the
current term structure and stochastic changes.
Interest rate= marginal productivity of capital and time preference in a risk free model.
In a risky model you have to take the marginal risk preferences into account.
The Central Bank has a strong influence on the term structure for the short term interest
rates. Another important element is expected rate of inflation. So we have to differ
between nominal and real interest rates.
Requirements for models:

    1. It should fit the economic understanding.

    2. The nominal interest rate should not be negative because the bank has no cost of
       storage.
       P0t (R0t )t = 100 yield to maturity in a discrete time model
       P0t exp(r0t t) = 100 in a time continous model
       So r0t ≥ 0



                                           34
3. All forward rates should be non-negative:
    t      t−1
   R0,t = R0,t−1 f0,t−1,t (Forward rate from t-1 to t)
                  t
                 R0,t
   f0,t−1,t =    t−1
                R0,t−1
                         ≥1
   The term structure for a long period cannot be downward sloping. It must become
   flat. This also holds for continous time models.

4. Stochastic changes in the term structure should fit the assumption of no arbitrage.
   Example: We assume parallel shifts of a flat term structure. This implies arbitrage
   opportunities.
   An arbitrage portfolio is a zero investment portfolio:

    (1) You buy a two year zero bond investing 100 Euro
    (2) You sell a one year zero bond receiving 200 Euro
    (3) You invest 100 Euro overnight (dt)

   This portfolio cost you nothing:

                                 0+dt       1              2
                                              ¯
                              100exp(rdt) −200R              ¯
                                                         +100R2

   At the end of 0+dt the term structure moves up or down.
                                ¯
   V=Value of the portfolio, R=term structure
                               ¯      ¯2
                            200R
   V0+dt = 100exp(rdt) − R1−dt + 100R
                                   R2−dt
   ∂V0+dt       ¯        ¯2
     ∂R
          = 2002R − 2 100R
             R      R3
                    > 0 ifR > R
                   
                                  ¯
   ignoring dt: = =0 ifR = R      ¯
                   
                   
                      < 0 ifR < R ¯




   ∂ 2 V0+dt        ¯        ¯2      ¯           ¯
       ∂R
           = −2 2003R + 6 100R = 4004R (−R + 1, 5R)
                  R        R4     R
                                         ¯
   We have a convexe curve until 1, 5R ⇒ Never model term structure changes by
   parallel shifts.

5. Current term structure should be consistent with the model.

                                           35
How do we satisfy this requirements: The Black- Scholes model describes the term struc-
ture changes.

One factor models
Assumption: Perfect capital markets in continous time
Factor: Instantaneous risk free rate ⇒ earned if you invest money for a very short period
of time. It is changing all the time. So we have to consider a diffusion model:
VoT =1 Euro*exp( 0T rt dt) Reinvestment all the time
drt = µr (·)dt + σr (·)dzt                         √
dzt Wiener process with E(dzt) = 0 and σ(dzt ) = dt
σr increases with r increasing because of inflation
How can you garantuee that r stays positive?

   - low r → high µ

   - σr → 0 if r → 0

   - µr → 0 if r → 0

Most models use a mean- reverting process:
µr (·) = Φ(¯ − r) with Φ > 0 and r= long term equilibrium level (Cox/ingersoll/Ross-
           r                      ¯
Model) √
σr (·) = r

Valuation of zero bonds:
Farkas Lemma tells us about non- negative shadow prices for state contingent claims.
  0
π0z = Arrow- Debreau price for a claim of 1 Euro to be paid in state z.
           0
          π0z
˜
π0z = P rob(z) = pricing kernel or stochastic discount factor. No arbitrage implies π0z > 0.
                                                                                    ˜
Itos Lemma:
Let Pt be the price of any asset realized by a diffusion process:
dPt = µP (·)dt + σP (·)dzt with µP (·) = µP (Pt , t) and σP (·) = σP (Pt , t).
and let gt = gt (Pt , t) be a twice differentiable function.
                                         2
                            ∂g
Then: dgt = ∂gt dt + ∂Ptt dPt + 1 ∂ g2 (dPt )2 where (dPt )2 is replaced by σ 2 (Pt t)dt =
                  ∂t                  2 ∂Pt
                                           t

  2
σP (Pt , t)dt.
Utility maximizing individuum:
U0 (C0 + E[U1 (C1 )] s.t. W0 = C0 + E[C1 π1 ]
First order conditions:
U0 (C0 ) = λ
U1 (C1z ) = λπ1z
⇒ U1 (C1z = U0 (C0 )π1z
                         1                 1
E[U1 (C1 )] = U0 (C0 ) 1+rf as E[π1 ] = 1+rf




                                            36
What is going on with this πt ?
dπt
πt
     = µπ (·)dt + σπ (·)dzt
    dπt
E[ πt ] = µπ (·)dt as E[dzt ] = 0 (Wiener Process)
                                                                              t
                                                                     ∂exp(−       rτ dτ
E[ πt+dt ] = 1 − rt dt since E[ πt+dt ] → exp(− 0t rτ ) dτ →
      πt                           πt                             ∂t
                                                                     0
                                                                          = −rt exp(−                t
                                                                                                     0 rτ dτ )
         πt
⇒ E[ πt ] = µπ dt = −rt dt.
Define gt (Pt , t) := lnπt and apply Itos Lemma:
                                     2 lnπ
dlnπt = ∂lnπt dt + ∂lnπt dπt + 1 ∂ ∂π2 t (dπt )2
            ∂t        ∂πt        2       t
=0 + πt [πt (−rt dt + σπ (·)dzt )] − 1 π12 πt σπ dt
          1
                                        2 t
                                            2 2

                          1 2
= −rt dt + σπ (·)dzt − 2 σπ dt
  t                                   πt      t         1 2
  0 dlnπτ dτ = lnπt − lnπ0 = ln( π0 ) = 0 (−rt dτ − 2 σπ dτ + σπ (·)dzt )
⇒ π0 = exp( 0t [−(r + 1 σπ )dτ + σπ (·)dzτ ])
     πt
                           2
                             2

Valuation: zero bond P0T = E0 [100πT ]
Coupon bond: A coupon bond is the same as a portfolio of zero bonds:
P01 = E0 [cπ01 ] where c is the coupon
P02 = E0 [cπ02 ]
.
.
.
P0T = E0 [(100 + c)π0T ]
⇒ market value= T P0i       i=1
Derivatives/ Options: same procedure
Example for a diffusion process where r remains positive (Cox/Ingersoll/Ross- Model):
                              √
drt = Φ(¯ − rt )dt + σr rt dzt (Mean reverting process)
            r
       ¯
with r: long term interest rate, rt : current interest rate and Φ > 0: speed of mean rever-
sion
if rt ↓ → σr ↓ and drift↑ ⇒ rt never becomes negative.
in general:
drt = µr (·)dt + σr (·)dzt
dπt
 πt
     = −rdt − σπ (·)dzt
                                  2
dPt = ∂Pt dt + ∂Pt dr + 1 ∂∂rPt σr (·)dt
          ∂t          ∂r       2    2
                                        2
                                 2P
dPt = ∂Pt dt + ∂pt dr + 2 ∂∂r2t σr (·)dt = Et [Pt+dt πt+dt ] = Et [(Pt + Pdt ) πt+dt ] = Pt + Pt Et [ dπtt ] +
          ∂t         ∂r
                             1        2
                                                         πt                     πt                    π
Et [dPt ] + Et [dPt dπtt ]
                         π
(where Et [ dπtt ] = −rdt and Et [dPt ] = E[drt ])
                π
⇒ 0 = Et [ dPtt dπtt ] + Et [ dPtt ] − rt dt
                P π             P
∂Pt        1 ∂ 2 Pt 2          ∂Pt                    Pt
 ∂t
    dt + 2 ∂r2 σ (·)dt + ∂r E[drt ] − Pt rdt = ∂r σr σπ dt
∂Pt        2P
 ∂t
     + 1 ∂∂r2t σr + ∂Pt µr − Pt r = ∂Pt σr σπ
        2
                  2
                         ∂r                ∂r
Boundary condition: PT,T = 1
Solution of Cox/Ingersoll/Ross- Model (solution for PDE with boundary condition):
PtT = exp[A(T − t) − B(T − t)rt ] ((T-t) is the time to maturity)
A(T − t) := Φ¯ (2ln Ψ(exp[γ(T2γ
                    r
                   σ2                −t)]−1)+2γ
                                                + ρ(T − t)
                 r

B(T − t) := (γ+Φ+σ2(1−exp[γ(T −t)])
                     r σπ )(exp[γ(T −t)]−1)+2γ
γ 2 = (Φ + σr σπ )2 + 2σr  2

ρ := Φ + σr σπ + γ

Connection of the process of pricing bonds and the process of the pricing kernel:
dPt = Pt [µp (·)dt + σp (·)dzt ]

                                                     37
    dπt
    πt
                       (·)−r
        = −rdt − µpσp (·) t dzt
    Proof that this equation holds given that the first equation holds:
    Drift term: Invest 1 euro for time period (t,t+dt):
    1 ∗ πt = Et [exp(rdt)πt+dt ] (Farkas- Lemma)
    = exp(rdt)Et [πt + dπt ] ⇐⇒ exp(−rdt) = 1 + Et [ dπtt ] ⇐⇒
                                                            π
    −1 + exp(−rdt) = Et [ dπtt ] =! −rdt
                               π
    Taylor expansion:
    1 + exp(0) + (−rdt)exp(0) = Et [ dπtt ] (Other terms can be ignored because dt is very small)
                                          π
    From earlier results:
    Et [dPt ] − Pt rdt = Et [dPt dπtt ]
                                  π
    Pt µp (·) − rPt dt = Pt σp (·)σπ (·)dt ⇐⇒
    (Because all other terms are too small and µpσ(·)−rt = σπ (·))
                                                      p (·)
    µp (·)−rt
     σp (·)
           = σπ (·) Sharpe ratio
    All other bonds have the same Sharpe ratio = risk premium per unit of risk ⇒ must be
    the same whether you invest for one year of for ten years

    Starting point of the one factor model: dr = µr (·)dt + σr (·)dzt . Only factor is the
    instantaneous risk free rate.
    Itos Lemma for the price of a zero bond:
                                   2
    dPt = ∂P dt + ∂P dr + 1 ∂ P σ 2 (·)dt
             ∂t      ∂r         2 ∂r2
                              2
    = [ ∂P + ∂P µr (·) + 1 ∂ P σr (·)]dt + ∂P σr (·)dzt
        ∂t      ∂r         2 ∂r2
                                     2
                                               ∂r
    = Pt [µr (·)dt + σp (·)dzt ]
                                          2
    Pt µr (·) = ∂P + ∂P µr (·) + 1 ∂ P σr (·)
                  ∂t   ∂r              2 ∂r2
                                             2
                  ∂P
    Pt σp(·) = ∂r σr (·)
    ⇒                          2
                ∂P
                     + ∂P µr (·)+ 1   ∂ P
                                            σr (·)−rt Pt
    σπ(·) = ∂t ∂r ∂P σ ∂r2
                         2
                           (·)
                      ∂r r
    One factor determines everything ⇒ risk free rate is the single risk factor.
    Is this reasonable to work with in realtiy?
    It is much too restrictive ⇒ you need at least two or three factors:

1. Factor Level factor
          ⇒ parallel shifts for example (problem: arbitrage violations)

2. Factor Twist factor
          ⇒ allows you to change a positive slope into a negative

3. Factor Hump factor




                                                           38
Steps for factor models:

  1. Designing a theoretical model

  2. Apply it to reality

  3. Change model over time

Affine multi- factor models
Affine models: Log price is linear in the risk factors
y = (y1 , y2 , . . . , yk )

 (1) dy = Φ(¯ − y)dt + Σdw (Σ= Covariance matrix)
            y
     dw = (dw1 , dw2 , . . . , dwk ) , Φ = (k × k) matrix
     For example: dy1 = l Φ1i (¯i − yi )dt + l σi1 dwi with yi − yi = equilibrium level
                                i=1      y           i=1         ¯
           √
 (2) dwi = αi + βi ydzi , E[dzi ] = 0 ∀i, E[dzi dzj ] = 0 ∀i = j
     We assume that the random stocks dwi are larger if y is greater ⇒ they are not
     independent

 (3) r = r(y) = σ0 + σy
       dπ
 (4)   π
            = −rdt − bπ dw

Instead of one Sharpe ratio you have k Sharpe ratios: lnPt,T = A(T − t) − B(T − t) y
Since we are still in a perfect capital market the model does not deal with liquidity premia
nor with taxation effects (only coupons are taxed after one year).

5.1     Concepts of asset- liability management
Banks are making money from maturity transformation. Maturities of liabilities and
assets differ:
Short term deposits for liabilities vs. long term for assets.
For a downward sloping term structure you revise this strategy. It is very attractive but
very dangerous.
Basic elements:

   - forecasting term structure changes

   - How do these changes effect business

Interest rate management:

  1) Cash flows: Interest income vs. interest expensure (interest cash cyclus)
     You have to destinguish between

            - residual maturity of our instruments
            - interest rate fixation period: interest is fixed for a certain period ⇒ floating
              rate note (More important issue)


                                             39
   Forecasting of your business is also very important. This cash flow analysis is also
   called gap analysis.
2) Market value analysis: You have to make sure that the equity is above a certain
   level
   Disadvantages: Many instruments are traded in illiquid markets
3) Profit/Losses:
   Trading book: Short- term trading
   Banking book: Long positions in trading
   Net amount of market values and cash flows for trading book
   Problem: Banking book ⇒ different international rules for accounting profit and
   losses. The question is whether you use the market value or the face value (for
   example: Niederstwertprinzip)
      - Banks
      - Insurance companies
      - IAS 39/ IFRS 39: Under which conditions is marcro hedging allowed ⇒ do
        you allow hedge accounting?
        example: Bond (∆ = 100) and Swap (∆ = −100)
        Are you allowed to add up these figures or are you not allowed?
        The german law does not allow you to do so. In Germany you have to come
        up with a loss of 100.
        IAS 39 allows you to do this microhedge but you are not allowed to do the
        macrohedge because of the risk of abuse.
4) Basel II:
   Example:
   Bank has a long term fixed rate bond rl
   Bank has a short term deposit at rs
    1) Maturity transformation (rl − rs ) gap analysis ⇒ you want to make sure that
       this gap stays positive. You have to predict rs :




        You have to come up with a probability distribution:

                                        40
           E(rl − rs ) and σ(rl − rs ) = σ(rs )
           VaR- Analysis: You also have to analyse this over several periods: V aRt+1 >
           V aRt; σ(rst+1 ) > σ(rst)
           You should look at a sequence of years.
       2) M Vbond −M Vdeposit (the M Vdeposit does almost not change ⇒ close to par value)
          M Vbond is determined by rl (termstructure)




Completely different approach:
Conditions for an optimal miopic policy (Metallgesellschaft):
Oil prices are iid and RRA is constant. But oil prices are strongely mean- reverting ⇒ so
miopic policy is very bad because it is only minimizing short term risk
Mean- reversion: A policy that is good in the short run is bad in long run. Forecasting
long term interest rates and look at short term rates.
The short term rate is driven by the Central Bank policy:

   - inflation

   - supporting business circle

The long term rate is independent of the Central Bank policy:

   - inflation expectation is important

   - international interest rates

If you come up with a model you should assume a multivariat normal distribution for
interest rate risk. But it relies on the condition that the risk factors are linear. We saw
already that only the log of the risk factors is linear. So bankers do a Monte Carlo simu-
lation. Most models only use two risk factors (short term rate and long term rate)

Another approach would be the stress test. Consider an event where all risk factors
are against you. It needs to be defined for your special scenario. It has to be adjusted all
the time.
What is a realistic scenario?

                                            41
Which fractile do you choose?




A different approach is back testing:
Take a probability distribution of P/L: Feed into the model the multivariat distribution
of the risk factors and undergo a Monte Carlo simulation.




Back testing: 215 days a year * 0,15= 35,5 ⇒ Frequency distribution at the end of the
year (ex post):




                                          42
If the probability function and the frequency distribution match it is okay, otherwise not.

5.2    Duration- based interest rate risk management
SKIPPED

5.3    Dynamic risk management in a diffusion model
Example: One factor model (Vasicek):
dr = Φ(¯ − r)dt + σr dz with Φ= speed of mean reverting and r= long term equilibrium
           r                                                     ¯
level
Mean- reverting model
Once you know the probability distribution of the risk factor you also know the probability
distribution of the pricing kernel.
With σ being a constant ⇒ Distribution if rt :
rt = (r0 − r)exp(−Φt) + r + σr 0t exp[−Φ(t − s)]dz(s)
               ¯           ¯
rt is normally distributed
E0 (rt ) = r + (r0 − r)exp(−Φt)
             ¯         ¯
  2      2 exp[−2Φt]−1
σ0 = σr        −2Φ
Effect of mean reversion:




Geometric Brwonian motion:
        √         2
σt = σ t and σt = σ 2 t
−→ r if t → ∞
      ¯
−→ r0 if t → 0
                            2
                           σr
σ0 (rt = σr exp[−2Φt]−1 −→ 2Φ if t → ∞
  2       2
                −2Φ
The problem is that the short term rt becomes negative because of a very high negative
dz. But it might be still acceptable. If you modell log- rates ln(1+r) you always have
positive short term rates.
We try to derive Vt at a given time horizon t and V0 .
dπt
 πt
    pricing kernel
dπt
 πt
    = −rdt + σπ dzt
σπ = Sharpe ratio

                                            43
Applying Itos Lemma:
πt
π0
   = exp[ 0t −(rt + 1 σπ dτ − σπ zt ]
                     2
                       2
                             πt
M ax E0 (u(Vt )) s.t. E0 (Vt π0 ) = V0 (Budget constraint) If we have a complete market ⇒
FOC:
          πt
u (Vt = λ π0
Demand function:




If the market is statically complete everything is fine. If it is only dynamically complete
you have to adjust your strategy.
Problems:
   - We assume that there is only one risk factor. But in reality there are more risk
     factors. So you have to monitor your strategy.
   - If you come up with a demand function you also want to know what happens in
     between (τ < t)
Adjusting to that:
       πt
Et (Vt πτ ) = Vτ




So you make sure that at an intermediate point of time your portfolio looks okay. Another

                                           44
adjustment would be changing the given horizon of t.

Many bankers still use miopic models:
They say term- structure is quite steep and then they do maturity transformation.
Short term policy:
∂V
 ∂r
     normally distributed
 ∂V
| ∂r | = a + bV
Depending on solvency and risk aversion. You completely ignore forecasts of changes in
term structure.
Example: long rate - short rate= 2%
Short term rate is very volantile. It matters exspecially in the case of transaction cost.


6     Credit risk valuation and management
If you buy a German bond it is riskless. This is not the case with an Argentine bond.
The loan business is fairly complex because there are various properties of the loan:

    - Short term or long term

    - There exists a call option for the lender

    - Checking accounts where you can change your position everyday. It is a three month
      loan which can be rolled over

    - Long term loans are colleteral loans for example morgages which are hopefully stable

6.1    Valuation of single credits
Rankings in bonds display default risk.
                          Bond            Percentage   Credit spread
                     Goverment bonds         4,5%           0%
                          AAA                4,8%          0,3%
                           AA               5,05%         0,505%
                            A                5,5%           1%
                          BBB                 6%           1,5%
                            .
                            .                  .
                                               .             .
                                                             .
                            .                  .             .
Credit spread: Differential you earn because of taking a higher risk
Bond pays you: e1 , e2 , . . . , eT
This could be:

    - actual payment

    - expected payment

    - claims on the bond:
                             eT
      P0 = T (1+rf +creditspread(ratingT ))T
              t=1
      If the maturity is short then the credit spread gets smaller

                                            45
KMV- Model:
It is an option priced model:
Loan for one year. V1 : Value of whole company at the end of year 1:




D: Claim of the loan you give to the company. The company is bancrupt if V1 falls below
D. If this is the case you get V1 .
       1
P0 = Rf (D- forward price of a put option) with P0 = Price of the loan
       1
P0 = Rf D if the probability of default is zero.
If there is a default, the company looses the triangle abc. It is like selling a put option to
the company owners.

6.2    Valuation of credit portfolios
Diversification is also important for loan valuation because otherwise you run into prob-
lems.
Examples:

    - Hypobank (only in real estates)

    - Berliner Bankgesellschaft (invested in East German real estates)

    - between 60-80 countries experienced a banking crises

⇒ Banks are not allowed to engage in many big loans (10% of the banks equity)
⇒ So you have to look at loan portfolios
Important: Overall portfolio risk and not single loan risk
Example:
Bank gives n loans each with a par value 1 Euro in states 0 and 1.
p: Probability of repayment
1-p: PD (Probability of default)
LDG: Loss given default




                                             46
   .




   .




   .
Loss portfolio= LP F = n Li
                          i=1
                                      1   n            1   n
Loss rate of the portfolio= LRP F =   n   i=1   Li =   n   i   LGDi
E[LP F ] = n(1 − p)LGD
E[LRP F ] = (1 − p)LGD

Single loan:
σ 2 (L1 ) = p[0−(1−p)LGD]2 +(1+p)[LGD −(1−p)LGD]2 = LGD2 p[(1−p)2 +p(1−p)] =
p(1 − p)LGD2
n losses:
σ 2 (LP F ) = np(1 − p)LGD2
                1
σ 2 (LRP F ) = n p(1 − p)LGD2
Variance of total losses goes to infinity (n → ∞)
Loss rate goes to zero (n → ∞) ⇒ do not only consider LRP F




   .




   .




     .
Thus, the credit spread must exceed the expected loss (60 basis points probability of loss
⇒ if 100 basis points ⇒ 40 basis points spread). The model is based on the assumption of
no correlation between defaults ⇒ highly misleading empirically (look at last four years:
recession) ⇒ macro factors drive defaults
Case of correlation:
σ 2 (LRP F ) = p(1 − p)LGD2 [ n n2 + n n ρij n2 ] (1)
                              i=1
                                  1
                                         i   j
                                                 1

Assumption: ρij = ρ (the same between all companies)
                       1  1
(1) = p(1 − p)LGD2 [ n + n2 ρn(n − 1)] (2)

                                                47
n(n-1)= Covariance matrix without diagonal
                       1       1
(2) = p(1 − p)LGD2 [ n + ρ(1 − n )] (3)
For n → ∞:
(3) → p(1 − p)LGD2 ρ
⇒ Lower limit is dependent on ρ
⇒ Danger for the bank ⇒ substantial variance for overall loss distribution (Losses for
overall portfolio)
σ 2 (LP F ) = p(1 − p)LGD2 ρn2

6.2.1   Credit metrics
Published by J.P.Morgan 1996/97 ⇒ default risk:
You have a loan portfolio and take it for given ⇒ what is the value of that portfolio after
12 month ⇒ random, so you have to come up with a probability distribution:
1. Step:
Change value of a single loan ⇒ come up with a transition metrics for ratings:
        rating today   rating after twelve month
                                  AAA              AA     A     ...   CCC   Default
           AAA                     0,9
            AA                                      0,9
            A                                             0,9
             .
             .
             .
           CCC                                                        0,9
          Default                  0                0     0     ...    0      1
0,9 describes the probability that you stay in the same rating in twelve month.
Rating agencies do not change ratings very often because they are proud of the stability
of their ratings so that they are more relyable. So the ratings are not as timely as market
values.
The essential input of credit metrics is the transition metrics.
Credit metrics ignores interest rate risk in forecasting values:
                             t
(1 + rt )t = (1 + st−1 )t−1 ft−1 = (1 + s1 )(1 + f orwardrate(1, t))t−1
⇒ So they use this forward rate as discount rate
(1 + rit )t (1)
rit = rt + ∆ti (Credit spread on rating i)
(→ it decreases with a decrease in time of maturity)




   .




   .


                                            48
    .
(1) = (1 + ri1 )(1 + f orwardrate(1, t(i))t−1
You estimate the term structure for each rating class.
V1i = T (1+f orwardrate(1,t(i)))t−1 with i = AAA, AA, . . . , CCC
        t=2
                       et

ViD = par value(1-LGD)




   .




   .




    .
                                               ˜         ˜
pi1 = transition probability from i to 1 → E0 (V1 ), σ0 (V1 )

2. Step: portfolio approach
Credit metrics assumes that the correlation of loan returns is the same as correlation
between stock returns. It is also assumed that all changes in loan values are given by a
multivariat normal distribution.
⇒ Expectation is the sum of the expectations of single loans.
     ˜
σ 2 (VP F i1 ) = i j σij




   .




   .




   .
You can come up with VaR
Advantages and disadvantages of credit metrics:

    - easy to handle

                                              49
       - correlation problems: you might even apply a mappig
Look at a bivariate normal distribution:
x1 , x2 and ρ ∈ (0, 1) ⇒ look at tails
x1 = µ1 − ασ1 , x2 = µ2 − ασ2 (bad tails)
You plot a simulation result:




       .




       .




   .
Solution:
Take a bivariate t- distribution

Copula functions C: F1 (x); F2 (x); joint distribution F (x1 , x2 ) ⇒ F (x1 , x2 ) = C[F (x1 , F (x2 )]
C connects the marginal functions and the local dependence everywhere
       - ignores term structure risks
       - it is fairly easy to derive a two- period model, a three- period model and so on.
 (1)
pij : Probability to migrate from rating i today to j in 1 year.
 (2)
pij : Probability to migrate from rating i today to j in 2 years.
Procedures: i → k → j
 (2)           (1) (1)
pij = n pik pkj ; (i,j)
          k=1
          (1)            (2)
p(1) = (pij ); p(2) = (pij ) transition metrics
⇒ p(2) = p(1) p(1)
˜
V2i = T (1+f orwardrate(2,t|i))t−2
                       et
         t=3




       .




       .

                                                 50
  .
      ˜         ˜
⇒ E0 (V2 ), σ0 (V2 )

Transition metrics assumes that the metrics is stationary ⇒ rating history does not play
a role.
In reality a company which has been downgraded has a higher probability of moving fur-
ther down than a company which has been remained in a rating class for several years.
With an upgrade this probability is not as high.
            todays situation   rating after twelve month
                                          AAA              AA   A   BBB. . .
                  AAA
                   AA−
                   AA
                    A−
                    A
                  BBB−
                     .
                     .
                     .

(·)− : Companies which have been downgraded over the last year.
This transition metrics seems to be more realistic.

6.2.2    McKinsey model
It is a factor model which captures developments of macro factors and ideosyncratic
factors.
V1k = value of a loan k at date 1
V0k = value of a loan k at date 0
V1k
V0k
    = ak + i bik Fi + k , ∀k (gross rate of return)
Linear regression on macro factors
Fi = macro factor i (realisation at date 1)
 k should be uncorrelated
Macro factors:

    - growth rate of GDP

    - level of term structure of interest rates

        a) short term level
        b) difference between long term and short term levels

    - effective exchange rate ⇒ weighted exchange rate by export shares for example

You might also come up with industry specific models
Problems:

    - it may be that you do not have enough data to undergo this regression

    - different factors for different countries


                                             51
The question is whether the default risk is included in that model.
E[V1k ] = V0k [a + i bik E[Fi ]]
bi = k bik V0k ; V0 = k V0k ⇒
            V0
E[ k V1k ] = V0 [a + i bi E[Fi ]]
σ 2 ( k Vik ) = V02 i j bi bj Cov(Fi , Fj ) + k V0k σ 2 ( k )
                                                  2

The first term on the right hand side describes the systematic risk and the second the
unsystematic risk
Correlation is only covered by systematic risk factors. So we look at normal times were
most of the loans do not default.

6.2.3   Credit risk+ model (CDFB)
It is much more concerned about default.
Structural models: Loans are derived from business and markets (economic story)
Reduced form models: Ignoring economic ideas completely. You just come up with prob-
abilities of default ⇒
Hazard rate:
ht = Prob(default in period t— non-default in previous periods)
Unconditional default probability in period t = ht t−1 (1 − hτ )
                                                      τ −1
This is the starting point for the credit risk+ model. The hazard rate model is modelled
equally to the McKinsey model:
hkt = akt + i bik Fit + kt , ∀k, t
The history of the macro factors appears to play a major role. An advantage is that you
can make many assumptions on the probability distribution ⇒ analytical solution (for
example Poisson process or Γ distribution)

6.2.4   Contagion effects
These effects say that the assumption that the k s are uncorrelatred is questionable.
Example: Parmalat ⇒ domino effect for suppliers and suppliers of suppliers etc.
So you have to look in balance sheets in what way companies are interlinked with each
other. We talk about quality problems of loans which means default risk. But you
also have to look at term structure changes. Most of the models do not have analytical
solutions.

6.3     Managing credit risk (Management of default risks)
6.3.1   Managing the laon portfolio given non- tradeability of default risk
Banking crisis has often been driven by loan defaults. If loans are tradeable you make
sure that further losses do not hit you anymore. It is almost the same as in maturity
transformation in asset liability management ⇒ interest rate swaps, closing maturity gap
We start with the assumption that default risks are non- tradeable. This is a crucial
assumption since in the US about half the bonds are securitized. Investment bank is an
intermediar between markets and clients. It does not keep bonds for example in its book.
In continental Europe only a quarter of the bonds is securitized. Bonds are not issued
by companies in Germany because of the Gewerbeertragssteuer. Some companies do it in
some remote places. So tradeability is very limited in Germany. So it is important that

                                          52
loans are viewed over many periods.
If default risk is high there are two possibilites:
   1. shorting maturity time for the bank ⇒ does not imply that the bank gets the money
      back surely.
   2. high colleterals for example morgages ⇒ relies on the value of the land (commercial
      vs. residential land) ⇒ banks prefer morgages on residential land
Diversification is the basic principle of loan policy. In Germany you can only put up to
30% of your equity in a single loan. It is implied by law ⇒ there are many rules restricting
the single loan size.
You restrict loans in terms of the rating of the borrower and in terms of the maturity of
the loan. The higher the rating the higher the inner limit. The shorter the maturity the
higher the inner limit. The same counts for colleterals.
Sensitivities are equivalent to the McKinsey model: bi V0 ≤ Bi , ∀i restricting securities
Bi is inversely related to σ(Fi ).
Some banks also come up with VaR limits:
P rob( k (V1k − V0k ) ≥ E1 ) ≥ α1 where the first term describes the profit and loss of the
total loan portfolio.
Date t:
P rob( k (Vtk − V0k ) ≥ Et ) ≥ αt
⇒ you should restrict term structure of loans in terms of loan maturities.
Problem:
No consideration of risk and return ⇒ you have to come up with a risk and return model:
Max E( k V1k ) − λσ 2 ( k V1k ) s.t. k V0k ≤ V0 ¯
So k becomes an index of an industry branch ⇒ long term policy for the bank ⇒ optimal
industry structure of loan portfolio .

6.3.2   Decisions about single loans
Positive decision about a loan:
           Effektive yield of the loan (IRR, bond on contractural paymants (ignore default risk)
 - refinancing cost of the bank with equal duration (we do not want to consider maturity transformati
         - expected annualized default loss per Euro (form expected default rate into an annuity)
                               - expected annualized transaction cost per Euro
                            - cost of unexpected default losses c∆V aR per Euro
                                                = net result
A new loan raises the VaR: ∆V aR ⇒ new VaR
    .
c = required rate of return which the bank wants to earn on equity capital.
Equity capital serves two purposes:
    - provides cash
    - provides a risk backup
⇒ So if 14% is demanded and the risk free rate is 5% ⇒ risk component equals 9%.
     cost of equity capital 14%
        - refinancing cost 5%
 compensation for risk bearing 9%=c

                                              53
     .
Assumption: Normal distribution:
σ 2 (P/L) = x2 σa + x2 σn + 2ρan σa σn xa xn with xa = volume of already existing loans and
               a
                 2
                       n
                          2

xn = volume of the new loan
                                                      σa
                    2                       2
∆σ 2 (P/L) = x2 σn + 2ρan σa σn xa xn = σn [x2 + 2(ρ σn xa )xn ] = σn [x2 + 2βxn ] = σn [(xn +
                 n                            n
                                                                    2
                                                                        n
                                                                                      2

β)2 − β 2 ]
∆V aR = α∆σ(P/L) = ασn (x + β)2 − β 2
∂∆V aR
  ∂x
            ασ
        = √ n 2(xn +β) 2 =
                   2
                             ασn
   n        (xn +β) −β      1−( xβ
                                     )2
                                n +β




   .




   .




   .




   .




   .




     .
t is declining in the size of the loan. The larger the loan, the lower the unexpected default
risk. It is not very reasonable because for very big loans we have a concentration risk.
This is not a good model unless you come up with some restrictions on the size of the
loan.
The higher the rating of the costumer the lower c∆V aR.




                                             54
6.4    Instruments for trading default risk
Most of the banks give a loan and have to keep it on their books ⇒ difficult to sell it
because others might be afraid of default risks. This is not a problem in the absense of
asymmetric information. This counts for continental Europe.
In the US you have either commercial banks or investment banks.
Investment bank: trades securities ⇒ offering circles for new bonds ⇒ lot of information
⇒ Underwriter is obliged to buy not sold bonds (investment bank) ⇒ no default risk
because it can sell it ⇒ strong contrast to commercial banks.
Problem for a commercial bank is that they cannot hedge default risk (Japanes banking
crisis in the 90ties)

First root of trading default risk:
1975 first morgage backed security in the US ⇒ sells morgange backed loans to investors
⇒ today credit card companies, leasers etc. do the same
Second root of trading default risk:
Derivative instruments in trading default risk of credits ⇒ for example German banks
and Kreditanstalt fuer Wiederaufbau

Credit default swap
It is like an option instrument




   .




   .




    .
You buy an insurance against default of an obligator.
For example:
Fiat bond, 7%, parvalue 100” Euro, repayable 2011
Maybe you are not happy with that anymore Protection seller and protection buyer; pro-
tection on 20” Euros and maturity five years
default definition: legal insolvency procedure is launched
damage payment: loss in market value of bonds




                                          55
   .




    .
These markets are fairly liquid now for single named credit default swaps.
⇒ single name for example Fiat
⇒ single loans
There is little trading on portfolio swaps.
The credit default swap only insures you during the maturity of the swap ⇒ no protection
against default after maturity.
The standardization of swaps is driven by the ISDA (International Swaps and Deriva-
tives Organization) ⇒ it recommends physical delivery: the protection buyer can deliver
any Fiat loan to the protection seller with a parvalue of 20”. ⇒ It makes much more
restrictions:

   - loan which you deliver must not be subordinate to some reference claim

   - maturity and coupons are not restricted

Default definition is very difficult. Example: 1987 Russia stopped paying the Rubel bonds
but not the Dollar bonds. Is this default or not?
ISDA definition:
Bancrupcy:

   - company files insolvency at the court

   - the court opens an insolvency case

   - owners have to pay for the company

   - the company fails to pay its obligation for at least 45 days

   - the company says it will not pay anymore

restrictions:
Agreement between compnay and creditors to reduce the obligation on retro payments or
a downgrade in rating ⇒ moral hazard problem
⇒ trigger default earlier for protection buyer
⇒ delay default for protection seller
⇒ So default has to be precisely defined.




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