# Interest Rate Risk Asset and Liability Management Chap 5

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```					         M35R Introduction to Asset Liability Management for Actuarial Science
Text: Fixed Income Securities (Fabozzi Chap 5)
Reference: BPP Course 6 Unit 9                 YJohns Lectures 5-6

Measuring Interest Rate Risk
1.   Immunize a portfolio using duration.
2.   Duration/convexity approach.
3.   Price volatility characteristics of option-free bonds.
4.   Criticisms of duration.
Reference: See Financial Institutions Management chap 9 (Handout)

Asset Iiability Management: is the measurement and management
of a firm’s interest rate risk.

Approaches to measuring interest rate risk

The value of a bond changes on the opposite direction of the change
in interest rates. There are two approaches to measuring interest rate
risk:

A. The full valuation approach
B. Duration /Convexity Approach

A. The full valuation approach: measure interest rate risk by
revaluing the bond using an interest rate change scenario. The
change in market value, expressed either as a dollar change or
percentage, indicates the interest rate risk.

B. The duration/convexity approach:

Duration:
– measure of approximate sensitivity of a bond’s value to rate
changes.
– First order approximation of the price change.
– “first derivative” of price as a function of yield
– To improve the estimate, a second order estimate using the
combined duration/convexity approach can be used.

1
M35R Introduction to Asset Liability Management for Actuarial Science
Text: Fixed Income Securities (Fabozzi Chap 5)
Reference: BPP Course 6 Unit 9                 YJohns Lectures 5-6

Economic Meaning of Duration:

C         C       C                F +C
P=           +          +         + ... +
(1 + R ) (1 + R ) (1 + R )
2         3
(1 + R )
n

dP        C       2C             n( F + C )
=−           −        − ... −
(1 + R ) (1 + R )          (1 + R )
2        3                  n +1
dR

dP      1  C             2C              n( F + C ) 
=−                +           + ... +            
dR    1 + R  (1 + R ) (1 + R ) 2         (1 + R )n 
                                        

dP     1                  1
dR
=−     ∑ t * PVt = − 1 + R [ P * D]
1+ R t

∑ t * PV ∑ t * PV
t           t
since   D=   t
=          t

∑ PV      t
P
t

Hence we can write

−dP
% price change
D= P =
dR  relative change in interest rates
1+ R

Duration is the interest-elasticity (sensitivity) of the security’s price to
small changes in interest rates.

dP                dR
% price change =                 x 100 = − D x      x 100
P               1+ R

dR
\$ price change =             dP = − D x P x
1+ R

2
M35R Introduction to Asset Liability Management for Actuarial Science
Text: Fixed Income Securities (Fabozzi Chap 5)
Reference: BPP Course 6 Unit 9                 YJohns Lectures 5-6

Duration approximation holds for small changes in interest rates of
order 1 basis point (1 b.p. = 0.01%)

Immunising a portfolio using duration (Chap 9)

dR
dP = − D x P x               fixed income security
1+ R

dR
dA = − D A x A x             asset
1+ R

dR
dL = − DL x L x              liability
1+ R

dR
dE = −( DA x A - DL x L) x           Market value of Equity
1+ R

L             dR
dE = −( DA -     DL ) x A x
A            1+ R

dR
dE = −( DA - kDL ) x A x
1+ R

where k = L/A the degree of leverage in the balance sheet

Immunisation against interest rate risk requires the bank to consider:
1. Duration (average life of the cash flows of the assets &
liabilities).
2. The degree of leverage in the balance sheet – proportion
of assets funded by liability.

3
M35R Introduction to Asset Liability Management for Actuarial Science
Text: Fixed Income Securities (Fabozzi Chap 5)
Reference: BPP Course 6 Unit 9                 YJohns Lectures 5-6

Macaulay duration: weighted average time to each cash flow

n

∑t.PV              t
D=         t =1
n

∑PV
t =1
t

Modified Duration: approximate percentage change in value of 100
basis point change in rates.

n

1        ∑                    t.PVt
D=              . t =n
1

1+ yield / k
∑ PV
t =1
t

k=1 (annual), 2 (semi-annual),….

Portfolio Duration: weighted average of the duration of the bonds in
the portfolio.
n
DA = ∑ wi Di
i =1
where

w     i    = the market value of bond i divided by the market value of the
portfolio

Di       = the duration of bond i

n        = number of bonds in the portfolio

4
M35R Introduction to Asset Liability Management for Actuarial Science
Text: Fixed Income Securities (Fabozzi Chap 5)
Reference: BPP Course 6 Unit 9                 YJohns Lectures 5-6

Price Volatility Characteristics of option-free bonds
For bonds without embedded options, the inverse price/yield
characteristic strictly applies.

However, this relationship is convex, not linear. Therefore a bond’s
price sensitivity to interest rates will change as its yield changes:

1. With a given change in required yield, the percentage price change is not the
same for all bonds.

2. For small changes in the required yield, the percentage price change for a given
bond is roughly the same, whether the required yield increases or decreases.

3. For large changes in the required yield, the percentage price change is not the
same for an increase in required yield as it is for a decrease in required yield.

4. For a given large change in basis points in the required yield, the percentage
price increase is greater than the percentage price decrease. {i.e. the price
appreciation for a decrease in required yield (- ∆R ) will be larger than the loss
realized for an increase in required yield of the same amount ( ∆R ) }.

We measure the price sensitivity of a bond in terms of a dollar price
change or a percentage price change.

Features of a bond that impact its degree of price sensitivity to
interest rate changes:

Maturity: the longer the bond’s maturity the greater the sensitivity
Coupon rate: the lower the coupon rate, the greater the sensitivity
Yield level: a 1% change in yield will have a more significant effect on
a bond with a low yield than on the price of a bond with a much
higher yield.

5
M35R Introduction to Asset Liability Management for Actuarial Science
Text: Fixed Income Securities (Fabozzi Chap 5)
Reference: BPP Course 6 Unit 9                 YJohns Lectures 5-6

Convexity:
Improvements to duration: Duration estimates the same percentage
change regardless of whether the interest rates increase or decrease.
(violates property 3 above)! Hence, duration is a good measure of
price change only for small changes in yield.

Duration is a first-order approximation of the price-yield curve. The
approximation can be improved by using a second-order
approximation, the duration/convexity approach.

Graph of price-yield relationship:

Price

Yield

For larger changes in yield (order 200bp=2%) the bond-shock
relationship exhibits convexity.

Rise in rates: duration overestimates price fall
Fall in rates: duration underestimates price increase

Greater convexity:
a) greater interest rate protection from increase in interest rates
(price fall slower than for linear relationship)
b) greater potential gains from interest rate falls
c) greater error in duration matching immunization strategy
Hence, convexity is desirable for assets.

6
M35R Introduction to Asset Liability Management for Actuarial Science
Text: Fixed Income Securities (Fabozzi Chap 5)
Reference: BPP Course 6 Unit 9                 YJohns Lectures 5-6

Taylor’s Expansion: Recall from calculus that the value of a function can
be approximated near a given point, using its Taylor Series around that
point. Using only the first two derivatives:

1
f ( x ) ≈ f ( x0 ) + f '( x 0) * ( x − x0 ) +
f "( x0 ) * ( x − x0 ) 2 + ... whichgives
2
1
f ( R + ∆ R ) ≈ f ( R) + f '( R) * ∆ R + f "( R)*( ∆R) 2 + O( ∆R) 3
2
2
dP            1d P
P + ∆P ≈ P +         * ∆R +          *( ∆R)
2

dR            2 dR 2

2
dP        1d P
∆P ≈    * ∆R +        *( ∆R)
2

dR        2 dR 2
∆P 1 dP
2
11 d P
≈       * ∆R +            *( ∆R)
2
2
P   P dR         2 P dR
∆P                      1                      
*100 ≈  −MD * ∆R + V *( ∆R) 2                *100     % price change
P                      2                      

1
∆P ≈ − MD * P * ∆R + V * P *( ∆ R) 2 price change
2
                  1               
P + ∆P ≈ P 1 − MD * P * ∆ R + V * P * (∆R) 2    new price
                  2               

ALTERNATIVE APPROACH TO FINDING DURATION & CONVEXITY
ESTIMATES FOR SHIFT IN YIELD CURVE OF ∆Y    (Fabozzi chap 5)

V− − V+     price if yields decline by ∆ y − price if yields rise by ∆y
duration estimate          MD =                 =
2V0 ( ∆Y )        2( initial price)( change in yieldindecimal , ?y)

V− + V+ − 2V0
convexity estimate            V=
V0 ( ∆Y ) 2

                  1            2
P + ∆P ≈ P 1 − MD * P * ∆ R + V * P *( ∆R)                       new price
                  2             

END

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