# Valuing Cash FlowsII

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```							Valuing Cash Flows II

Contingent Payments
Valuing Cash Flows
   So far, we have been pricing securities with cash
flows that are predefined and fixed.
   Given the stream of cash flows and an expected path
for interest rates, the price of the instrument is simply
the discounted value

N 
    CFt     

P0    t             
t 1     Fi  
1
 i 1       
Valuing Cash Flows
   Suppose that the cash flows are still predefined,
but not fixed. Instead, each cash flow is
dependant on some future “state of the world”
   Given the stream of (state dependant) cash flows and
an expected path for (state dependant) interest rates,
the price of the instrument is still the discounted value

N   
    CFt i     

P0    t                 
t 1     Fi i  
 i 1 1         
Example
Assume that the required real (inflation adjusted)
rate of return is 2%. Each period there are two
possible states of the world

State #1                         State #2
Inflation = 0%                   Inflation = 8%
r = 2%                           r = 2%
i = r + Inflation = 2%           i = r + Inflation = 10%
Each state has an equal chance of occurring.

E(Inflation) = (.5)(0%) + (.5)(8%) = 4%
E(i) = (.5)(2%) + (.5)(10%) = 6%
Now, consider a 1 year STRIP with a face value of
\$100. How should this security be priced?

State #1                         \$100
P=        = \$98.04
Inflation = 0%                   (1.02)
i = 2% + 0% = 2%

State #2
\$100
Inflation = 8%               P=        = \$90.91
(1.10)
i = 2% + 8% = 10%

E(P) = (.5)(\$98.04) + (.5)(\$90.91) = \$94.48

Shouldn’t this equal 6%
\$100 - \$94.48                     (the expected nominal
YTM =                   *100 = 5.84%      return)?
\$94.48
Remember, the bond pricing
Price                is non-linear! Therefore,

E(f(x)) = f(E(x)) (Jenson’s Inequality)

\$98.04

\$94.48

\$90.91
Pricing
Function
Yield
2%         10%

E(Y(P)) = 5.84%   = 6% = E(Y)
Pricing TIPS
   In 1996, the US Government introduced Treasury
Inflation Protected Securities. These are bonds with
state contingent payouts. Each future payment is
indexed to the inflation rate.

State #1                 State #2
Inflation = 0%           Inflation = 8%
i = r + Inflation = 2%   i = r + Inflation = 10%
CF = \$100                CF = \$108
Now, consider a 1 year STRIP with a face value of
\$100. How should this security be priced?

State #1                        \$100
P=        = \$98.12
Inflation = 4%                  (1.02)
i = 2% + 0% = 2%

State #2
\$108
Inflation = 8%              P=        = \$98.12
(1.10)
i = 2% + 8% = 10%

E(P) = (.5)(\$98.12) + (.5)(\$98. 12) = \$98.12

\$100 - \$98.12
YTM =                   = 2%
\$98.12
TIPS vs. STRIPS

P(TIPS) - P(STRIP) = \$98.12 - \$94.48 = \$3.64
Y(STRIP) – Y(TIPS) = 5.84% - 2% = 3.84%

Here, the price reflects (approximately),
the expected inflation rate over the
coming year.

Is this the only factor
influencing the spread
between STRIPS and TIPS?
The STRIP has a larger amount of risk associated
with it.

STRIP
P (Inflation = 0%) = \$98.04   Std Dev = 5
P (Inflation = 8%) = \$90.91
Shouldn’t this
TIP                                           risk be worth
something?
P(Inflation = 0%) = \$98.12    Std Dev = 0
P(Inflation = 8%) = \$98.12

TIP Price = STRIP Price + Inflation Premium + Risk Premium
Consider the following Prices (for a 3.875% annual
coupon) on Treasuries vs. comparable TIPS

Maturity Date       Non-Indexed                TIPS
Treasury
1/09                \$101.56                    \$111.75
(3.47%)                    (.823%)

3.47% - .823% = 2.647%

This would be a reasonable starting point
for market expectations of inflation, but
there could be other risk factors built in!!
Default rates on corporate debt
tend to be countercyclical while
interest rates are procyclical

% Change

GDP Interest
Rates

Default
Rates

Time
Expansion          Recession
Suppose that default rates are 2% in expansions and 12% during
recessions. Consider two bonds with \$100 of Face Value

State           Treasury           Corporate        Interest Rate

Expansion       \$100               \$98              6%
Recession       \$100               \$88              3%

Expansion                       \$98
P=        = \$92.45
Payment = \$98                  (1.06)
i = 6%
Std. Dev = 4.95
Recession
\$88
Payment = \$88              P=        = \$85.44
(1.03)
i = 3%
\$100 - \$88.95
YTM =                   = 12.42%
E(P) = (.5)(\$92.45) + (.5)(\$85.44) = \$88.95             \$88.95
A Treasury Bill Pays out \$100 Regardless of the State

State           Treasury            Corporate           Interest Rate

Expansion       \$100                \$98                 6%
Recession       \$100                \$88                 3%

Expansion                      \$100
P=        = \$94.34
Payment = \$100                 (1.06)
i = 6%
Std. Dev = 1.93
Recession
\$100
Payment = \$100             P=        = \$97.08
(1.03)
i = 3%
\$100 - \$95.71
YTM =                     = 4.48%
E(P) = (.5)(\$94.34) + (.5)(\$97.08) = \$95.71                 \$95.71
Consider the following Prices (for a 3.625% annual
coupon) on Treasuries vs. Corporates

Maturity Date        Treasury                  Corporate
1/08                 \$108.11                   \$99.24
(3.26%)                   (3.77%)

This difference reflects default risk as well
as additional risk based on the timing of
payments

STRIP Price = Corporate Price + “Default” Premium + Risk Premium
Equities
We can think of stocks as simply bonds with state
contingent payments:

Excess Returns
Excess Returns
to Asset i
to the Market

Ri  R f   i Rm  R f 
Covi, m 
A stock’s Beta measures
it’s movements relative to
i 
Varm 
the market
High Beta Stocks move with the
market while low beta stocks move
against the market

% Change
High Beta
Stocks
Interest
Rates

Low Beta
Stocks

Time
Expansion        Recession
State            High Beta          Low Beta         Interest Rate
Stock              Stock
Expansion        \$120               \$80              6%
Recession        \$80                \$120             3%

High Beta stocks pay out larger amounts during good times

Expansion                     \$120
P=        = \$113.20
Payment = \$120                (1.06)
i = 6%
Std. Dev = 25.10
Recession
\$80
Payment = \$80              P=        = \$77.70
(1.03)
i = 3%
\$100 - \$95.43
YTM =                   = 4.8%
E(P) = (.5)(\$113.20) + (.5)(\$77.70) = \$95.43             \$95.43
State            High Beta          Low Beta         Interest Rate
Stock              Stock
Expansion        \$120               \$80              6%
Recession        \$80                \$120             3%

Low Beta stocks pay out larger amounts during bad times

Expansion                      \$80
P=        = \$75.47
Payment = \$80                 (1.06)
i = 6%
Std. Dev = 28.63
Recession
\$120
Payment = \$120             P=        = \$116.50
(1.03)
i = 3%
\$100 - \$96.01
YTM =                   = 4.41%
E(P) = (.5)(\$75.47) + (.5)(\$116.50) = \$96.01             \$96.01
Some assets have maturity dates that can vary
based on interest rate movements…

Now           5yr         10yrs       15yrs         20yrs        25yrs

Face Value can be repaid
anytime in this period
Bond Issued

Some corporate bonds are callable. That is, after
some initial period, the company is able to pay
off the face value early. For example, a 25 year
bond – callable after 15 yrs

If interest rates drop low enough in the “call period”, the firm will
pay the bond off early
MBS/ABS
   In the early eighties, many types of cash
flows were “securitized” into bonds
 Home/Commercial Mortgages
 Car Loans
 Student Loans
 Credit Card Debt

All these bonds have one thing in common…the
loans on which these bonds are based have the
ability to be refinanced!
As homeownership rates increases
worldwide, the mortgage market has
grown….

Germany
Denmark
Netherlands
France
Japan
Canada
United States
United Kingdom
Australia
Spain

0%    20%      40%         60%           80%   100%
2002 Top 10 Mortgage
Markets                                                  \$11.3T worth of
mortgages
outstanding
worldwide in 2002
Netherlands   Australia          Spain
2%      France    2%
3%
Canada                             2%              Denmark
3%                                                 1%

Germany
8%                                              United States
58%
United
Kingdom
8%

Japan
13%
Funding Mortgages

The bank can
either keep the
loan on its
books or
replenish its
funds by selling
off the loan
Home buyer goes to
a mortgage provider
for a loan

These companies will either hold the loans on their
books or package the loans into Mortgage Backed
Securities to sell to private investors
Mortgage Payments
The bank
collects the
payment and
passes it along
to the MBS
creator (they
collect a fee for
this service)
Home buyer makes
monthly mortgage
payments

MBS issuers pass along mortgage payments to
individual investors
Fannie Mae (Federal National Mortgage Association)
was created in 1938 by the Federal Housing Authority
to promote home ownership by creating liquidity in the
home mortgage market

Fannie Mae raises funds                           Fannie Mae uses these
through the issuance of                           funds to purchase
Agency bonds – these                              mortgages
are implicitly backed by
the US government

Fannie Mae will convert
some of these
mortgages to issue MBS
Fannie Mae currently holds around \$1T worth of
mortgages on its books in addition to issuing close to
\$2T in MBS (roughly 40% of all mortgages)
Fannie Mae is the largest participant in the \$4T MBS
Market
Constructing a MBS

You purchase a \$200,000 house by taking
out a 30yr mortgage with a 6% fixed annual
interest rate. Your monthly payment will be
\$1200

Principal                   Remaining
Year
Month   Payment      Applied       Interest       Balance
1       1     \$1,199.10     \$199.10     \$1,000.00     \$199,800.90
1       2     \$1,199.10     \$200.10      \$999.00      \$199,600.80
1       3     \$1,199.10     \$201.10      \$998.00      \$199,399.71
1       4     \$1,199.10     \$202.10      \$997.00      \$199,197.60
1       5     \$1,199.10     \$203.11      \$995.99      \$198,994.49
1       6     \$1,199.10     \$204.13      \$994.97      \$198,790.36
1       7     \$1,199.10     \$205.15      \$993.95      \$198,585.21
1       8     \$1,199.10     \$206.17      \$992.93      \$198,379.04
1       9     \$1,199.10     \$207.21      \$991.90      \$198,171.83
30 Year, 6% APR
(Fixed) = \$1200/mo                           \$24,000/Mo

Fannie Mae Purchases
your loan plus 19
other identical loans
Available Funds
\$24,000/Mo

A

These funds are then         B
divided up into Tranches
(Claims to different parts
of the available funds)      D       C
Available Funds

E
F
Sequential-Pay Example
    Suppose the collateral is a 30-year, \$100M,
9% coupon mortgage portfolio
A                                   A

B
B
D       C
C
Each Trance becomes the
basis for a mortgage
backed security             D

E
E
F
Why not just divide up         F
the payments equally?
(i.e. why have different
tranches?)
Prepayment Risk
Suppose that shortly after you buy your
house, interest rates drop dramatically. You
have the ability to refinance you mortgage at
a lower interest rate
Principal                   Remaining
Year
Month      Payment      Applied       Interest       Balance
1         1       \$1,199.10     \$199.10      \$1,000.00     \$199,800.90
1         2       \$1,199.10     \$200.10       \$999.00      \$199,600.80
1         3       \$1,199.10     \$201.10       \$998.00      \$199,399.71
1         4       \$1,199.10     \$202.10       \$997.00      \$199,197.60
1         5       \$1,199.10     \$203.11       \$995.99      \$198,994.49
1         6       \$1,199.10     \$204.13       \$994.97      \$198,790.36
1         7       \$1,199.10     \$205.15       \$993.95      \$198,585.21
1         8       \$1,199.10     \$206.17       \$992.93      \$198,379.04
1         9       \$1,199.10     \$207.21       \$991.90      \$198,171.83

For example, in the 9th month, you could take out a new loan and pay
off the \$198,171 outstanding on your original mortgage.
Loan pools are characterized by Constant Prepayment Rates
(CPR) the Public Securities Association assumes that , for a
given interest rate, CPRs start at zero and reach a maximum of
6% per year in the 30th month. As interest rates fall, CPR’s
increase.

CPR (%)

9.0                                                 150% PSA
Falling interest rates
6.0                                                 100% PSA
Rising interest rates
3.0                                                     50% PSA

month
0         30                                  360
30 Year, 6% APR
(Fixed) = \$1200/mo

\$18,000/Mo

As loans are
refinances, the
available pool shrinks
Available Funds

Payments made to the various MBS
are altered accordingly to reflect
these prepayments
Mortgage backed securities are Path Dependant. That is, the cash flows vary
based on interest rate movements. For example, suppose that the average
household refinances when interest rates fall below 4.5%

7
6.5
6
5.5
5
4.5
4
3.5
3
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Refinanced in the 19th month
Refinanced in the 4th month
Consider a \$30,000 mortgage with a 5% annual interest
rate paid over three years (three equal installments).
Each annual payment equals \$11,000

Year                     Payment (BOY)            Principal
Outstanding
(Year End)
1                        \$11,000                  \$20,500
2                        \$11,000                  \$10,525
3                        \$11,000                  \$0
Assume that households refinance
whenever the interest rate hits 4.5%
Cash Flows

5.8% Path 1: \$11,000 \$11,000 \$11,000

5.4%
Path 2: \$11,000 \$11,000 \$11,000
5.0%                5.0%
Path 3: \$11,000 \$20,500
4.5%

4.4% Path 4: \$11,000 \$20,500

Given the above paths for interest rates, there
are two paths in which the mortgage is paid off
early
To price this asset, calculate the value over each
possible path, then average them.

Path #1                \$11,000    \$11,000           \$11,000
P=         +              +
P = \$29,809             (1.05)   (1.05)(1.054) (1.05)(1.054)(1.058)

Path #2                \$11,000    \$11,000           \$11,000
P=         +              +
P = \$29,881             (1.05)   (1.05)(1.054) (1.05)(1.054)(1.05)

Path #3                \$11,000    \$20,500
P=         +
P = \$29,159             (1.05)   (1.05)(1.045)

Path #4                \$11,000    \$20,500
P=         +
P = \$29,159             (1.05)   (1.05)(1.045)

(\$29,809 + \$29,881 + \$29,159 + \$29,159)
E(P) =                                             = \$29,502
4
Interest Rate Risk
   When the payments of an asset are
variable, how do we asses interest rate
risk?
Cash Flows

6.8% Path 1: \$11,000 \$11,000 \$11,000

6.4%
Path 2: \$11,000 \$11,000 \$11,000
6.0%                    6.0%
Path 3: \$11,000 \$11,000 \$11,000
5.5%

5.4%    Path 2: \$11,000 \$11,000 \$11,000

At sufficiently high interest rates, the bond will never
prepay. Therefore, we can treat this bond like a non-
contingent bond
Any bond with equal monthly payments has a Macaulay
duration equal to the median payment date

\$11,000   \$11,000     \$11,000
P(Y=6%) =           +           +                = \$29,402
(1.06)    (1.06) 2    (1.06) 3

\$10,377     \$9,790          \$9,235

\$10,377             \$9,790             \$9,235
-1 +              -2 +               -3 = 2
\$29,402             \$29,402            \$29,402

-2
Macaulay Duration = -2                  Modified Duration =        = -1.89
1.06
Cash Flows

5.8% Path 1: \$11,000 \$11,000 \$11,000

5.4%
Path 2: \$11,000 \$11,000 \$11,000
5.0%                5.0%
Path 3: \$11,000 \$20,500
4.5%

4.4% Path 4: \$11,000 \$20,500

However, the value of this bond will be very
sensitive at interest rates near 4.5% (the
prepayment “trigger”)
Effective Duration
   To compute an effective duration, calculate the price
of a bond for a 50 basis point increase as well as a 50
basis point decrease (around an initial value)
   Note: for a given path, all interest rates rise by 50
basis points!

 P50  P50 
ED                *100
      P      
P(5.5%) = \$29,682

6.3%

5.9%

5.5%                5.5%

5.0%

4.9%

P(-50) = \$29,502                              P (+50) = \$29,408

5.8%
6.8%
5.4%
6.4%
5.0%                 5.0%
6.0%                6.0%
4.5%
5.5%
4.4%
5.4%
Effective Duration
 \$29,408  \$29,502 
ED(5.5%)                      *100  .31
      \$29,682      

Note, that this is significantly
lower than this bonds modified
duration of -1.85
For non-contingent cash flows, modified
Price         duration and effective duration yield similar
results

Effective
Duration
Pricing
Function
Yield
6%
Modified
Duration
For contingent cash flows, the curvature of
Price              the pricing function changes near the
“trigger point”. Modified duration will be very
different from effective duration in this area!!

Effective
Duration

Modified
Duration
Pricing
Function
Yield
5.0% 5.5% 6%

```
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