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
Covi, m
A stock’s Beta measures
it’s movements relative to
i
Varm
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!
P50 P50
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%
