Valuing Cash Flows IIValuing Cash Flows II

Contingent PaymentsContingent Payments

Valuing Cash FlowsValuing Cash Flows

So far, we have been pricing securities with cash So far, we have been pricing securities with cash flows that are predefined and fixed. flows that are predefined and fixed. Given the stream of cash flows and an expected path Given the stream of cash flows and an expected path

for interest rates, the price of the instrument is simply for interest rates, the price of the instrument is simply the discounted valuethe discounted value

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Valuing Cash FlowsValuing Cash Flows

Suppose that the cash flows are still predefined, Suppose that the cash flows are still predefined, but not fixed. Instead, each cash flow is but not fixed. Instead, each cash flow is dependant on some future “state of the world”dependant on some future “state of the world” Given the stream of (state dependant) cash flows and Given the stream of (state dependant) cash flows and

an expected path for (state dependant) interest rates, an expected path for (state dependant) interest rates, the price of the instrument is still the discounted valuethe price of the instrument is still the discounted value

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ExampleExample

Assume that the required real (inflation adjusted) rate of return is 2%. Each period there are two possible states of the world

State #1

Inflation = 0% r = 2% i = r + Inflation = 2%

State #2

Inflation = 8% r = 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

Inflation = 0% i = 2% + 0% = 2%

State #2

Inflation = 8% i = 2% + 8% = 10%

P = $100

(1.02)= $98.04

P = $100

(1.10)= $90.91

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

YTM = $100 - $94.48

$94.48= 5.84%

Shouldn’t this equal 6% (the expected nominal return)?*100

Price

Yield

$90.91

5.84%

10%2%

$98.04

$94.48

Pricing Function

Remember, the bond pricing is non-linear! Therefore,

= 6%E(Y(P)) = = E(Y)

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

Pricing TIPSPricing TIPS

In 1996, the US Government introduced In 1996, the US Government introduced Treasury reasury Inflation nflation Protected rotected Securities. These are bonds with ecurities. These are bonds with state contingent payouts. Each future payment is state contingent payouts. Each future payment is indexed to the inflation rate. indexed to the inflation rate.

State #1

Inflation = 0% i = r + Inflation = 2% CF = $100

State #2

Inflation = 8% i = r + Inflation = 10% CF = $108

Now, consider a 1 year STRIP with a face value of $100. How should this security be priced?

State #1

Inflation = 4% i = 2% + 0% = 2%

State #2

Inflation = 8% i = 2% + 8% = 10%

P = $100

(1.02)= $98.12

P = $108

(1.10)= $98.12

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

YTM = $100 - $98.12

$98.12= 2%

TIPS vs. STRIPSTIPS vs. STRIPS

P(TIPS) - P(STRIP) = $98.12 - $94.48 = $3.64P(TIPS) - P(STRIP) = $98.12 - $94.48 = $3.64Y(STRIP) – Y(TIPS) = 5.84% - 2% = 3.84%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 P (Inflation = 8%) = $90.91TIP

P(Inflation = 0%) = $98.12 P(Inflation = 8%) = $98.12

Std Dev = 5

Shouldn’t this risk be worth something?

Std Dev = 0

TIP Price = STRIP Price + Inflation Premium + Risk Premium

Consider the following Prices (for a 3.875% annual coupon) on Treasuries vs. comparable TIPS

Maturity DateMaturity Date Non-Indexed Non-Indexed TreasuryTreasury

TIPSTIPS

1/091/09 $101.56 $101.56 (3.47%)

$111.75 $111.75 (.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

GDP

Time

Interest Rates

% Change

Default Rates

RecessionExpansion

StateState TreasuryTreasury Corporate Interest RateInterest Rate

ExpansionExpansion $100$100 $98 6%6%

RecessionRecession $100$100 $88 3%3%

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

Expansion

Payment = $98 i = 6%

Recession

Payment = $88 i = 3%

P = $98

(1.06)= $92.45

P = $88

(1.03)= $85.44

E(P) = (.5)($92.45) + (.5)($85.44) = $88.95 YTM = $100 - $88.95

$88.95= 12.42%

Std. Dev = 4.95

StateState Treasury CorporateCorporate Interest RateInterest Rate

ExpansionExpansion $100 $98 $98 6%6%

RecessionRecession $100 $88$88 3%3%

A Treasury Bill Pays out $100 Regardless of the State

Expansion

Payment = $100 i = 6%

Recession

Payment = $100 i = 3%

P = $100

(1.06)= $94.34

P = $100

(1.03)= $97.08

E(P) = (.5)($94.34) + (.5)($97.08) = $95.71 YTM = $100 - $95.71

$95.71= 4.48%

Std. Dev = 1.93

Consider the following Prices (for a 3.625% annual coupon) on Treasuries vs. Corporates

Maturity DateMaturity Date TreasuryTreasury CorporateCorporate

1/081/08 $108.11 $108.11 (3.26%)

$99.24 $99.24 (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

EquitiesEquities

We can think of stocks as simply bonds with state We can think of stocks as simply bonds with state contingent payments: contingent payments:

mVar

miCov

RRRR

i

fmifi

,

Excess Returns to Asset i Excess Returns

to the Market

A stock’s Beta measures it’s movements relative to the market

High Beta Stocks move with the market while low beta stocks move against the market

High Beta Stocks

Time

Interest Rates

% Change

Low Beta Stocks

RecessionExpansion

StateState High Beta Stock

Low Beta Low Beta StockStock

Interest RateInterest Rate

ExpansionExpansion $120 $80 $80 6%6%

RecessionRecession $80 $120$120 3%3%

High Beta stocks pay out larger amounts during good times

Expansion

Payment = $120 i = 6%

Recession

Payment = $80 i = 3%

P = $120

(1.06)= $113.20

P = $80

(1.03)= $77.70

E(P) = (.5)($113.20) + (.5)($77.70) = $95.43 YTM = $100 - $95.43

$95.43= 4.8%

Std. Dev = 25.10

StateState High Beta High Beta StockStock

Low Beta Stock

Interest RateInterest Rate

ExpansionExpansion $120$120 $80 6%6%

RecessionRecession $80$80 $120 3%3%

Expansion

Payment = $80 i = 6%

Recession

Payment = $120 i = 3%

P = $80

(1.06)= $75.47

P = $120

(1.03)= $116.50

E(P) = (.5)($75.47) + (.5)($116.50) = $96.01 YTM = $100 - $96.01

$96.01= 4.41%

Std. Dev = 28.63

Low Beta stocks pay out larger amounts during bad times

Some assets have maturity dates that can vary based on interest rate movements…

Now 5yr 10yrs 20yrs15yrs 25yrs

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

Bond Issued

Face Value can be repaid anytime in this period

If interest rates drop low enough in the “call period”, the firm will pay the bond off early

MBS/ABSMBS/ABS In the early eighties, many types of cash In the early eighties, many types of cash

flows were “securitized” into bondsflows were “securitized” into bondsHome/Commercial MortgagesHome/Commercial MortgagesCar LoansCar LoansStudent LoansStudent LoansCredit Card DebtCredit 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 As homeownership rates increases worldwide, the mortgage market has worldwide, the mortgage market has grown….grown….

SpainAustralia

United StatesCanada

JapanFrance

NetherlandsDenmark

Germany

0% 20% 40% 60% 80% 100%

United Kingdom

Canada3%

Germany8%

United Kingdom

8%

Australia2%

United States58%

Japan13%

France2%

Spain2%

Netherlands3%

Denmark1%

2002 Top 10 Mortgage 2002 Top 10 Mortgage MarketsMarkets $11.3T worth of

mortgages outstanding worldwide in 2002

Funding Mortgages

Home buyer goes to a mortgage provider for a loan

The bank can either keep the loan on its books or replenish its funds by selling off the 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

Home buyer makes monthly mortgage payments

The bank collects the payment and passes it along to the MBS creator (they collect a fee for this service)

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 through the issuance of Agency bonds – these are implicitly backed by the US government

Fannie Mae uses these funds to purchase mortgages

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

YearMonth Payment

PrincipalApplied Interest

RemainingBalance

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

Fannie Mae Purchases your loan plus 19 other identical loans

30 Year, 6% APR (Fixed) = $1200/mo $24,000/Mo

Available Funds

$24,000/Mo

Available Funds

These funds are then divided up into Tranches (Claims to different parts of the available funds)

A

B

CD

EF

Sequential-Pay ExampleSequential-Pay Example

Suppose the collateral is a 30-year, $100M, Suppose the collateral is a 30-year, $100M, 9% coupon mortgage portfolio9% coupon mortgage portfolio

Each Trance becomes the basis for a mortgage backed security

A

B

CD

EF

Why not just divide up the payments equally? (i.e. why have different tranches?)

A

B

F

E

D

C

Prepayment RiskSuppose that shortly after you buy your house, interest rates drop dramatically. You have the ability to refinance you mortgage at a lower interest rate

YearMonth Payment

PrincipalApplied Interest

RemainingBalance

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.

0 360month

30

6.0

3.0

9.0

CPR (%)

100% PSA

50% PSA

150% PSA

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.

Rising interest rates

Falling interest rates

As loans are refinances, the available pool shrinks

30 Year, 6% APR (Fixed) = $1200/mo

$18,000/Mo

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%

3

3.5

4

4.5

5

5.5

6

6.5

7

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Refinanced in the 4th month

Refinanced in the 19th 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

YearYear Payment (BOY)Payment (BOY) Principal Principal Outstanding Outstanding (Year End)(Year End)

11 $11,000$11,000 $20,500$20,500

22 $11,000$11,000 $10,525$10,525

33 $11,000$11,000 $0$0

Assume that households refinance whenever the interest rate hits 4.5%

5.4%

5.0%

4.5%

Path 1: $11,000 $11,000 $11,000Path 1: $11,000 $11,000 $11,000

Path 2: $11,000 $11,000 $11,000Path 2: $11,000 $11,000 $11,000

4.4%

5.0%

5.8%

Path 3: $11,000 $20,500Path 3: $11,000 $20,500

Path 4: $11,000 $20,500Path 4: $11,000 $20,500

Cash Flows

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

P = $29,809

Path #2

P = $29,881

P = $11,000

(1.05)

($29,809 + $29,881 + $29,159 + $29,159)

Path #3

P = $29,159

Path #4

P = $29,159

$11,000

(1.05)(1.054)(1.058)

$11,000

(1.05)(1.054)+ +

P = $11,000

(1.05)

$11,000

(1.05)(1.054)(1.05)

$11,000

(1.05)(1.054)+ +

P = $11,000

(1.05)

$20,500

(1.05)(1.045)+

P = $11,000

(1.05)

$20,500

(1.05)(1.045)+

E(P) =4

= $29,502

Interest Rate RiskInterest Rate Risk

When the payments of an asset are When the payments of an asset are variable, how do we asses interest rate variable, how do we asses interest rate risk?risk?

6.4%

6.0%

5.5%

Path 1: $11,000 $11,000 $11,000Path 1: $11,000 $11,000 $11,000

Path 2: $11,000 $11,000 $11,000Path 2: $11,000 $11,000 $11,000

5.4%

6.0%

6.8%

Cash Flows

At sufficiently high interest rates, the bond will never prepay. Therefore, we can treat this bond like a non-contingent bond

Path 3: $11,000 $11,000 $11,000Path 3: $11,000 $11,000 $11,000

Path 2: $11,000 $11,000 $11,000Path 2: $11,000 $11,000 $11,000

Any bond with equal monthly payments has a Macaulay duration equal to the median payment date

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

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

$10,377

$29,402+-1

Macaulay Duration = -2

$9,790

$29,402+-2

$9,235

$29,402= 2-3

Modified Duration =-2

1.06 = -1.89

5.4%

5.0%

4.5%

Path 1: $11,000 $11,000 $11,000Path 1: $11,000 $11,000 $11,000

Path 2: $11,000 $11,000 $11,000Path 2: $11,000 $11,000 $11,000

4.4%

5.0%

5.8%

Path 3: $11,000 $20,500Path 3: $11,000 $20,500

Path 4: $11,000 $20,500Path 4: $11,000 $20,500

Cash Flows

However, the value of this bond will be very sensitive at interest rates near 4.5% (the prepayment “trigger”)

Effective DurationEffective Duration

To compute an effective duration, calculate the price To compute an effective duration, calculate the price of a bond for a 50 basis point increase as well as a 50 of a bond for a 50 basis point increase as well as a 50 basis point decrease (around an initial value)basis point decrease (around an initial value)

Note: for a given path, all interest rates rise by 50 Note: for a given path, all interest rates rise by 50 basis points!basis points!

100*5050

P

PPED

5.4%

5.0%

4.5%

4.4%

5.0%

5.8%

6.4%

6.0%

5.5%

5.4%

6.0%

6.8%

5.9%

5.5%

5.0%

4.9%

5.5%

6.3%

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

P(5.5%) = $29,682

Effective DurationEffective Duration

31.100*682,29$

502,29$408,29$%)5.5(

ED

Note, that this is significantly lower than this bonds modified duration of -1.85

Price

Yield6%

Pricing Function

Modified Duration

Effective Duration

For non-contingent cash flows, modified duration and effective duration yield similar results

Price

Yield6%5.5%

Pricing Function

Modified Duration

Effective Duration

For contingent cash flows, the curvature of the pricing function changes near the “trigger point”. Modified duration will be very different from effective duration in this area!!

5.0%