Valuing Cash Flows II Contingent Payments

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

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” 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 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

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 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 $ % 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 TIPS In 1996, the US Government introduced reasury nflation rotected ecurities. These are bonds with state contingent payouts. Each future payment is indexed to the inflation rate. 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 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. STRIPS P(TIPS) - P(STRIP) = $ $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 P (Inflation = 8%) = $90.91 TIP 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 Date Non-Indexed Treasury TIPS 1/09 $ $ (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 GDP Time Interest Rates % Change Default Rates RecessionExpansion

StateTreasuryCorporate Interest Rate Expansion$100$986% Recession$100$883% 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

StateTreasuryCorporate Interest Rate Expansion$100$986% Recession $883% 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 Date TreasuryCorporate 1/08 $ $ (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

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

StateHigh Beta Stock Low Beta Stock Interest Rate Expansion$120$806% Recession$80$1203% High Beta stocks pay out larger amounts during good times Expansion Payment = $120 i = 6% Recession Payment = $80 i = 3% P = $120 (1.06) = $ 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

State High Beta Stock Low Beta Stock Interest Rate Expansion$120$806% Recession$80$1203% Expansion Payment = $80 i = 6% Recession Payment = $120 i = 3% P = $80 (1.06) = $75.47 P = $120 (1.03) = $ E(P) = (.5)($75.47) + (.5)($116.50) = $96.01 YTM = $100 - $96.01 $96.01 = 4.41% Std. Dev = Low Beta stocks pay out larger amounts during bad times

Some assets have maturity dates that can vary based on interest rate movements… Now5yr10yrs20yrs15yrs25yrs 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/ABS In the early eighties, many types of cash flows were “securitized” into bonds In the early eighties, many types of cash flows were “securitized” into bonds Home/Commercial Mortgages Home/Commercial Mortgages Car Loans Car Loans Student Loans Student Loans Credit Card Debt 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…. United Kingdom

2002 Top 10 Mortgage Markets $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 Year MonthPayment Principal AppliedInterest Remaining Balance 11$1,199.10$199.10$1,000.00$199, $1,199.10$200.10$999.00$199, $1,199.10$201.10$998.00$199, $1,199.10$202.10$997.00$199, $1,199.10$203.11$995.99$198, $1,199.10$204.13$994.97$198, $1,199.10$205.15$993.95$198, $1,199.10$206.17$992.93$198, $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 C D E F

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

Each Trance becomes the basis for a mortgage backed security A B C D E F Why not just divide up the payments equally? (i.e. why have different tranches?) A B F E D C

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 Year MonthPayment Principal AppliedInterest Remaining Balance 11$1,199.10$199.10$1,000.00$199, $1,199.10$200.10$999.00$199, $1,199.10$201.10$998.00$199, $1,199.10$202.10$997.00$199, $1,199.10$203.11$995.99$198, $1,199.10$204.13$994.97$198, $1,199.10$205.15$993.95$198, $1,199.10$206.17$992.93$198, $1,199.10$207.21$991.90$198, For example, in the 9 th month, you could take out a new loan and pay off the $198,171 outstanding on your original mortgage.

0360 month 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 30 th 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% Refinanced in the 4 th month Refinanced in the 19 th 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%

5.4% 5.0% 4.5% Path 1: $11,000 $11,000 $11,000 Path 2: $11,000 $11,000 $11, % 5.0% 5.8% Path 3: $11,000 $20,500 Path 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 Risk When the payments of an asset are variable, how do we asses interest rate risk? When the payments of an asset are variable, how do we asses interest rate risk?

6.4% 6.0% 5.5% Path 1: $11,000 $11,000 $11,000 Path 2: $11,000 $11,000 $11, % 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 $11,000 Path 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 =++ (1.06) 23 P(Y=6%)= $29,402 $10,377$9,235$9,790 $10,377 $29,402 + Macaulay Duration = -2 $9,790 $29, $9,235 $29,402 = 2-3 Modified Duration = = -1.89

5.4% 5.0% 4.5% Path 1: $11,000 $11,000 $11,000 Path 2: $11,000 $11,000 $11, % 5.0% 5.8% Path 3: $11,000 $20,500 Path 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 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) 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! Note: for a given path, all interest rates rise by 50 basis points!

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

Price Yield 6% Pricing Function Modified Duration Effective Duration For non-contingent cash flows, modified duration and effective duration yield similar results

Price Yield 6%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%