Valuing Cash Flows Non-Contingent Payments Non-Contingent Payouts

Valuing Cash Flows Non-Contingent Payments

Non-Contingent Payouts l Given an asset with fixed payments (i. e. independent of the state of the world), the asset’s price should equal the present value of the cash flows.

Treasury Notes US Treasuries notes have maturities between 2 and ten years. l Treasury notes make biannual interest payments and then a repayment of the face value upon maturity l US Treasury notes can be purchased in increments of $1, 000 of face value. l

Consider a 3 year Treasury note with a 6% annual coupon and a $1, 000 face value. $30 6 mos Now F(0, 1) $30 1 yrs F(1, 1) $30 1. 5 yrs F(2, 1) F(3, 1) F(1, 1) = 2. 75% F(2, 1) = 2. 8% F(3, 1) = 3% F(4, 1) = 3. 1% F(5, 1) = 4. 1% F(4, 1) $1, 030 2. 5 yrs 2 yrs F(0, 1) = 2. 25% You have a statistical model that generates the following set of (annualized) forward rates $30 3 yrs F(5, 1)

$30 6 mos Now 2. 25% $30 1 yrs 2. 75% $30 1. 5 yrs 2. 8% $30 2 yrs 3% $1, 030 2. 5 yrs 3. 1% 3 yrs 4. 1% Given an expected path for (annualized) forward rates, we can calculate the present value of future payments. P= + $30 $30 + + (1. 01125)(1. 01375)(1. 014) … + $1, 030 (1. 01125)…………. (1. 0205) = $1, 084. 90 + …

Forward Rate Pricing Current Asset Price Cash Flow at time t Interest rate between periods t-1 and t

Alternatively, we can use current spot rates from the yield curve $30 Now $30 6 mos 1 yrs $30 1. 5 yrs $30 2 yrs $30 $1, 030 2. 5 yrs 3 yrs

The yield curve produces the same bond price…. . why? $30 6 mos Now P = $30 1 yrs 2 = $30 1. 5 yrs 2 yrs $30 $1, 030 2. 5 yrs 3 yrs $30 $30 $30 $1, 030 + + + (1. 0125) 2 (1. 0135) 3 (1. 0135) 4 (1. 015) 5 (1. 015) 6 S(1) P $30 $1, 084. 90 S(2) S(3) 2 2

Spot Rate Pricing Current Asset Price Current spot rate for a maturity of t periods Cash flow at period t

Alternatively, given the current price, what is the implied (constant) interest rate. $30 6 mos Now P P = = $30 1 yrs $30 (1+i) + $1, 084. 90 $30 + 2 (1+i) $30 1. 5 yrs $30 + 3 (1+i) $30 2 yrs $30 + 4 (1+i) $30 $1, 030 2. 5 yrs 3 yrs $30 $1, 030 + (1+i)5 (1+i) 6 (1+i) = 1. 015 (1. 5%) Given the current , market price of $1, 084. 90, this Treasury Note has an annualized Yield to Maturity of 3%

Yield to Maturity Cash flow at time t Yield to Maturity Current Market Price

l Yield to maturity measures the total performance of a bond from purchase to expiration. Consider $1, 000, 2 year STRIP selling for $942 . 5 $942 = $1, 0002 (1+Y) $1, 000 (1+Y) = $942 For a discount (one payment) bond, the YTM is equal to the expected spot rate For coupon bonds, YTM is cash flow specific = 1. 03 (3%)

Consider a 5 year Treasury Note with a 5% annual coupon rate (paid annually) and a face value of $1, 000 The one year interest rate is currently 5% and is expected to stay constant. Further, there is no liquidity premium Yield 5% Term P $50 $50 $50 + + = = $1, 000 (1. 05) 2 (1. 05) 3 (1. 05) 4 (1. 05) 5 This bond sells for Par Value and YTM = Coupon Rate

Consider a 5 year Treasury Note with a 5% annual coupon rate (paid annually) and a face value of $1, 000 Now, suppose that the current 1 year rate rises to 6% and is expected to remain there Yield 6% 5% Term P $50 $50 $50 + + = = $958 (1. 06) 2 (1. 06) 3 (1. 06) 4 (1. 06) 5 This bond sells at a discount and YTM > Coupon Rate

Price $1, 000 $958 A 1% rise in yield is associated with a $42 (4. 2%) drop in price $42 5% 6% Yield

Consider a 5 year Treasury Note with a 5% annual coupon rate (paid annually) and a face value of $1, 000 Now, suppose that the current 1 year rate falls to 4% and is expected to remain there Yield 5% 4% Term P $50 $50 $50 + + = = $1045 (1. 04) 2 (1. 04) 3 (1. 04) 4 (1. 04) 5 This bond sells at a premium and YTM < Coupon Rate

Price A 1% drop in yield is associated with a $45 (4. 5%) rise in price $1, 045 $1, 000 $958 $42 4% 5% 6% Yield

A bond’s pricing function shows all the combinations of yield/price Price 1) The bond pricing is non-linear 2) The pricing function is unique to a particular stream of cash flows $1, 045 $1, 000 $958 $42 Pricing Function 4% 5% 6% Yield

Duration l Recall that in general the price of a fixed income asset is given by the following formula l Note that we are denoting price as a function of yield: P(Y).

For the 5 year, 5% Treasury, we had the following: Yield 5% Term P(Y=5%) = $50 $50 $50 + + = $1, 000 2 3 4 5 (1. 05) This bond sells for Par Value and YTM = Coupon Rate

Price $1, 000 Pricing Function 5% Yield

Suppose we take the derivative of the pricing function with respect to yield For the 5 year, 5% Treasury, we have

Now, evaluate that derivative at a particular point (say, Y = 5%, P = $1, 000) For every 100 basis point change in the interest rate, the value of this bond changes by $43. 29 This is the dollar duration DV 01 is the change in a bond’s price per basis point shift in yield. This bond’s DV 01 is $. 43

Price Duration predicted a $43 price change for every 1% change in yield. This is different from the actual price Error = $2 $1, 045 $1, 000 Error = - $1 $958 Pricing Function 4% 5% 6% Yield Dollar Duration

Dollar duration depends on the face value of the bond (a $1000 bond has a DD of $43 while a $10, 000 bond has a DD of $430) modified duration represents the percentage change in a bonds price due to a 1% change in yield For the 5 year, 5% Treasury, we have Every 100 basis point shift in yield alters this bond’s price by 4. 3%

Macaulay's Duration Macaulay’ duration measures the percentage change in a bond’s price for every 1% change in (1+Y) (1. 05)(1. 01) = 1. 0605 For the 5 year, 5% Treasury, we have

For bonds with one payment, Macaulay duration is equal to the term Example: 5 year STRIP Dollar Duration Modified Duration Macaulay Duration

Think of a coupon bond as a portfolio of STRIPS. Each payment has a Macaulay duration equal to its date. The bond’s Macaulay duration is a weighted average of the individual durations Back to the 5 year Treasury P(Y=5%) = $50 $50 $50 + + = $1, 000 2 3 4 5 (1. 05) $47. 62 $1, 000 1+ $45. 35 $1, 000 $45. 35 2+ Macaulay Duration = 4. 55 $43. 19 $1, 000 3+ $41. 14 $822. 70 4 + 5 $1, 000

Macaulay Duration = 4. 55 Macaulay Duration Modified Duration = (1+Y) 4. 55 = 4. 3 Modified Duration = 1. 05 Dollar Duration = Modified Duration (Price) Dollar Duration = 4. 3($1, 000) = $4, 300

Duration measures interest rate risk (the risk involved with a parallel shift in the yield curve) This almost never happens.

Yield curve risk involves changes in an asset’s price due to a change in the shape of the yield curve

Key Duration l In order to get a better idea of a Bond’s (or portfolio’s) exposure to yield curve risk, a key rate duration is calculated. This measures the sensitivity of a bond/portfolio to a particular spot rate along the yield curve holding all other spot rates constant.

Returning to the 5 Year Treasury A Key duration for the three year spot rate is the partial derivative with respect to S(3) Evaluated at S(3) = 5%

Key Durations X 100 Note that the individual key durations sum to $4329 – the bond’s overall duration

Yield Curve Shifts +1% 0% - 2% +1% - 4%

+1% 0% - 2% - 4% +1% $. 4535 1 + $. 8638 1 + $. 12341 0 + $. 15671 (-2) + $39. 81 (-4) = $161 This yield curve shift would raise a five year Treasury price by $161

Suppose that we simply calculate the slope between the two points on the pricing function Price Slope = $1, 045 - $958 4% - 6% = $43. 50 or $1, 045 Slope = $958 $1, 045 - $958 $1, 000 *100 = 4. 35 4% - 6% 4% 6% Yield

Effective duration measures interest rate sensitivity using the actual pricing function rather that the derivative. This is particularly important for pricing bonds with embedded options!! Price $1, 045 Effective Duration $958 Pricing Function 4% 6% Dollar Duration Yield

Value At Risk Suppose you are a portfolio manager. The current value of your portfolio is a known quantity. Tomorrow’s portfolio value us an unknown, but has a probability distribution with a known mean and variance Profit/Loss = Tomorrow’s Portfolio Value – Today’s portfolio value Known Distribution Known Constant

Probability Distributions 1 Std Dev = 65% 2 Std Dev = 95% 3 Std Dev = 99% One Standard Deviation Around the mean encompasses 65% of the distribution

Remember, the 5 year Treasury has a MD 0 f 4. 3 $1, 000, 5 Year Treasury (6% coupon) Interest Rate Mean = 6% Dev. = 2% Mean = $1, 000 Std. Dev. = $86 Std. Profit/Loss Mean = $0 Dev. = $86 Std.

The VAR(65) for a $1, 000, 5 Year Treasury (assuming the distribution of interest rates) would be $86. The VAR(95) would be $172 In other words, there is only a 5% chance of losing more that $172 One Standard Deviation Around the mean encompasses 65% of the distribution 1 Std Dev = 65% 2 Std Dev = 95% 3 Std Dev = 99%

A 30 year Treasury has a MD of 14 $1000, 30 Year Treasury (6% coupon) Interest Rate Mean = 6% Dev. = 2% Mean = $1, 000 Std. Dev. = $280 Std. Profit/Loss Mean = $0 Dev. = $280 Std.

The VAR(65) for a $1, 000, 30 Year Treasury (assuming the distribution of interest rates) would be $280. The VAR(95) would be $560 In other words, there is only a 5% chance of losing more that $560 One Standard Deviation Around the mean encompasses 65% of the distribution

Example: Orange County In December 1994, Orange County, CA stunned the markets by declaring bankruptcy after suffering a $1. 6 B loss. l The loss was a result of the investment activities of Bob Citron – the county Treasurer – who was entrusted with the management of a $7. 5 B portfolio l

Example: Orange County l Actually, up until 1994, Bob’s portfolio was doing very well.

Example: Orange County l Given a steep yield curve, the portfolio was betting on interest rates falling. A large share was invested in 5 year FNMA notes.

Example: Orange County l Ordinarily, the duration on a portfolio of 5 year notes would be around 4 -5. However, this portfolio was heavily leveraged ($7. 5 B as collateral for a $20. 5 B loan). This dramatically raises the VAR

Example: Orange County l In February 1994, the Fed began a series of six consecutive interest rate increases. The beginning of the end!

Risk vs. Return l As a portfolio manager, your job is to maximize your risk adjusted return Risk Adjusted Return = Nominal Return – “Risk Penalty” You can accomplish this by 1 of two methods: 1) Maximize the nominal return for a given level of risk 2) Minimize Risk for a given nominal return

Again, assume that the one year spot rate is currently 5% and is expected to stay constant. There is no liquidity premium, so the yield curve is flat. Yield 5% Term P $5 $5 + + = 2 3 4 (1. 05) … = $100 All 5% coupon bonds sell for Par Value and YTM = Coupon Rate = Spot Rate = 5%. Further, bond prices are constant throughout their lifetime.

Available Assets 1 Year Treasury Bill (5% coupon) l 3 Year Treasury Note (5% coupon) l 5 Year Treasury Note (5% coupon) l 10 Year Treasury Note (5% coupon) l 20 Year Treasury Bond (5% coupon) l STRIPS of all Maturities l How could you maximize your risk adjusted return on a $100, 000 Treasury portfolio?

Suppose you buy a 20 Year Treasury $100, 000 P(Y=5%) = 20 Year $5000/yr $105, 000 $5000 + + + 2 3 (1. 05) $4, 762 1+ $100, 000 $4, 535 2 + $100, 000 Macaulay Duration = 12. 6 … + $4, 319 $100, 000 $105, 000 (1. 05) 20 $39, 573 3+ …+ $82, 270 $100, 000 20

Alternatively, you could buy a 20 Year Treasury and a 5 year STRIPS $50, 000 20 Year $2500/yr 5 Year $63, 814 $52, 500 5 Year $63, 814 (Remember, STRIPS have a Macaulay duration equal to their Term) Portfolio Duration = $50, 000 12. 6 + $100, 000 $50, 000 $100, 000 5 = 8. 8

Alternatively, you could buy a 20 Year Treasury and a 5 year Treasury $50, 000 20 Year $50, 000 5 Year $2500/yr 5 Year $52, 500 (5 Year Treasuries have a Macaulay duration equal to 4. 3) Portfolio Duration = $50, 000 12. 6 + $100, 000 $50, 000 $100, 000 4. 3 = 8. 5

Even better, you could buy a 20 Year Treasury, and a 1 Year T-Bill $2500/yr $50, 000 20 Year $50, 000 1 Year $52, 500 … (1 Year Treasuries have a Macaulay duration equal to 1) Portfolio Duration = $50, 000 12. 6 + $100, 000 $50, 000 $100, 000 1 = 6. 3

Alternatively, you could buy a 20 Year Treasury, a 10 Year Treasury, 5 year Treasury, and a 3 Year Treasury $25, 000 D = 12. 6 20 Year $25, 000 D = 7. 7 10 Year $25, 000 D = 4. 3 5 Year $25, 000 D = 2. 7 3 Year $25, 000 $100, 000 $1250/yr 12. 6 + $25, 000 $100, 000 Portfolio Duration = 6. 08 7. 7 + $25, 000 $100, 000 4. 3 + $25, 000 $100, 000 2. 7

Obviously, with a flat yield curve, there is no advantage to buying longer term bonds. The optimal strategy is to buy 1 year T-Bills $100, 000 1 Year … $105, 000 Portfolio Duration = 1 However, the yield curve typically slopes up, which creates a risk/return tradeoff

Also, with an upward sloping yield curve, a bond’s price will change predictably over its lifetime

A Bond’s price will always approach its face value upon maturity, but will rise over its lifetime as the yield drops Pricing Date Coupon YTM Price ($) Issue 2005 2006 2007 2008 2009 2010 2011 3. 75% 3. 69 3. 48 3. 28 3. 04 2. 78 2. 55 Matures 100. 00 100. 96 101. 77 102. 20 102. 35 102. 11 101. 29 100. 00 3. 75% 3. 75 3. 75

Also, the change is a bond’s duration is also a non-linear function Length of Bond Initial Duration Percentage after 5 Years Change 30 Year 20 Year 15. 5 12. 6 7. 8 14. 2 10. 5 4. 4 -8% -17% -44% As a bond ages, its duration drops at an increasing rate