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# Valuing Cash Flows

Valuing Cash Flows. Non-Contingent Payments. Non-Contingent Payouts. 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.

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## Valuing Cash Flows

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1. Valuing Cash Flows Non-Contingent Payments

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

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

4. Consider a 3 year Treasury note with a 6% annual coupon and a \$1,000 face value. \$30 \$30 \$30 \$30 \$30 \$1,030 Now 6mos 1yrs 1.5 yrs 2yrs 2.5yrs 3yrs F(0,1) F(1,1) F(2,1) F(3,1) F(4,1) F(5,1) F(0,1) = 2.25% You have a statistical model that generates the following set of (annualized) forward rates F(1,1) = 2.75% F(2,1) = 2.8% F(3,1) = 3% F(4,1) = 3.1% F(5,1) = 4.1%

5. \$30 \$30 \$30 \$30 \$30 \$1,030 Now 6mos 1yrs 1.5 yrs 2yrs 2.5yrs 3yrs 2.25% 2.75% 2.8% 3% 3.1% 4.1% Given an expected path for (annualized) forward rates, we can calculate the present value of future payments. + … \$30 \$30 \$30 P = + + (1.01125) (1.01125)(1.01375) (1.01125)(1.01375)(1.014) + … \$1,030 = \$1,084.90 + (1.01125)………….(1.0205)

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

7. Alternatively, we can use current spot rates from the yield curve \$30 \$30 \$30 \$30 \$30 \$1,030 Now 6mos 1yrs 1.5 yrs 2yrs 2.5yrs 3yrs

8. The yield curve produces the same bond price…..why? \$30 \$30 \$30 \$30 \$30 \$1,030 Now 6mos 1yrs 1.5 yrs 2yrs 2.5yrs 3yrs \$30 \$30 \$30 \$30 \$30 \$1,030 P = + + + + + 2 3 4 5 6 (1.0125) (1.0125) (1.0135) (1.0135) (1.015) (1.015) S(1) S(2) S(3) 2 2 2 P = \$1,084.90

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

10. Alternatively, given the current price, what is the implied (constant) interest rate. \$30 \$30 \$30 \$30 \$30 \$1,030 Now 6mos 1yrs 1.5 yrs 2yrs 2.5yrs 3yrs \$30 \$30 \$30 \$30 \$30 \$1,030 P + + + + + = 2 3 4 5 6 (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) = 1.015 (1.5%) P = \$1,084.90 Given the current ,market price of \$1,084.90, this Treasury Note has an annualized Yield to Maturity of 3%

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

12. Yield to maturity measures the total performance of a bond from purchase to expiration. Consider \$1,000, 2 year STRIP selling for \$942 .5 \$1,000 \$1,000 1.03 (3%) \$942 = (1+Y) = = \$942 2 (1+Y) For a discount (one payment) bond, the YTM is equal to the expected spot rate For coupon bonds, YTM is cash flow specific

13. 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 \$50 \$50 \$50 \$50 \$50 P + + + + = = \$1,000 2 3 4 5 (1.05) (1.05) (1.05) (1.05) (1.05) This bond sells for Par Value and YTM = Coupon Rate

14. 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 \$50 \$50 \$50 \$50 \$50 P + + + + = = \$958 2 3 4 5 (1.06) (1.06) (1.06) (1.06) (1.06) This bond sells at a discount and YTM > Coupon Rate

15. Price A 1% rise in yield is associated with a \$42 (4.2%) drop in price \$1,000 \$42 \$958 Yield 5% 6%

16. 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 \$50 \$50 \$50 \$50 \$50 P + + + + = = \$1045 2 3 4 5 (1.04) (1.04) (1.04) (1.04) (1.04) This bond sells at a premium and YTM < Coupon Rate

17. Price A 1% drop in yield is associated with a \$45 (4.5%) rise in price \$1,045 \$45 \$1,000 \$42 \$958 Yield 4% 5% 6%

18. A bond’s pricing function shows all the combinations of yield/price Price • The bond pricing is non-linear • The pricing function is unique to a particular stream of cash flows \$1,045 \$45 \$1,000 \$42 \$958 Pricing Function Yield 4% 5% 6%

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

20. For the 5 year, 5% Treasury, we had the following: Yield 5% Term \$50 \$50 \$50 \$50 \$50 P(Y=5%) = + + + + = \$1,000 2 3 4 5 (1.05) (1.05) (1.05) (1.05) (1.05) This bond sells for Par Value and YTM = Coupon Rate

21. Price \$1,000 Pricing Function Yield 5%

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

23. 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 DV01 is the change in a bond’s price per basis point shift in yield. This bond’s DV01 is \$.43

24. 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 Yield 4% 5% 6% Dollar Duration

25. 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%

26. 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

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

28. 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 \$50 \$50 \$50 \$50 \$50 P(Y=5%) = + + + + = \$1,000 (1.05) 2 3 4 5 (1.05) (1.05) (1.05) (1.05) \$47.62 \$45.35 \$43.19 \$41.14 \$822.70 \$47.62 \$45.35 \$43.19 \$41.14 \$822.70 1 + 2 + 3 + 4 + 5 \$1,000 \$1,000 \$1,000 \$1,000 \$1,000 Macaulay Duration = 4.55

29. Macaulay Duration = 4.55 Macaulay Duration Modified Duration = (1+Y) 4.55 Modified Duration = = 4.3 1.05 Dollar Duration = Modified Duration (Price) Dollar Duration = 4.3(\$1,000) = \$4,300

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

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

32. Key Duration • 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.

33. 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%

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

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

36. +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

37. Suppose that we simply calculate the slope between the two points on the pricing function Price \$1,045 - \$958 Slope = = \$43.50 4% - 6% \$1,045 or \$1,045 - \$958 *100 \$1,000 = 4.35 Slope = \$958 4% - 6% Yield 4% 6%

38. 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 Yield 4% 6% Dollar Duration

39. 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

40. 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

41. Remember, the 5 year Treasury has a MD 0f 4.3 \$1,000, 5 Year Treasury (6% coupon) Interest Rate Mean = \$1,000 Std. Dev. = \$86 Mean = 6% Std. Dev. = 2% Profit/Loss Mean = \$0 Std. Dev. = \$86

42. 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 1 Std Dev = 65% 2 Std Dev = 95% 3 Std Dev = 99% One Standard Deviation Around the mean encompasses 65% of the distribution

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

44. 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

45. Example: Orange County • In December 1994, Orange County, CA stunned the markets by declaring bankruptcy after suffering a \$1.6B loss. • The loss was a result of the investment activities of Bob Citron – the county Treasurer – who was entrusted with the management of a \$7.5B portfolio

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

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

48. Example: Orange County • Ordinarily, the duration on a portfolio of 5 year notes would be around 4-5. However, this portfolio was heavily leveraged (\$7.5B as collateral for a \$20.5B loan). This dramatically raises the VAR

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

50. Risk vs. Return • 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

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