# Bond Price Volatility. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 71

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1 Bond Price Volatility c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 71

2 Well, Beethoven, what is this? Attributed to Prince Anton Esterházy c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 72

3 Price Volatility Volatility measures how bond prices respond to interest rate changes. It is key to the risk management of interest-rate-sensitive securities. Assume level-coupon bonds throughout. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 73

4 Price Volatility (concluded) What is the sensitivity of the percentage price change to changes in interest rates? Define price volatility by P y P. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 74

5 Price Volatility of Bonds The price volatility of a coupon bond is (C/y) n ( C/y 2) ( (1 + y) n+1 (1 + y) ) nf (C/y) ((1 + y) n+1 (1 + y)) + F (1 + y). F is the par value. C is the coupon payment per period. For bonds without embedded options, P y P > 0. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 75

6 Macaulay Duration The Macaulay duration (MD) is a weighted average of the times to an asset s cash flows. The weights are the cash flows PVs divided by the asset s price. Formally, MD 1 P n i=1 ic i (1 + y) i. The Macaulay duration, in periods, is equal to MD = (1 + y) P y 1 P. (7) c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 76

7 MD of Bonds The MD of a coupon bond is [ MD = 1 n ic P (1 + y) i + nf (1 + y) n i=1 ]. (8) It can be simplified to MD = c(1 + y) [ (1 + y)n 1 ] + ny(y c) cy [ (1 + y) n 1 ] + y 2, where c is the period coupon rate. The MD of a zero-coupon bond equals its term to maturity n. The MD of a coupon bond is less than its maturity. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 77

8 Finesse Equations (7) on p. 76 and (8) on p. 77 hold only if the coupon C, the par value F, and the maturity n are all independent of the yield y. That is, if the cash flow is independent of yields. To see this point, suppose the market yield declines. The MD will be lengthened. But for securities whose maturity actually decreases as a result, the MD (as originally defined) may actually decrease. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 78

9 How Not To Think about MD The MD has its origin in measuring the length of time a bond investment is outstanding. The MD should be seen mainly as measuring price volatility. Many, if not most, duration-related terminology cannot be comprehended otherwise. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 79

10 Conversion For the MD to be year-based, modify Eq. (8) on p. 77 to [ n ] 1 i C ( ) P k i=1 1 + y i + n F ( ) k 1 + y n, k k where y is the annual yield and k is the compounding frequency per annum. Equation (7) on p. 76 also becomes MD = ( 1 + y ) P k y 1 P. By definition, MD (in years) = MD (in periods) k. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 80

11 Modified Duration Modified duration is defined as modified duration P y By Taylor expansion, 1 P = MD (1 + y). (9) percent price change modified duration yield change. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 81

12 Example Consider a bond whose modified duration is with a yield of 10%. If the yield increases instantaneously from 10% to 10.1%, the approximate percentage price change will be = = 1.154%. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 82

13 Modified Duration of a Portfolio The modified duration of a portfolio equals ω i D i. D i is the modified duration of the ith asset. i ω i is the market value of that asset expressed as a percentage of the market value of the portfolio. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 83

14 Effective Duration Yield changes may alter the cash flow or the cash flow may be so complex that simple formulas are unavailable. We need a general numerical formula for volatility. The effective duration is defined as P P + P 0 (y + y ). P is the price if the yield is decreased by y. P + is the price if the yield is increased by y. P 0 is the initial price, y is the initial yield. y is small. See plot on p. 85. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 84

15 P + P 0 P - y - y y + c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 85

16 Effective Duration (concluded) One can compute the effective duration of just about any financial instrument. Duration of a security can be longer than its maturity or negative! Neither makes sense under the maturity interpretation. An alternative is to use P 0 P + P 0 y. More economical but less accurate. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 86

17 The Practices Duration is usually expressed in percentage terms call it D % for quick mental calculation. The percentage price change expressed in percentage terms is approximated by D % r when the yield increases instantaneously by r%. Price will drop by 20% if D % = 10 and r = 2 because 10 2 = 20. In fact, D % equals modified duration as originally defined (prove it!). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 87

18 Hedging Hedging offsets the price fluctuations of the position to be hedged by the hedging instrument in the opposite direction, leaving the total wealth unchanged. Define dollar duration as modified duration price (% of par) = P y. The approximate dollar price change per \$100 of par value is price change dollar duration yield change. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 88

19 Convexity is defined as Convexity convexity (in periods) 2 P y 2 1 P. The convexity of a coupon bond is positive (prove it!). For a bond with positive convexity, the price rises more for a rate decline than it falls for a rate increase of equal magnitude (see plot next page). Hence, between two bonds with the same duration, the one with a higher convexity is more valuable. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 89

20 Price Yield c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 90

21 Convexity (concluded) Convexity measured in periods and convexity measured in years are related by convexity (in years) = when there are k periods per annum. convexity (in periods) k 2 c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 91

22 Use of Convexity The approximation P /P duration yield change works for small yield changes. To improve upon it for larger yield changes, use P P P y 1 P y P y 2 1 P ( y)2 = duration y convexity ( y)2. Recall the figure on p. 90. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 92

23 The Practices Convexity is usually expressed in percentage terms call it C % for quick mental calculation. The percentage price change expressed in percentage terms is approximated by D % r + C % ( r) 2 /2 when the yield increases instantaneously by r%. Price will drop by 17% if D % = 10, C % = 1.5, and r = 2 because = 17. In fact, C % equals convexity divided by 100 (prove it!). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 93

24 Effective Convexity The effective convexity is defined as P + + P 2P 0 P 0 (0.5 (y + y )) 2, P is the price if the yield is decreased by y. P + is the price if the yield is increased by y. P 0 is the initial price, y is the initial yield. y is small. Effective convexity is most relevant when a bond s cash flow is interest rate sensitive. Numerically, choosing the right y is a delicate matter. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 94

25 Approximate d 2 f(x) 2 /dx 2 at x = 1, Where f(x) = x 2 The difference of ((1 + x) 2 + (1 x) 2 2)/( x) 2 and 2: Error dx c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 95

26 Term Structure of Interest Rates c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 96

27 Why is it that the interest of money is lower, when money is plentiful? Samuel Johnson ( ) If you have money, don t lend it at interest. Rather, give [it] to someone from whom you won t get it back. Thomas Gospel 95 c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 97

28 Term Structure of Interest Rates Concerned with how interest rates change with maturity. The set of yields to maturity for bonds forms the term structure. The bonds must be of equal quality. They differ solely in their terms to maturity. The term structure is fundamental to the valuation of fixed-income securities. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 98

29 Yield (%) Year c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 99

30 Term Structure of Interest Rates (concluded) Term structure often refers exclusively to the yields of zero-coupon bonds. A yield curve plots yields to maturity against maturity. A par yield curve is constructed from bonds trading near par. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 100

31 Four Typical Shapes A normal yield curve is upward sloping. An inverted yield curve is downward sloping. A flat yield curve is flat. A humped yield curve is upward sloping at first but then turns downward sloping. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 101

32 Spot Rates The i-period spot rate S(i) is the yield to maturity of an i-period zero-coupon bond. The PV of one dollar i periods from now is [ 1 + S(i) ] i. The one-period spot rate is called the short rate. Spot rate curve: Plot of spot rates against maturity. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 102

33 Problems with the PV Formula In the bond price formula, n i=1 C (1 + y) i + F (1 + y) n, every cash flow is discounted at the same yield y. Consider two riskless bonds with different yields to maturity because of their different cash flow streams: n 1 i=1 n 2 i=1 C (1 + y 1 ) i + F (1 + y 1 ) n, 1 C (1 + y 2 ) i + F (1 + y 2 ) n. 2 c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 103

34 Problems with the PV Formula (concluded) The yield-to-maturity methodology discounts their contemporaneous cash flows with different rates. But shouldn t they be discounted at the same rate? c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 104

35 Spot Rate Discount Methodology A cash flow C 1, C 2,..., C n is equivalent to a package of zero-coupon bonds with the ith bond paying C i dollars at time i. So a level-coupon bond has the price P = n i=1 C [ 1 + S(i) ] i + F [ 1 + S(n) ] n. (10) This pricing method incorporates information from the term structure. Discount each cash flow at the corresponding spot rate. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 105

36 Discount Factors In general, any riskless security having a cash flow C 1, C 2,..., C n should have a market price of P = n C i d(i). i=1 Above, d(i) [ 1 + S(i) ] i, i = 1, 2,..., n, are called discount factors. d(i) is the PV of one dollar i periods from now. The discount factors are often interpolated to form a continuous function called the discount function. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 106

37 Extracting Spot Rates from Yield Curve Start with the short rate S(1). Note that short-term Treasuries are zero-coupon bonds. Compute S(2) from the two-period coupon bond price P by solving P = C 1 + S(1) + C [ 1 + S(2) ] 2. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 107

38 Extracting Spot Rates from Yield Curve (concluded) Inductively, we are given the market price P of the n-period coupon bond and S(1), S(2),..., S(n 1). Then S(n) can be computed from Eq. (10) on p. 105, repeated below, P = n i=1 C [ 1 + S(i) ] i + F [ 1 + S(n) ] n. The running time is O(n) (see text). The procedure is called bootstrapping. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 108

39 Some Problems Treasuries of the same maturity might be selling at different yields (the multiple cash flow problem). Some maturities might be missing from the data points (the incompleteness problem). Treasuries might not be of the same quality. Interpolation and fitting techniques are needed in practice to create a smooth spot rate curve. Any economic justifications? c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 109

40 Yield Spread Consider a risky bond with the cash flow C 1, C 2,..., C n and selling for P. Were this bond riskless, it would fetch P = n t=1 C t [ 1 + S(t) ] t. Since riskiness must be compensated, P < P. Yield spread is the difference between the IRR of the risky bond and that of a riskless bond with comparable maturity. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 110

41 Static Spread The static spread is the amount s by which the spot rate curve has to shift in parallel to price the risky bond: P = n t=1 C t [ 1 + s + S(t) ] t. Unlike the yield spread, the static spread incorporates information from the term structure. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 111

42 Of Spot Rate Curve and Yield Curve y k : yield to maturity for the k-period coupon bond. S(k) y k S(k) y k S(k) y k normal). S(k) y k inverted). if y 1 < y 2 < (yield curve is normal). if y 1 > y 2 > (yield curve is inverted). if S(1) < S(2) < (spot rate curve is if S(1) > S(2) > (spot rate curve is If the yield curve is flat, the spot rate curve coincides with the yield curve. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 112

43 Shapes The spot rate curve often has the same shape as the yield curve. If the spot rate curve is inverted (normal, resp.), then the yield curve is inverted (normal, resp.). But this is only a trend not a mathematical truth. a a See a counterexample in the text. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 113

44 Forward Rates The yield curve contains information regarding future interest rates currently expected by the market. Invest \$1 for j periods to end up with [ 1 + S(j) ] j dollars at time j. The maturity strategy. Invest \$1 in bonds for i periods and at time i invest the proceeds in bonds for another j i periods where j > i. Will have [ 1 + S(i) ] i [ 1 + S(i, j) ] j i dollars at time j. S(i, j): (j i)-period spot rate i periods from now. The rollover strategy. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 114

45 When S(i, j) equals Forward Rates (concluded) f(i, j) [ (1 + S(j)) j (1 + S(i)) i ] 1/(j i) 1, (11) we will end up with [ 1 + S(j) ] j dollars again. By definition, f(0, j) = S(j). f(i, j) is called the (implied) forward rates. More precisely, the (j i)-period forward rate i periods from now. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 115

46 Time Line f(0, 1) f(1, 2) f(2, 3) f(3, 4) Time 0 S(1) S(2) S(3) S(4) c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 116

47 Forward Rates and Future Spot Rates We did not assume any a priori relation between f(i, j) and future spot rate S(i, j). This is the subject of the term structure theories. We merely looked for the future spot rate that, if realized, will equate two investment strategies. f(i, i + 1) are instantaneous forward rates or one-period forward rates. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 117

48 Spot Rates and Forward Rates When the spot rate curve is normal, the forward rate dominates the spot rates, f(i, j) > S(j) > > S(i). When the spot rate curve is inverted, the forward rate is dominated by the spot rates, f(i, j) < S(j) < < S(i). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 118

49 c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 119

50 Forward Rates Spot Rates Yield Curve The FV of \$1 at time n can be derived in two ways. Buy n-period zero-coupon bonds and receive [ 1 + S(n) ] n. Buy one-period zero-coupon bonds today and a series of such bonds at the forward rates as they mature. The FV is [ 1 + S(1) ][ 1 + f(1, 2) ] [ 1 + f(n 1, n) ]. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 120

51 Forward Rates Spot Rates Yield Curves (concluded) Since they are identical, S(n) = {[ 1 + S(1) ][ 1 + f(1, 2) ] [ 1 + f(n 1, n) ]} 1/n 1. (12) Hence, the forward rates, specifically the one-period forward rates, determine the spot rate curve. Other equivalencies can be derived similarly, such as f(t, T + 1) = d(t ) d(t + 1) 1. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 121

52 Locking in the Forward Rate f(n, m) Buy one n-period zero-coupon bond for 1/(1 + S(n)) n. Sell (1 + S(m)) m /(1 + S(n)) n bonds. m-period zero-coupon No net initial investment because the cash inflow equals the cash outflow 1/(1 + S(n)) n. At time n there will be a cash inflow of \$1. At time m there will be a cash outflow of (1 + S(m)) m /(1 + S(n)) n dollars. This implies the rate f(n, m) between times n and m. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 122

53 1 n m (1 + S(m)) m /(1 + S(n)) n c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 123

54 Forward Contracts We generated the cash flow of a financial instrument called forward contract. Agreed upon today, it enables one to borrow money at time n in the future and repay the loan at time m > n with an interest rate equal to the forward rate f(n, m). Can the spot rate curve be an arbitrary curve? a a Contributed by Mr. Dai, Tian-Shyr (R , D ) in c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 124

55 Spot and Forward Rates under Continuous Compounding The pricing formula: P = n Ce is(i) + F e ns(n). i=1 The market discount function: d(n) = e ns(n). The spot rate is an arithmetic average of forward rates, S(n) = f(0, 1) + f(1, 2) + + f(n 1, n). n c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 125

56 Spot and Forward Rates under Continuous Compounding (concluded) The formula for the forward rate: f(i, j) = The one-period forward rate: js(j) is(i). j i f(t ) f(j, j + 1) = ln d(j + 1). d(j) S lim f(t, T + T ) = S(T ) + T T 0 T. f(t ) > S(T ) if and only if S/ T > 0. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 126

57 Unbiased Expectations Theory Forward rate equals the average future spot rate, f(a, b) = E[ S(a, b) ]. (13) Does not imply that the forward rate is an accurate predictor for the future spot rate. Implies the maturity strategy and the rollover strategy produce the same result at the horizon on the average. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 127

58 Unbiased Expectations Theory and Spot Rate Curve Implies that a normal spot rate curve is due to the fact that the market expects the future spot rate to rise. f(j, j + 1) > S(j + 1) if and only if S(j + 1) > S(j) from Eq. (11) on p So E[ S(j, j + 1) ] > S(j + 1) > > S(1) if and only if S(j + 1) > > S(1). Conversely, the spot rate is expected to fall if and only if the spot rate curve is inverted. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 128

59 More Implications The theory has been rejected by most empirical studies with the possible exception of the period prior to Since the term structure has been upward sloping about 80% of the time, the theory would imply that investors have expected interest rates to rise 80% of the time. Riskless bonds, regardless of their different maturities, are expected to earn the same return on the average. That would mean investors are indifferent to risk. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 129

60 A Bad Expectations Theory The expected returns on all possible riskless bond strategies are equal for all holding periods. So (1 + S(2)) 2 = (1 + S(1)) E[ 1 + S(1, 2) ] (14) because of the equivalency between buying a two-period bond and rolling over one-period bonds. After rearrangement, 1 E[ 1 + S(1, 2) ] = 1 + S(1) (1 + S(2)) 2. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 130

61 A Bad Expectations Theory (continued) Now consider two one-period strategies. Strategy one buys a two-period bond and sells it after one period. The expected return is E[ (1 + S(1, 2)) 1 ] (1 + S(2)) 2. Strategy two buys a one-period bond with a return of 1 + S(1). The theory says the returns are equal: [ 1 + S(1) (1 + S(2)) 2 = E S(1, 2) ]. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 131

62 A Bad Expectations Theory (concluded) Combine this with Eq. (14) on p. 130 to obtain [ ] 1 1 E = 1 + S(1, 2) E[ 1 + S(1, 2) ]. But this is impossible save for a certain economy. Jensen s inequality states that E[ g(x) ] > g(e[ X ]) for any nondegenerate random variable X and strictly convex function g (i.e., g (x) > 0). Use g(x) (1 + x) 1 to prove our point. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 132

63 Local Expectations Theory The expected rate of return of any bond over a single period equals the prevailing one-period spot rate: E [ (1 + S(1, n)) (n 1) ] (1 + S(n)) n = 1 + S(1) for all n > 1. This theory is the basis of many interest rate models. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 133

64 Duration Revisited To handle more general types of spot rate curve changes, define a vector [ c 1, c 2,..., c n ] that characterizes the perceived type of change. Parallel shift: [ 1, 1,..., 1 ]. Twist: [ 1, 1,..., 1, 1,..., 1 ]. Let P (y) i C i/(1 + S(i) + yc i ) i be the price associated with the cash flow C 1, C 2,.... Define duration as P (y)/p (0) y. y=0 c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 134

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