ZeroCoupon Bonds (Pure Discount Bonds)


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1 ZeroCoupon Bonds (Pure Discount Bonds) The price of a zerocoupon bond that pays F dollars in n periods is F/(1 + r) n, where r is the interest rate per period. Can meet future obligations without reinvestment risk. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 54
2 Example The interest rate is 8% compounded semiannually. A zerocoupon bond that pays the par value 20 years from now will be priced at 1/(1.04) 40, or 20.83%, of its par value. It will be quoted as If the bond matures in 10 years instead of 20, its price would be c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 55
3 Coupon rate. LevelCoupon Bonds Par value, paid at maturity. F denotes the par value, and C denotes the coupon. Cash flow: C C C C + F n Coupon bonds can be thought of as a matching package of zerocoupon bonds, at least theoretically. a a You see, Daddy didn t bake the cake, and Daddy isn t the one who gets to eat it. But he gets to slice the cake and hand it out. And when he does, little golden crumbs fall off the cake. And Daddy gets to eat those, wrote Tom Wolfe (1931 ) in Bonfire of the Vanities (1987). c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 56
4 Pricing Formula P = n i=1 C ( ) 1 + r i + m = C 1 ( 1 + r m r m ) n F ( 1 + r + m ) n F ( ) 1 + r n. (5) m n: number of cash flows. m: number of payments per year. r: annual rate compounded m times per annum. C = F c/m when c is the annual coupon rate. Price P can be computed in O(1) time. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 57
5 Yields to Maturity It is the r that satisfies Eq. (5) on p. 57 with P being the bond price. For a 15% BEY, a 10year bond with a coupon rate of 10% paid semiannually sells for 1 [ 1 + (0.15/2) ] /2 = percent of par [ 1 + (0.15/2) ] 2 10 c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 58
6 Price Behavior (1) Bond prices fall when interest rates rise, and vice versa. Only 24 percent answered the question correctly. a a CNN, December 21, c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 59
7 A levelcoupon bond sells Price Behavior (2) at a premium (above its par value) when its coupon rate is above the market interest rate; at par (at its par value) when its coupon rate is equal to the market interest rate; at a discount (below its par value) when its coupon rate is below the market interest rate. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 60
8 9% Coupon Bond Yield (%) Price (% of par) c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 61
9 Terminology Bonds selling at par are called par bonds. Bonds selling at a premium are called premium bonds. Bonds selling at a discount are called discount bonds. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 62
10 Price Behavior (3): Convexity Price Yield c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 63
11 Day Count Conventions: Actual/Actual The first actual refers to the actual number of days in a month. The second refers to the actual number of days in a coupon period. The number of days between June 17, 1992, and October 1, 1992, is days in June, 31 days in July, 31 days in August, 30 days in September, and 1 day in October. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 64
12 Day Count Conventions: 30/360 Each month has 30 days and each year 360 days. The number of days between June 17, 1992, and October 1, 1992, is days in June, 30 days in July, 30 days in August, 30 days in September, and 1 day in October. In general, the number of days from date D 1 (y 1, m 1, d 1 ) to date D 2 (y 2, m 2, d 2 ) is 360 (y 2 y 1 ) + 30 (m 2 m 1 ) + (d 2 d 1 ). Complications: 31, Feb 28, and Feb 29. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 65
13 Full Price (Dirty Price, Invoice Price) In reality, the settlement date may fall on any day between two coupon payment dates. Let ω number of days between the settlement and the next coupon payment date number of days in the coupon period. (6) The price is now calculated by PV = n 1 i=0 C ( ) 1 + r ω+i + m F ( ) 1 + r ω+n 1. (7) m c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 66
14 Accrued Interest The buyer pays the quoted price plus the accrued interest the invoice price: C number of days from the last coupon payment to the settlement date number of days in the coupon period = C (1 ω). The yield to maturity is the r satisfying Eq. (7) when P is the invoice price. The quoted price in the U.S./U.K. does not include the accrued interest; it is called the clean price or flat price. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 67
15 C(1 ω) coupon payment date coupon payment date (1 ω)% ω% c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 68
16 Example ( 30/360 ) A bond with a 10% coupon rate and paying interest semiannually, with clean price The maturity date is March 1, 1995, and the settlement date is July 1, There are 60 days between July 1, 1993, and the next coupon date, September 1, c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 69
17 Example ( 30/360 ) (concluded) The accrued interest is (10/2) = per $100 of par value. The yield to maturity is 3%. This can be verified by Eq. (7) on p. 66 with ω = 60/180, m = 2, C = 5, PV= , r = c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 70
18 Price Behavior (2) Revisited Before: A bond selling at par if the yield to maturity equals the coupon rate. But it assumed that the settlement date is on a coupon payment date. Now suppose the settlement date for a bond selling at par (i.e., the quoted price is equal to the par value) falls between two coupon payment dates. Then its yield to maturity is less than the coupon rate. The short reason: Exponential growth is replaced by linear growth, hence overpaying the coupon. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 71
19 Bond Price Volatility c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 72
20 Well, Beethoven, what is this? Attributed to Prince Anton Esterházy c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 73
21 Price Volatility Volatility measures how bond prices respond to interest rate changes. It is key to the risk management of interest ratesensitive securities. Assume levelcoupon bonds throughout. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 74
22 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. YuhDauh Lyuu, National Taiwan University Page 75
23 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. YuhDauh Lyuu, National Taiwan University Page 76
24 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. (8) c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 77
25 MD of Bonds The MD of a coupon bond is [ MD = 1 n ic P (1 + y) i + nf (1 + y) n i=1 ]. (9) 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 zerocoupon bond equals its term to maturity n. The MD of a coupon bond is less than its maturity. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 78
26 Remarks Equations (8) on p. 77 and (9) on p. 78 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. YuhDauh Lyuu, National Taiwan University Page 79
27 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, durationrelated terminology cannot be comprehended otherwise. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 80
28 Conversion For the MD to be yearbased, modify Eq. (9) on p. 78 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 (8) on p. 77 also becomes MD = ( 1 + y ) P k y 1 P. By definition, MD (in years) = MD (in periods) k. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 81
29 Modified Duration Modified duration is defined as modified duration P y By Taylor expansion, 1 P = MD (1 + y). (10) percent price change modified duration yield change. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 82
30 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. YuhDauh Lyuu, National Taiwan University Page 83
31 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. YuhDauh Lyuu, National Taiwan University Page 84
32 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. 86. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 85
33 P + P 0 P  y  y y + c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 86
34 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. YuhDauh Lyuu, National Taiwan University Page 87
35 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. YuhDauh Lyuu, National Taiwan University Page 88
36 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. YuhDauh Lyuu, National Taiwan University Page 89
37 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. YuhDauh Lyuu, National Taiwan University Page 90
38 Price Yield c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 91
39 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. YuhDauh Lyuu, National Taiwan University Page 92
40 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. 91. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 93
41 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. YuhDauh Lyuu, National Taiwan University Page 94
42 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. YuhDauh Lyuu, National Taiwan University Page 95
43 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. YuhDauh Lyuu, National Taiwan University Page 96
44 Term Structure of Interest Rates c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 97
45 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. YuhDauh Lyuu, National Taiwan University Page 98
46 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 fixedincome securities. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 99
47 Yield (%) Year c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 100
48 Term Structure of Interest Rates (concluded) Term structure often refers exclusively to the yields of zerocoupon bonds. A yield curve plots yields to maturity against maturity. A par yield curve is constructed from bonds trading near par. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 101
49 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. YuhDauh Lyuu, National Taiwan University Page 102
50 Spot Rates The iperiod spot rate S(i) is the yield to maturity of an iperiod zerocoupon bond. The PV of one dollar i periods from now is [ 1 + S(i) ] i. The oneperiod spot rate is called the short rate. Spot rate curve: Plot of spot rates against maturity. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 103
51 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. YuhDauh Lyuu, National Taiwan University Page 104
52 Problems with the PV Formula (concluded) The yieldtomaturity methodology discounts their contemporaneous cash flows with different rates. But shouldn t they be discounted at the same rate? c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 105
53 Spot Rate Discount Methodology A cash flow C 1, C 2,..., C n is equivalent to a package of zerocoupon bonds with the ith bond paying C i dollars at time i. So a levelcoupon bond has the price P = n i=1 C [ 1 + S(i) ] i + F [ 1 + S(n) ] n. (11) This pricing method incorporates information from the term structure. Discount each cash flow at the corresponding spot rate. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 106
54 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. YuhDauh Lyuu, National Taiwan University Page 107
55 Extracting Spot Rates from Yield Curve Start with the short rate S(1). Note that shortterm Treasuries are zerocoupon bonds. Compute S(2) from the twoperiod coupon bond price P by solving P = C 1 + S(1) + C [ 1 + S(2) ] 2. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 108
56 Extracting Spot Rates from Yield Curve (concluded) Inductively, we are given the market price P of the nperiod coupon bond and S(1), S(2),..., S(n 1). Then S(n) can be computed from Eq. (11) on p. 106, 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. YuhDauh Lyuu, National Taiwan University Page 109
57 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. YuhDauh Lyuu, National Taiwan University Page 110
58 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. YuhDauh Lyuu, National Taiwan University Page 111
59 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. YuhDauh Lyuu, National Taiwan University Page 112
60 Of Spot Rate Curve and Yield Curve y k : yield to maturity for the kperiod 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. YuhDauh Lyuu, National Taiwan University Page 113
61 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. YuhDauh Lyuu, National Taiwan University Page 114
62 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. YuhDauh Lyuu, National Taiwan University Page 115
63 When S(i, j) equals Forward Rates (concluded) f(i, j) [ (1 + S(j)) j (1 + S(i)) i ] 1/(j i) 1, (12) 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. YuhDauh Lyuu, National Taiwan University Page 116
64 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. YuhDauh Lyuu, National Taiwan University Page 117
65 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 oneperiod forward rates. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 118
66 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. YuhDauh Lyuu, National Taiwan University Page 119
67 c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 120
68 Forward Rates Spot Rates Yield Curve The FV of $1 at time n can be derived in two ways. Buy nperiod zerocoupon bonds and receive [ 1 + S(n) ] n. Buy oneperiod zerocoupon 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. YuhDauh Lyuu, National Taiwan University Page 121
69 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. (13) Hence, the forward rates, specifically the oneperiod 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. YuhDauh Lyuu, National Taiwan University Page 122
70 Locking in the Forward Rate f(n, m) Buy one nperiod zerocoupon bond for 1/(1 + S(n)) n. Sell (1 + S(m)) m /(1 + S(n)) n bonds. mperiod zerocoupon 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. YuhDauh Lyuu, National Taiwan University Page 123
71 1 n m (1 + S(m)) m /(1 + S(n)) n c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 124
72 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, TianShyr (R , D ) in c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 125
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