Investments: Bonds Business Administration 365 Professor Scott Hoover 1
n Definitions: q q Bond a security with pre-specified cash flows to be paid on pre-specified dates Yield-to-maturity annual return if a bond is bought today and all promised CFs are received n quoted as an APR 2
n Managing Bond Portfolios q Relationship between yields and bond values. n Recall that the value of a fixed coupon bond is where y yield-to-maturity n <see Price-Yield spreadsheet> 3
n Lessons Learned q Assuming all else equal… § Higher coupon rates lower price sensitivity to interest rate changes. Why? § Longer maturities higher price sensitivity to interest rate changes. Why? § Higher yields lower price sensitivity to interest rate changes. Why? q Of course, “all else equal” doesn’t always hold. In particular, default risk tends to be higher for… § …higher coupon rates § …shorter maturities § …higher yields q 4
n Macaulay Duration q q q We want a quick way to estimate the impact of interest rate changes on bond prices. Why? § We can quickly approximate the change in price when rates change. § It also is useful for another reason that we will discuss shortly. one appropriate measure: slope 5
q Duration “weighted average” maturity of a bond’s cash flows. q § Weight fraction of the bond’s present value contributed by the given cash flow. q Taking the first derivative (i. e. , slope) of the general value function with respect to y and rearranging gives q This gives us the percentage change in bond value as a function of the change in yield-to-maturity. 6
q We sometimes use the “modified duration, ” which gives q Note that the greater the convexity of the price-yield curve, the greater the error. 7
q example: § 5 years to maturity § 7% annual coupons § yield-to-maturity = 7. 8%. Date Cash Flow PV(CF) t 1 $70 $64. 935 2 $70 $60. 237 120. 473 3 $70 $55. 878 167. 634 4 $70 $51. 835 207. 340 5 $1070 $735. 004 3675. 022 $967. 89 4235. 40 Totals § § D = 4235. 40 / 967. 89 = 4. 376 years Dm = 4. 376/1. 078 = 4. 059 years 8
§ So… 1% change in bond yield ~ 4. 059% change in value. § Suppose that the yield changes to 8. 4% (an increase of 0. 6%). V/V = -4. 059 0. 6% = -2. 436% new value $967. 89 (1 -0. 02436) = $944. 31 actual price = $944. 69 § § § 9
n Miscellaneous Comments on Duration q q q not particularly useful for individual bonds quite useful for bond portfolios. portfolio duration = weighted average of individual bond durations § The weights are the fractions of total market value. linear approximation to a convex curve § Note that we might consider a higher order approximation (Taylor series expansion) to get a better estimate. Do other assets (stocks, etc. ) have durations? § Yes! § How might we estimate them? 10
q Immunization n example: We want to invest for 4 years. Two bonds are available. q q q 6 -year zero coupon bond, y-t-m = 5. 5% § V = $725. 25 § D = 6 years 4 -year 6% coupon bond , y-t-m = 5. 5% § V = $1, 017. 53 § D = 3. 68 Suppose further that we buy one 6 -year bond and six 4 year bonds § Total investment = $725. 25 + 6×$1, 017. 53 = $6, 830. 40 11
q q q Suppose interest rates stay at 5. 5%… Payments on 4 -year bond: § $60 in one year $60 1. 0553 = $70. 45 in 4 years. § $60 in two years $60 1. 0552 = $66. 78 in 4 years. § $60 in three years $60 1. 055 = $63. 30 in 4 years. § $1060 in four years Proceeds from sale of 6 -year bond § $1000/1. 0552 = $898. 45 Total value in 4 years § 6 (70. 45+66. 78+63. 30+1060)+898. 45 = $8, 461. 67 § Notice that this is $6, 830. 40 1. 0554 = $8, 461. 67 12
q q q Suppose, instead, that interest rates increase to 6. 5%… Payments on 4 -year bond: § $60 in one year…. reinvested to get $60 1. 0653 = $72. 48 in 4 years. § $60 in two years…. reinvested to get $60 1. 0652 = $68. 05 in 4 years. § $60 in three years…. reinvested to get $60 1. 065 = $63. 90 in 4 years. § $1060 in four years Proceeds from sale of 6 -year bond § $1000/1. 0652 = $881. 66 Total value in 4 years § 6 (72. 48+68. 05+63. 90+1060)+881. 66 = $8, 468. 24 § Notice that this is nearly identical to the outcome with a 5. 5% interest rate! 13
q q q Suppose, instead, that interest rates drop to 4. 5%… Payments on 4 -year bond: § $60 in one year $60 1. 0453 = $68. 47 in 4 years. § $60 in two years $60 1. 0452 = $65. 52 in 4 years. § $60 in three years $60 1. 045 = $62. 70 in 4 years. § $1060 in four years Proceeds from sale of 6 -year bond § $1000/1. 0452 = $915. 73 Total value in 4 years § 6 (68. 47+65. 52+62. 70+1060)+915. 73 = $8, 455. 88 § Notice that this is nearly identical to the outcome with a 5. 5% interest rate! 14
q q The net payoff is virtually the same in all three scenarios!! § Our portfolio is said to be immunized against changes in interest rates. Why does this work? § Duration of portfolio = (725. 25/6830. 40)× 6 + (6× 1, 017. 53/6830. 40)× 3. 68 = 4. 00 years! Portfolio duration = investment horizon! In that case… …higher interest rates higher returns on reinvested funds but lower price for sold securities § …lower interest rates lower returns on reinvested funds but higher price for sold securities. § In both cases, the effects offset each other. Implication: We can nearly eliminate interest rate risk by matching portfolio durations to our cash flow needs. <see spreadsheet> § q q 15
n example: You wish to invest for a period of 4. 5 years, at which time you will use the money to finance your own company. Furthermore, you wish to immunize your portfolio against interest rate risk. q q q Two bonds are available: § 7 years, 8% annual coupons, ytm=9% D = 5. 58 § 4 years, 10% annual coupons, ytm=9% D = 3. 50 To ensure an immunized portfolio, you must invest a certain fraction of your money in each bond so that the portfolio duration is 4. 5 years. § w 7 5. 58 + w 4 3. 50 = 4. 5 § w 7 5. 58 + (1 -w 7) 3. 50 = 4. 5 § Solving gives w 7 = 0. 48 and hence w 4 = 0. 52 What annual return do you expect to receive over the 4. 5 years? 16
n Bond portfolios are often managed in this way. q q n We choose an investment horizon. Then we choose a portfolio with a duration equal to that investment horizon. Problems with immunization q q q may need to rebalance frequently high transactions costs Which portfolio should be chosen (there are literally an infinite number of possible ones) ? § one that suits our risk preference § one that most closely resembles the cash needs The duration calculation assumes no embedded options (call, conversion, etc. ) and no default. § How might we get around this? 17
n Applying the general concept of immunization… q Example: Consider the following sensitivities n n Suppose we have $1, 000 to invest. b: market; d: interest rates; g: $/€ exchange rate. q q We believe Ford is undervalued relative to GM. We choose (arbitrarily) to invest $500, 000 in Ford and short $500, 000 of GM. We have identified four other fairly-priced assets to use. d g Wt. A 1. 2 0. 1 -0. 4 w. A B 0. 6 0. 3 0 w. B C 0. 7 -0. 4 0. 9 w. C D -0. 2 0. 8 0. 5 w. D Ford 0. 9 1. 1 -0. 8 0. 5 GM n b 1 1. 3 -0. 7 -0. 5 Can we form a portfolio that is immunized against all three types of risk? 18
n Equations: q q n w. A× 1. 2+w. B× 0. 6+w. C× 0. 7+w. D×(-0. 2)+0. 5× 0. 9+(-0. 5)× 1. 0 = 0 w. A× 0. 1+w. B× 0. 3+w. C×(-0. 4 )+w. D×(0. 8)+0. 5× 1. 1+(-0. 5)× 1. 3 = 0 w. A×(-0. 4)+w. B× 0. 0+w. C× 0. 9+w. D×(0. 5)+0. 5×(-0. 8)+(-0. 5)×(-0. 7) = 0 w. A+w. B+w. C+w. D+0. 5+(-0. 5) = 1 Solving these using matrix algebra gives q q w. A = -4. 660 short $4, 660, 000 of A w. B = 8. 947 invest $8, 947, 000 in B w. C = -0. 426 short $426, 000 of C w. D = -2. 861 short $2, 861, 000 of D 19
n Of course, our desire is to control for risk while taking advantage of mispricings. n Adding more assets gives us degrees of freedom. q n This allows us to take advantage of relative mispricings while controlling for various risk factors. Note: Matrix algebra is not fair game on the final exam. 20
n Bond Provisions q q Convertibility n give the purchaser the right to exchange the bond for a pre-specified number of shares of stock during a pre-specified period. Warrants n often "attached" to a bond n gives the purchaser the right to buy a pre-specified number of shares of stock for a pre-specified price during a pre-specified period. 21
q Call provision n q gives the seller the right to buy back the bond for a pre-specified price during a pre-specified period Other Covenants n force the issuer to maintain a certain level of firm stability q q q usually specified through financial ratio restrictions covenant violated holder may have the right to demand immediate repayment. How are each of these likely to affect bond values? 22
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