ACC 471 Practice Problem Set #2 Fall Suggested Solutions

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1 ACC 471 Practice Problem Set #2 Fall 2002 Suggested Solutions 1. Text Problems: 11-6 a. i. Current ield: %. ii. Yield to maturit: solving for gives a ield to maturit of 4% semi-annuall (or 8% annual bond equivalent ield). iii. Start b calculating the future value of receiving $35 ever 6 months for the next three ears: FV $ Three ears from now, the bond will be selling at par (since the coupon is equal to the forecasted ield to maturit). Therefore total proceeds in three ears will be $1, The realized compound ield (on a semi-annual basis) is therefore: $ $ % or 8.332% annual bond equivalent ield. b. i. Current ield does not account for capital gains or losses on bonds bought at prices other than par value. It also does not account for reinvestment income on coupon paments. ii. Yield to maturit assumes that the bond is held until maturit and that all coupon income can be reinvested at a rate equal to the ield to maturit. iii. Realized compound ield is affected b the forecast of reinvestment rates, holding period, and ield of the bond at the end of the investor s holding period The reported bond price is 0.02 percent of par, or $1, However, 15 das have passed since the last semi-annual coupon was paid, so accrued interest equals $ $ The invoice price is the reported price plus accrued interest, or $1, The price schedule is: Year Remaining Maturit (T ) Constant ield value ( T ) Imputed interest n.a n.a , Note that the imputed interest is simpl the increase in the constant ield value a. The bond price is: Yield to call: P 0 $40 $ $ $1 000 $ $ % 1

2 b. In this case ield to call is: $ $ $ % c. Now ield to call is: $ $ $ % The promised ield to maturit is: $900 $ $ % However, the expected ield to maturit is: $900 $ $ % Zero coupon bonds provide no coupons to be reinvested. Therefore, the investor s proceeds from the bond are independent of the rate at which coupons could be reinvested (if the were paid). There is no reinvestment rate uncertaint with zeros Factors which might make the ABC debt more attractive to investors, therefore justifing a lower coupon rate and ield to maturit are: The ABC debt is a larger issue and thus ma sell with more liquidit. An option to extend the term from ears to 20 ears is favorable if interest rates in ears are lower than toda. In contrast, if interest rates increase, the investor can present the bond for pament and reinvest the mone for better returns. In the event of financial distress, the ABC debt is a more senior claim. It has more underling securit in the form of a first claim against real propert. The call feature on the XYZ bonds makes the ABC bonds relativel more attractive since ABC bonds cannot be called from the investor. The XYZ bond has a sinking fund requiring XYZ to retire part of the issue each ear. Since most sinking funds give the firm the option to retire this amount at the lower of par or market value, the sinking fund can work to the detriment of bondholders a. The call provision requires the firm to offer a higher coupon (or higher promised ield to maturit) on the bond to compensate the investor for the firm s option to call back the bond at a specified price if interest rates fall sufficientl. Investors are willing to grant this option to issuers, but onl for a price that reflects the possibilit that the bond will be called. That price is the higher promised ield at which the are willing to bu the bond. b. The call option will reduce the expected life of the bond. If interest rates fall substantiall and the likelihood of call increases, investors will begin to treat the bond as if it will mature and be paid off at the call date, not at the stated maturit date. On the other hand, if interest rates rise, the bond must be paid off at the maturit date, not later. This asmmetr means that the expected life of the bond will be less than the stated maturit. c. The advantage of a callable bond is the higher coupon (and a higher promised ield to maturit) when the bond is issued. If the bond turns out not to be called, then one will earn a higher realized compound ield on a callable bond than an otherwise identical but non-callable bond issued on the same date. The disadvantage of the callable bond is the risk of call. If rates fall and the bond is called, the investor will receive the call price and will have to reinvest the proceeds at rates lower than the ield to maturit at which the bond was originall issued. In this event, the firm s savings in interest paments is the investor s loss. 2

3 12-1 i. Expectations hpothesis: The ields on long-term bonds are geometric averages of present and expected future short rates. An upward-sloping ield curve is explained b expected future short rates being higher than the current short rate. A downward-sloping curve implies expected future short rates are lower than the current short rate. Thus bonds of different maturities have different ields if expectations of future short rates are different from the current short rate. ii. Liquidit preference hpothesis: Yields on long-term bonds are greater than the expected return from rolling over short-term bonds in order to compensate investors in long-term bonds for bearing interest rate risk. Thus bonds of different maturities can have different ields even if expected future short rates are all equal to the current short rate. An upward-sloping ield curve can be consistent even with expectations of falling short rates if liquidit premiums are high enough. If, however, the ield curve is downward-sloping and liquidit premiums are assumed to be positive, then we can conclude that future short rates are expected to be lower than the current short rate. iii. Segmentation hpothesis: This hpothesis would explain a sloping ield curve as an imbalance between suppl and demand for bonds of different maturities. An upward-sloping curve is evidence of suppl pressure in the long-term market and demand pressure in the short-term market. According to the segmentation hpothesis, expectations of future rates have little to do with the shape of the ield curve True. Under the expectations hpothesis, there are no risk premia built into bond prices. The onl reason for long-term ields to exceed short-term ields is an expectation of higher short-term rates in the future Uncertain. Lower inflation will usuall lead to lower nominal interest rates, but if the liquidit premium is sufficientl high, long-term ields ma exceed short-term ields despite expectations of falling short rates Maturit Price YTM Forward Rates 1 $ % n.a. 2 $ % % 3 $ % % 4 $ % % 12-6 The expected price path of the 4 ear zero coupon bond is as follows. Note that we discount the face value b the appropriate sequence of forward rates implied b the current ield curve. Start of Year Expected Price Expected Rate of Return 1 $ % ( ) 2 $ ( ) 5.00% ( ) 3 $ ( ) 6.00% ( ) 4 $ ( ) 7.00% ( ) 12-8 You should expect it to lie about the curve since the bond must offer a premium to investors to compensate them for the option granted to the issuer a. We have: b. We have: $1 86 P $ % c. The forward rate for next ear is f %. Thus the forecasted bond price is P $ d. If the liquidit premium is 1%, then the forecasted interest rate is f %. Thus P $

4 12-16 a. To calculate the 5 ear spot rate: $ % Then the 5 ear forward rate is %. b. Yield to maturit is the single discount rate that equates the present value of a series of cash flows to a current price. It is the internal rate of return. The spot rate for a given period is the ield to maturit on a zero coupon bond which matures at the end of the period. A spot rate is the discount rate for each period. Spot rates are used to discount each cash flow of a coupon bond to calculate a current price. Spot rates are the rates appropriate for discounting future cash flows of different maturities. A forward rate is the implicit rate that links an two spot rates. Forward rates are directl related to spot rates, and therefore ield to maturit. Some would argue (as in the expectations theor) that forward rates are market expectations of future interest rates. Regardless, a forward rate represents a break-even rate that links two spot rates. It is important to note that forward rates link spot rates, not ields to maturit. Yield to maturit is not unique for an particular maturit. In other words, two different bonds with the same maturit but different coupon rates ma have different ields to maturit. In contrast, spot and forward rates for each date are unique. c. The 4 ear spot rate is 7.16%. Thus, 7.16% is the expected ield to maturit for the zero coupon bond. The price of the zero is $ $ The price of the coupon bond is: $ If the coupons were stripped and sold as zeros, the could be sold separatel for: $ The arbitrage strateg is to bu zeros with face values of $120 and $1,120 and respective maturities of one and two ears, and simultaneousl sell the coupon bond. The profit equals $2.91 on each bond a. We have: Maturit (Years) Price Yield to Maturit (Spot Rate) Forward Rate % n.a % % % % % % % % b. You can create the loan b selling 3 ear zeros toda and buing 4 ear zeros. For each 3 ear zero sold, ou can use the proceeds to bu ear zeros. Your cash flows are: Time Cash Flow 0 0 Your purchase of 4 ear zeros is financed b the sale of 3 ear zeros. 3 -$1,000 The 3 ear zero that ou sell matures and ou pa out the face value. 4 +$1,095 The 4 ear zeros that ou bu mature and ou collect the face value on each one. 4

5 This is a snthetic loan starting at time 3, with a rate of 9.5%, precisel the forward rate for ear 4. c. For each 4 ear zero sold, ou can bu ear zeros. Your cash flows are: Time Cash Flow 0 0 Your purchase of 5 ear zeros is financed b the sale of 4 ear zeros. 4 -$1,000 The 4 ear zero that ou sell matures and ou pa out the face value. 5 +$1,0 The 5 ear zeros that ou bu mature and ou collect the face value on each one. This is a snthetic loan starting at time 4, with a rate of %, precisel the forward rate for ear a. For each 3 ear zero that ou bu toda, ou need to sell ear zeros to make our initial cash flow equal to zero. b. Your cash flows are: Time Cash Flow 0 0 Your purchase of 3 ear zeros is financed b the sale of 5 ear zeros. 3 +$1,000 The 3 ear zero that ou bu matures and ou collect the face value. 5 -$1, The 5 ear zeros that ou sell mature and ou pa out the face value on each one. This is a snthetic two ear loan, starting at time 3. c. The two ear forward rate on the forward loan is %. d. The one ear forward rates for ears 4 and 5 and 9.5% and % respectivel. Notice that , which equals the two ear forward rate on the three ear ahead forward loan Recall that the duration of a coupon bond is: 1 1 T c c 1 T 1 where is the bond s ield per pament period, c is its coupon rate per pament period, and T is the number of pament periods. Moreover, this can be simplified to T for bonds selling at par. Therefore: 06 D D ears ears 1 Note that we can also calculate duration directl, as follows. The price of the bond is its par value when 06, and it is $ when. Then 06 D ears D ears

6 13-3 When 06, we have: D half-ears 2 79 ears while when, we have: D half-ears ears 13-4 a. Bond B has a higher ield to maturit than Bond A since its coupons and time to maturit are the same as for A, while its price is lower. Therefore, its duration must be shorter. b. Bond A has a lower ield and a lower coupon, both of which cause it to have a longer duration than Bond B. Morever, A cannot be called, which makes its maturit at least as long as that of B, which generall increases duration a. The present value of the firm s liabilities is: Therefore the duration of the liabilities is: D $ ears Thus the required maturit of the zero coupon bond is ears. b. The market value of the zero must be $11,574,594.38, the same as the market value of the obligations. Therefore, the face value must be $ a. The call feature provides a valuable option to the issuer, since it can bu back the bond at a given call price even if the present value of the scheduled remaining paments exceeds the call price. The investor will demand, and the issuer will be willing to pa, a higher ield on the issue as compensation for this feature. b. The call feature will reduce both the duration (interest rate sensitivit) and the convexit of the bond. The bond will not experience as large a price increase if interest rates fall. Moreover, the usual curvature that would characterize a straight bond will be reduced b a call feature. The price-ield curve flattens out (see text Figure 13.7) as the interest rate falls and the option to call the bond becomes more attractive. In fact, at ver low interest rates, the bond exhibits negative convexit. 13- a. D D ears. b. For option-free coupon bonds, modified duration is better than maturit as a measure of the bond s sensitivit to changes in interest rates. Maturit considers onl the final cash flow, while modified duration includes other factors such as the size and timing of coupon paments and the level of interest rates (ield to maturit). Modified duration, unlike maturit, tells us the approximate percentage change in bond price for a given change in ield to maturit. c. i. Modified duration increases as the coupon decreases. ii. Modified duration decreases as the maturit decreases. d. Convexit measures the curvature of the bond s price-ield curve. Such curvature means that the duration rule for bond price change (which is based onl on the slope of the ield curve at the original 6

7 ield) is onl an approximation. Adding a term to account for the convexit of the bond will increase the accurac of the approximation. That convexit adjustment is the last term in the following equation: P P D 1 2 Convexit a. In an interest rate swap, one firm exchanges or swaps a fixed pament for another pament that is tied to the level of interest rates. One part in the swap agreement must pa a fixed interest rate on the notional principal of the swap. The other part pas the floating interest rate (tpicall based on LIBOR) on the same notional principal. b. There are several applications of interest rate swaps. For example, a portfolio manager who is holding a portfolio of long-term bonds, but is worried that interest rates might increase, causing a capital loss on the portfolio, can enter a swap to pa a fixed rate and receive a floating rate, thereb converting the holdings into a snthetic floating rate portfolio. Or, a pension fund manager might identif some mone market securities that are paing excellent ields compared to other comparable risk short term securities. However, the manager might believe that such short-term assets are inappropriate for the portfolio. The fund can hold these securities and enter a swap where it receives a fixed rate and pas a floating rate. It thus captures the benefit of the advantageous relative ields on these securities, but still establishes a portfolio with characteristics more like those of long-term bonds The firm should enter a swap where it pas a fixed rate of 7% and receives LIBOR on $ million of notional principal. The firm s combined position will be LIBOR LIBOR 08, i.e. it will be paing an interest rate of 8% a. The present value of the obligation is 2 16 $12 5 million. The duration of the obligation is ears. Let w be the weight on the 5 ear bond (which has a duration of 4 ears). Then: w 4 1 w w 5357 Therefore, $6 696 million should be invested in the 5 ear bonds and $5 804 million in the 20 ear bonds. b. The price of the 20 ear bond is: $ Therefore, the bond sells for times its par value, and so the par value needed for the position is $ million a. The duration of the perpetuit is ears. Let w be the weight of the zero coupon bond. Then: w 5 1 w 21 w 6875 so ou would have 68.75% invested in the zero and 31.25% in the perpetuit. b. The zero coupon bond will now have a duration of 4 ears, while the perpetuit s duration will remain at 21 ears. Then: w 4 1 w 21 9 w 7059 so the percentage invested in the zero increases to 70.59%, while that in the perpetuit falls to 29.41% a. Recall that the duration of an annuit is: 1 T 1 T 1 Therefore, if the annuit were to start in 1 ear, its duration would be ears 1 Because the pament stream starts in 5 ears, instead of 1 ear, we must add 4 ears to the duration, resulting in duration of ears. 7

8 b. The present value of the deferred annuit is: $ $ Let w be the weight of the portfolio in the 5 ear zero. Then: w 5 1 w w 7516 Therefore the investment in the 5 ear zero is 7516 $ $ and the investment in the 20 ear zero is 2484 $ $ 425. These are the present values of each investment. The face value of the 5 ear zeros is $ $50 800, while that of the 20 ear zeros is $ $ We can calculate the exact prices of the bonds as follows: % 120 8% 120 8% $ $ $ Using the duration rule, if the ield falls to 7%, we would predict a price change of: $ $ so our predicted new price would be $ $ The actual price is $1,620.45, impling a percentage error of %. Using the duration rule, if the ield rises to 9%, we would predict a price change of: $ $ so our predicted new price would be $ $ The actual price is $1,308.21, impling a percentage error of %. Using the duration-with-convexit rule, if the ield drops to 7% we would predict a price change of: $ $ so our estimated new price would be $ $ , with a percentage error of %. Using the duration-with-convexit rule, if the ield increases to 9% we would predict a price change of: $ $ so our estimated new price would be $ $ , with a percentage error of 0.083%. The duration-with-convexit rule provides more accurate approximations to the true change in price. In this example, the percentage error using convexit with duration is less than one-tenth the error using onl duration to estimate the price change The economic climate is one of impending rate increases (i.e. falling bond prices), so we will want to shorten portfolio duration. a. Choose the short maturit (2001) bond. b. Choose the Arizona bond since it likel has lower duration (it has slightl lower coupons, but substantiall higher ield). 8

9 c. Choose the 15 3/8 coupon bond the maturities are about the same, but it has a much higher coupon, impling a lower duration. d. Choose the Shell bond. Although it has a little lower coupon, it has a higher ield to maturit and earlier start to the sinking fund repaments. e. Choose the floating rate bond. It has a duration approximatel equal to the adjustment period of 6 months The minimum terminal value that the manager is willing to accept is determined b the requirement for a 3% annual return on the initial investment. Therefore, the floor is $1 16 million. Three ears after the initial investment, onl two ears remain until the horizon date, and the interest rate has risen to 8%. Therefore, at this time, the manager needs a portfolio worth $ million to be assured that the target value can be attained. This is the trigger point. 2. a. The promised ield to maturit is: $1 000 $ % b. The expected ield to maturit is: $ $1 000 $ % 3. a. Since the bond is selling at par, its ield to maturit is equal to the coupon rate of 12%. Its ield to call is: $ % 1 4 b. The ield to call is higher. This is because the issuer s call option reduces the value of the bond to the holder, impling a lower price and therefore a higher ield. The ield to maturit calculation ignores this option feature, resulting in a lower ield. 4. a. For bonds A and B we have: r r r r 2 2 Buing two of bond B and selling one of bond A therefore costs $ and returns $1,000 in two ears, so: r 2 2 r %. Then r r % and (from bond C) r 3 3 r %. b. Under the expectations hpothesis, forward rates are equal to expected future spot rates. We have: f 2 f 3 1 r % 1 r 1 1 r r % Therefore the one ear spot rate is expected to be 9.85% in one ear and.25% in two ears. The usual assumption for the liquidit preference hpothesis is that the market is dominated b investors with a short term horizon. For such investors, long term bonds are riskier than short term bonds and thus long term bonds command a risk premium. This implies that forward rates will exceed expected future spot rates. On the other hand, if the market is dominated b investors with a long term horizon, short term bonds are riskier and forward rates are less than expected future spot rates. 9

10 5. Let x be the spread in the fixed rate market (in this case 140 basis points) and let be the spread in the floating rate market (50 basis points). The total gain available is x 90 basis points. The bank will take basis points, so that leaves 80 basis points to be split equall between A and B. Therefore, A must end up paing LIBOR - 30 basis points and B must end up paing 13%. One wa to accomplish this is as follows: A Pa 12% to outside lenders Pa LIBOR - 0.1% to the bank in the swap Receive 12.2% from the bank in the swap Total 12% + LIBOR - 0.1% % = LIBOR - 0.3% B Pa LIBOR + 0.6% to outside lenders Pa 12.2% to the bank in the swap Receive LIBOR - 0.2% from the bank in the swap Total LIBOR + 0.6% % - (LIBOR - 0.2%) = 13% Note that the bank s profits of basis points come from receiving LIBOR - basis points from A and paing LIBOR - 20 basis points to B. 6. Let x be the spread in the en market (in this case 150 basis points) and let be the spread in the dollar market (40 basis points). The total gain available is x 1 basis points. The bank will take 50 basis points, so that leaves 60 basis points to be split equall between X and Y. Therefore, X must end up paing 9.3% in dollars and B must end up paing 6.2% in en. One wa to accomplish this is as follows: X Total Y Total Pa 5% in en to outside lenders Pa 9.3% in dollars to the bank in the swap Receive 5% in en from the bank in the swap 9.3% in dollars Pa % in dollars to outside lenders Pa 6.2% in en to the bank in the swap Receive % in dollars from the bank in the swap 6.2% in en Note that the bank s profits of 50 basis points come from receiving 6.2% and paing 5% in en (thereb gaining 120 basis points in en) while receiving 9.3% and paing % in dollars (thus losing 70 basis points in dollars).