Chapter 16 - Immunization

Transcription

1 Chapter 16 - Immunization ANSWERS TO QUESTIONS & PROBLEMS 16-1: A. The duration of the assets can be set to a minimum of 1 year (100% in 1 year 7% bonds) and to a maximum of 2.78 years (100% in 3 year 8% bonds). There is no way John can set the duration of the assets to an average of 3 years. B. The present value of $1,000,000 in three years at 10% is $751, The $1000 8% bond trades at $ so John needs to buy 791 $1000 bonds. t=1 Interest Rate in Second Year 9% 10% 11% Coupon from the 791 8% bonds $63, $63, $63, % bonds with 2 years left @$ Purchase bonds 64 bonds 65 bonds 66 bonds Expenditure ($62,874.24) ($62,743.85) ($62,608.92) Number of 8% bonds in the portfolio Market Value $839, $826, Cash $ $ $ TOTAL PORTFOLIO VALUE $840, $826, $813, Interest Rate in Third Year 9% 10% 11% t=2 Number of 8% bonds in the portfolio Coupon from the 8% bonds $68, $68, $68, reinvested residual cash $ $ $ TOTAL CASH $68, $69, $69, % bonds with 1 year left to maturity $ $ $ Purchase bonds Expenditure ($68,367.27) ($68,727.40) ($69,080.87) Number of 8% bonds in the portfolio Market Value $915, $909, $902, Cash $ $ $ TOTAL PORTFOLIO VALUE $916, $909, $903,140.19

2 OLTHETEN & WASPI 2012 Interest Rate 9% 10% 11% t=3 Number of 8% bonds in the portfolio Coupon from the 8% bonds $73, $74, $74, % bonds mature $924, $926, $928, Reinvested residual cash $ $ $ TOTAL PORTFOLIO VALUE $998, $1,000, $1,002, In this case the duration of the assets is shorter than the duration of the liability: the re-investment risk is too large. When rates go down the decrease in reinvestment is too large to be offset by the increase in prices. This is the opposite of the example on pages In that example the duration of the assets was longer than the duration of the liability; the price risk was too large. When rates increased the decrease in price wiped out any gains from reinvestment. 16-2: There are only two cash flows scheduled by the Trust: one payment of $1,000,000 in three years - so six semi-annual periods down the time line; and one payment of $2,000,000 in five years - so ten semi-annual periods down the time line. time Cash Flow NPV NPV/P NPV/P * time 6 $1,000,000. $813, $2,000,000. $1,417, ,231, A. The Net Present Value of the liability is $2,231, If we don't have at least that much in the fund now then Investments Unlimited is in trouble. The duration is years B. Since the durations of the assets is too long, years compared to years,- we should buy shorter duration bonds (bonds with a duration less than 4.27 years)

3 ANSWERS TO QUESTIONS & PROBLEMS 16-3: Let x be the weight in IBM. Thus (1 x) is the weight in GE Duration of Assets Duration of Liabilities 7.56 x 4.00 (1 x) 5.00 years x IBM 28.09% 1 x GE 71.91% IBM: $1,000, $ bonds GE: 1) Round both down IBM: 302 * $ = $280, GE: 616 * $1, = $718, This leaves $1, $1,000, $1, bonds We would prefer to buy another GE bond because it is better to round up the of a GE bond than to round up the of the IBM bond, but we cannot afford another GE bond. However we can afford another IBM bond. So we buy 303 IBM bonds and 616 GE bonds. This leaves us with $71.90 left over. A. To calculate a yield we need a price. We are given prices for both the IBM and the GE bonds. Feeding this information into our calculator should generate a yield of 7.000% As a reality check we notice that the 6% coupon bond is trading at a discount (the yield must be greater than 6%) and the 11% coupon bond is trading at a premium (the yield must be less than 11%), so 7% is reasonable. BOND enter calculate BOND enter calculate SDT = SDT = CPN = CPN = RDT = RDT = RV = RV = /Y 2/Y YLD = YLD = PRI = PRI = AI = AI =

4 OLTHETEN & WASPI : time Cash Flow NPV NPV/P Duration = (NPV/P)*t 2 $250,000. $250,000./(1.02) 2 = $240, $250,000. $250,000/(1.02) 4 = $230, $250,000. $250,000/(1.02) 6 = $221, $250,000. $250,000/(1.02) 8 = $213, A. The duration is semi-annual periods or years. B. $906, $906, C. The prevailing yield is 4%. This means that investors demand a fixed income investment of this risk profile to yield 4%. The 6% bond has a coupon that is based on 6% of the face value. Although there are lots of investors who would be willing to buy a 6% bond in a 4% environment, no-one would be willing to sell this bond unless he could sell it for more than the par value; the bond will sell at a premium sufficient to render the return on investment equivalent to a 4% bond. The 3% bond has a coupon that is based on 3% of the face value. No one would be willing to buy this bond unless he could sell it for less than the par value; the bond will sell at a discount sufficient to render the return on investment equivalent to a 4% bond. D % and 101 6% bonds 3% bond: 6% bond: $906, $ $906, $1, % bonds % bonds Number Issue price Market Value 101 6% bonds $1, $113, % bonds $ $793, You will ask the client to start the trust with 906,823.70

5 ANSWERS TO QUESTIONS & PROBLEMS E. Number Issue Coupon Payment Total Payment 101 6% bonds $30.00 $3, % bonds $ , Total Income paid into the trust 15, F. Number Issue Payment Total Payment 101 6% bonds $ , % bonds $ , cash $15, * (1.015) 1 $15, Cash Position 30, G. Asset price Market Value 101 6% bonds $1, , % bonds $ , Cash $30, TOTAL 943, Rate of Return $906, Out of the $943, we must pay out the first $250,000 and we will have $693, left to finance the remaining three $250,000 payments. H. We must pay out the $250,000 in scholarships but we have only $30, in the account. Obviously we must sell some bonds. But which ones? And how many? To figure this out we must re-immunize. Calculate the value and duration of the remaining payments. time Cash Flow NPV NPV/P 1 $0 Duration (NPV/P)*t 2 $250,000 $250,000./(1.02) 2 = $240, $0 4 $250,000 $250,000/(1.02) 4 = $230, $0 6 $250,000 $250,000/(1.02) 6 = $221, $693, We breathe a sigh of relief because we have $693,382.83,- which is slightly more than the $693, we calculate as required to meet the remaining obligations of the trust.

6 OLTHETEN & WASPI 2012 We recalculate the investment weights: % bond: 6% bond: $693, $ $693, $1, % bonds % bonds and sell bonds accordingly. Cash Reconciliation Price Cash Flow Cash in the account at the end of year 1 $ 30, Sell 273 3% bonds (we have 809 and we only need 536) $ $ 270, Buy 46 6% bonds (we have 101 and we need 147) $ 1, $ -50, SubTotal $ 250, Pay out scholarships $ -250, Cash in the account when we begin year 2 $ leaving the portfolio in the following situation: Asset price Market Value 147 6% bonds $1, , % bonds $ , Cash TOTAL 693,382.83