Ps Bonds Solutions

8/3/2019 Ps Bonds Solutions

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Problem 1)

Part 1)

You have to set money aside for two payments, one at the end of the first and one at the end of the second year.For the first year's payment you need the present value of $20,000 discounted seminannually at 8%:

18,491.12$

This is just the present value of $20,000 discounted at 8% semiannually. For the second payment we use:20,000 / (1.04)^4 =

17,096.08$

Hence, the total amount you need is:

35,587.21$

Part 2 -- Invest in 4 year Zero Coupon Bonds

Since the value today is $35,587.21 then at 8% we need to buy a par amount equal to35,587.21*1.04^8 = $48,703.55

We sell $20,000 worth of bonds at the end of the first year and make a second payment at

the end of the second year.

Hence, at the end of year 1 and after making the payment the amount left in the bond account is:

Year 1=$48,703.55/(1+yield/2)^6-$20,000

The question assumes only a one-off change in yield, hence the value of the portfolio at the end of

year 2 is (Year1)*((1+yield/2))^2. Deducting the 20,000 fee payment gives:

Rate Year 1 Year 2 Year 4 7% 19,620.37 1,017.83 48,703.55$ Rates went down, we get extra cash8% 18,491.12 0.00 48,703.55$ Rates stay constant, amount is right.9% 17,399.25 -999.58 48,703.55$ Rates went up, we need more cash

Your bonds are too long, hence you are subject toprice risk.

Hence, if interest rates increase by 1%, you are $999.59 short, otherwise you have $1017.83 left over.You could also do the computation another way. Work out the par value of the bonds you need to sellat the end of year 1, then work out the remaining par value. Finally, discount this at the prevailing yield:

Rate Par valuesold

Par valueremaining

Value at theend of year 2

Balanceafter 2ndpayment

7% 24,585.11 24,118.45 21,017.83 1,017.838% 25,306.38 23,397.17 20,000.00 0.009% 26,045.20 22,658.35 19,000.42 -999.58

Discussion:

Think of a time line with 8 periods:

The intuition is that you need to save the present value of your future cash outlays:20,000 in year 1 (period 2) and 20,000 in year 2 (period 4). When we invest in the 4 yearbond we can compute the future value of the bond using the interest rate that we havelocked in (8%). We own the bond today (time 0). One year from today we're going toneed to sell a fraction of our bond holding to make the first payment, to figure outthe price of our bond in year one, we must discount the final cash flow to the bond (48,703.55)back to year one (discount 6 periods). After paying out 20,000, we need to future value what'sleft over, one year (2 periods) ahead to the next payment of 20,000 in year 2.The interest rate change does not affect the future value of our 4 year bond. We have locked in8% for four years by purchasing the bond. What the interest rate change does affect is therate at which we discount the future cash flow to the bond. When the interest rate falls, we're

0 1 2 3 4 5 6 7 8



happy because we have locked in a higher rate with the bond. So, we'll end up with a bitextra after our expenses. The exact opposite occurs when the interest rate increases becauusenow we have locked in at a rate that is lower than the prevailing market rate and we wind up witha shortfall.

Part 3 -- Invest in One Year Zero Coupon Bonds (then reinvest in one year)

Since the value today is $35,587.21 then the value of the portfolio at the end of year 1 is: 35,587.21*1.04^2.Hence, the par value is equal to $38,491.13. Of these, $20,000 have to be withdrawn forthe fee payment, the remainder is reinvested at the current yield.

ReinvestmentRate

Year 1Value

Year 2Value

7% 18,491.13$ ($191.84) Rates went down, we need more $8% 18,491.13$ $0.00 Rates stay constant, amount is just right.9% 18,491.13$ $192.77 Rates went up, we have extra cash.

Your bonds are too short, hence you are subject toreinvestment risk.

Hence, if the yield at which you reinvest is only 7%, you are $191.84 short at the end of year 2, otherwise youhave $192.77 left over if yields have increased.

Discussion:Note that the result is the exact opposite as part 2. Here we haven't locked in a rate beyond year 1. So when interestrates increase, we can take advantage by reinvesting at the end of the first year to take advantage of the higher rate.When interest rates fall, however, we're going to come up short because we have only secured the higher rate of 8%for one year.



Problem 2)

We have a choice of loans.

1. Receive $2,000,000 now. Pay back loan in one year at 8% interest.2. Receive $2,000,000 now. Pay back loan in two years, where interest accrues at 7.5% per year.

In one year, we receive enough money to pay off the one year loan.We have the option to lock in a one year rate of 7% in one year.

Which method should we use?

Look at the cash flows. Under option 1, we have a positive cash flow of $2,000,000 now,and no cash flows in the future (in one year, payment equals income).

Transaction/Period 0 1 2

Short term loan 2.000 -2.160 0.000Cash flow 2.160 0.000

Total 2.000 0.000 0.000

Under option 2, we have the following cash flows:

In year 0, we have a cash flow of $2,000,000In year 1, we get $2,160,000 ($2MM at 8%)-- but we reinvest it at 7%

In year 2, we get the money we invested the year before = $2,160,000 * 1.07 = $2,311,200.In year 2, we also have to pay back the $2MM plus interest. This is $2MM * 1.075^2 = $2,311,250

Transaction/Period 0 1 2

Long-term loan 2 0 -2.31125Forward 0.000 -2.160 2.311

Cash flow 2.160 0.000Total 0.000 0.000 -0.00005

This means we have to pay an extra $50 in two years under this option, so we should take option 1 .

The difference is really small, but it is important to notice that earning 7% over one and 8% over anotherperiod is not the same as earning the average, i. e. 7.5%, over both periods.

We could solve this problem another way. We could find the one year forward rate.We know that (1+r1)(1+f) = (1+r2)^2

So -- (1.08)(1+f) = (1.075)^2 or (1+f) = (1.155625)/1.08 = 1.07002315

This means that the forward rate is 7.002315%. This is slightly above the 7% we can lock in.Since we would have to pay a higher rate than we can get, we are better off taking the shortterm loan & paying it off when we get our money.



Problem 3)

Using the formula for the valuation of the coupon bond we get:Semiannual coupon 3%Principal 100Semiannual yield 6%

No. of periods 24

3*[(1-1.06-24)/0.06] + 100/1.0624 = $62.35



Problem 4)

We need to find the future value that is equivalent to receiving $10,000 every month for 12 months.

Using the formula for the future value of an annuity we get:

year-end salary = ((1.0112 - 1)/0.01)*$10,000 = $126,825.03

where the annuity is received monthly and 0.01 is the monthly periodic interest rate. Hence the year-end salary you demand is equal to $126,825.



Problem 5

Problem 5)

5a.

(1) par 1000

C 5.00%t 7m 2

(2) c 2.50%n 14R 6.00%

(3) i 3.00%

(4) Z 0.970873786(5) Z_14 0.661117806

(6) A_14=(1/i)*(1-Z_14) 11.29607314

(7) Annuity Value 282.40$ (6)*(2) $282.40

(8) Principal Value 661.12$ (5)*(1) $661.12

Bond Value 943.52$ (7)+(8) $943.52

5b.(1+0.03)^2-1 = 6.090%

5c.

Here, only the principal repayment matters:

Z_14(r=6%)*Par = 661.12$

Note that this has to be the same as the calculation of the principal value in (8) (or C17) above.

5d.

(1) par 1000

C 5.00%t 7m 2

(2) c 2.50%n 14R 7.00%

(3) i 3.50%

(4) Z 0.966183575(5) Z_14 0.61778179

(6) A_14 10.92052028

(7) Annuity Value 273.01$ (6)*(2)

(8) Principal Value 617.78$ (5)*(1)

Bond Value 890.79$ (7)+(8)

and the zero coupon bond is worth:

Z_14(r=7%)*Par = 617.78$

Use formula Use EXCEL functions



Problem 6

Problem 6)

6a

Par 400,000.00$ (1)t 15

m 12n 180R 9.50%i 0.7917%Z 0.9921Z_180 0.2419A_180 95.7648 (2) Use EXCEL functions

Mth. Payment 4,176.90$ $400,000/A_180=(1)/(2) 4,176.90$

6bInterest Monthly Principal Interest

Period Principal Owed Payment Reduction Period Principal Owed0 400,000.00$1 398,989.77$ 3,166.67$ 4,176.90$ 1,010.23$ 91 266,079.28$ 2,122.72$ 2 397,971.54$ 3,158.67$ 4,176.90$ 1,018.23$ 92 264,008.85$ 2,106.46$ 3 396,945.25$ 3,150.61$ 4,176.90$ 1,026.29$ 93 261,922.02$ 2,090.07$

4 395,910.83$ 3,142.48$ 4,176.90$ 1,034.42$ 94 259,818.67$ 2,073.55$ 5 394,868.23$ 3,134.29$ 4,176.90$ 1,042.60$ 95 257,698.67$ 2,056.90$ 6 393,817.37$ 3,126.04$ 4,176.90$ 1,050.86$ 96 255,561.88$ 2,040.11$ 7 392,758.19$ 3,117.72$ 4,176.90$ 1,059.18$ 97 253,408.18$ 2,023.20$ 8 391,690.63$ 3,109.34$ 4,176.90$ 1,067.56$ 98 251,237.43$ 2,006.15$ 9 390,614.61$ 3,100.88$ 4,176.90$ 1,076.01$ 99 249,049.50$ 1,988.96$

10 389,530.08$ 3,092.37$ 4,176.90$ 1,084.53$ 100 246,844.24$ 1,971.64$ 11 388,436.96$ 3,083.78$ 4,176.90$ 1,093.12$ 101 244,621.52$ 1,954.18$ 12 387,335.19$ 3,075.13$ 4,176.90$ 1,101.77$ 102 242,381.21$ 1,936.59$ 13 386,224.69$ 3,066.40$ 4,176.90$ 1,110.50$ 103 240,123.16$ 1,918.85$ 14 385,105.41$ 3,057.61$ 4,176.90$ 1,119.29$ 104 237,847.24$ 1,900.98$ 15 383,977.26$ 3,048.75$ 4,176.90$ 1,128.15$ 105 235,553.30$ 1,882.96$ 16 382,840.18$ 3,039.82$ 4,176.90$ 1,137.08$ 106 233,241.20$ 1,864.80$

17 381,694.10$ 3,030.82$ 4,176.90$ 1,146.08$ 107 230,910.79$ 1,846.49$ 18 380,538.95$ 3,021.74$ 4,176.90$ 1,155.15$ 108 228,561.94$ 1,828.04$ 19 379,374.65$ 3,012.60$ 4,176.90$ 1,164.30$ 109 226,194.49$ 1,809.45$ 20 378,201.13$ 3,003.38$ 4,176.90$ 1,173.52$ 110 223,808.29$ 1,790.71$ 21 377,018.32$ 2,994.09$ 4,176.90$ 1,182.81$ 111 221,403.21$ 1,771.82$ 22 375,826.15$ 2,984.73$ 4,176.90$ 1,192.17$ 112 218,979.09$ 1,752.78$ 23 374,624.55$ 2,975.29$ 4,176.90$ 1,201.61$ 113 216,535.77$ 1,733.58$ 24 373,413.42$ 2,965.78$ 4,176.90$ 1,211.12$ 114 214,073.12$ 1,714.24$ 25 372,192.72$ 2,956.19$ 4,176.90$ 1,220.71$ 115 211,590.96$ 1,694.75$ 26 370,962.34$ 2,946.53$ 4,176.90$ 1,230.37$ 116 209,089.16$ 1,675.10$ 27 369,722.23$ 2,936.79$ 4,176.90$ 1,240.11$ 117 206,567.55$ 1,655.29$ 28 368,472.30$ 2,926.97$ 4,176.90$ 1,249.93$ 118 204,025.98$ 1,635.33$ 29 367,212.47$ 2,917.07$ 4,176.90$ 1,259.83$ 119 201,464.29$ 1,615.21$ 30 365,942.67$ 2,907.10$ 4,176.90$ 1,269.80$ 120 198,882.3$ 1,594.93$ 31 364,662.82$ 2,897.05$ 4,176.90$ 1,279.85$ 121 196,279.9$ 1,574.48$

32 363,372.83$ 2,886.91$ 4,176.90$ 1,289.98$ 122 193,656.9$ 1,553.88$ 33 362,072.64$ 2,876.70$ 4,176.90$ 1,300.20$ 123 191,013.1$ 1,533.12$ 34 360,762.15$ 2,866.41$ 4,176.90$ 1,310.49$ 124 188,348.4$ 1,512.19$ 35 359,441.28$ 2,856.03$ 4,176.90$ 1,320.87$ 125 185,662.6$ 1,491.09$ 36 358,109.96$ 2,845.58$ 4,176.90$ 1,331.32$ 126 182,955.5$ 1,469.83$ 37 356,768.10$ 2,835.04$ 4,176.90$ 1,341.86$ 127 180,227.0$ 1,448.40$ 38 355,415.61$ 2,824.41$ 4,176.90$ 1,352.48$ 128 177,476.9$ 1,426.80$ 39 354,052.42$ 2,813.71$ 4,176.90$ 1,363.19$ 129 174,705.0$ 1,405.03$ 40 352,678.44$ 2,802.92$ 4,176.90$ 1,373.98$ 130 171,911.2$ 1,383.08$ 41 351,293.58$ 2,792.04$ 4,176.90$ 1,384.86$ 131 169,095.3$ 1,360.96$ 42 349,897.75$ 2,781.07$ 4,176.90$ 1,395.82$ 132 166,257.1$ 1,338.67$ 43 348,490.88$ 2,770.02$ 4,176.90$ 1,406.87$ 133 163,396.4$ 1,316.20$ 44 347,072.87$ 2,758.89$ 4,176.90$ 1,418.01$ 134 160,513.0$ 1,293.55$ 45 345,643.63$ 2,747.66$ 4,176.90$ 1,429.24$ 135 157,606.8$ 1,270.73$ 46 344,203.07$ 2,736.35$ 4,176.90$ 1,440.55$ 136 154,677.7$ 1,247.72$

47 342,751.12$ 2,724.94$ 4,176.90$ 1,451.96$ 137 151,725.3$ 1,224.53$ 48 341,287.66$ 2,713.45$ 4,176.90$ 1,463.45$ 138 148,749.6$ 1,201.16$ 49 339,812.62$ 2,701.86$ 4,176.90$ 1,475.04$ 139 145,750.3$ 1,177.60$ 50 338 325 91$ 2 690 18$ 4 176 90$ 1 486 72$ 140 142 727 2$ 1 153 86$



Problem 6

64 316,232.64$ 2,516.65$ 4,176.90$ 1,660.25$ 154 97,803.4$ 801.00$ 65 314,559.25$ 2,503.51$ 4,176.90$ 1,673.39$ 155 94,400.8$ 774.28$ 66 312,872.61$ 2,490.26$ 4,176.90$ 1,686.64$ 156 90,971.2$ 747.34$ 67 311,172.62$ 2,476.91$ 4,176.90$ 1,699.99$ 157 87,514.5$ 720.19$ 68 309,459.17$ 2,463.45$ 4,176.90$ 1,713.45$ 158 84,030.5$ 692.82$

69 307,732.16$ 2,449.89$ 4,176.90$ 1,727.01$ 159 80,518.8$ 665.24$ 70 305,991.47$ 2,436.21$ 4,176.90$ 1,740.69$ 160 76,979.3$ 637.44$ 71 304,237.01$ 2,422.43$ 4,176.90$ 1,754.47$ 161 73,411.9$ 609.42$ 72 302,468.65$ 2,408.54$ 4,176.90$ 1,768.36$ 162 69,816.1$ 581.18$ 73 300,686.30$ 2,394.54$ 4,176.90$ 1,782.36$ 163 66,192.0$ 552.71$ 74 298,889.83$ 2,380.43$ 4,176.90$ 1,796.47$ 164 62,539.1$ 524.02$ 75 297,079.14$ 2,366.21$ 4,176.90$ 1,810.69$ 165 58,857.3$ 495.10$ 76 295,254.12$ 2,351.88$ 4,176.90$ 1,825.02$ 166 55,146.3$ 465.95$ 77 293,414.65$ 2,337.43$ 4,176.90$ 1,839.47$ 167 51,406.0$ 436.58$ 78 291,560.62$ 2,322.87$ 4,176.90$ 1,854.03$ 168 47,636.1$ 406.96$ 79 289,691.91$ 2,308.19$ 4,176.90$ 1,868.71$ 169 43,836.3$ 377.12$ 80 287,808.40$ 2,293.39$ 4,176.90$ 1,883.50$ 170 40,006.4$ 347.04$ 81 285,909.99$ 2,278.48$ 4,176.90$ 1,898.42$ 171 36,146.3$ 316.72$ 82 283,996.54$ 2,263.45$ 4,176.90$ 1,913.44$ 172 32,255.5$ 286.16$ 83 282,067.95$ 2,248.31$ 4,176.90$ 1,928.59$ 173 28,334.0$ 255.36$ 84 280,124.09$ 2,233.04$ 4,176.90$ 1,943.86$ 174 24,381.4$ 224.31$ 85 278,164.84$ 2,217.65$ 4,176.90$ 1,959.25$ 175 20,397.5$ 193.02$ 86 276,190.08$ 2,202.14$ 4,176.90$ 1,974.76$ 176 16,382.1$ 161.48$ 87 274,199.68$ 2,186.50$ 4,176.90$ 1,990.39$ 177 12,334.9$ 129.69$ 88 272,193.53$ 2,170.75$ 4,176.90$ 2,006.15$ 178 8,255.6$ 97.65$ 89 270,171.50$ 2,154.87$ 4,176.90$ 2,022.03$ 179 4,144.1$ 65.36$ 90 268,133.46$ 2,138.86$ 4,176.90$ 2,038.04$ 180 (0.0)$ 32.81$

6c

annual bonus = 200,000Interest Monthly Principal Interest

Period Principal Owed Payment Reduction Period Principal Owed0 400,000.00$1 398,989.77$ 3,166.67$ 4,176.90$ 1,010.23$2 397,971.54$ 3,158.67$ 4,176.90$ 1,018.23$

3 396,945.25$ 3,150.61$ 4,176.90$ 1,026.29$4 395,910.83$ 3,142.48$ 4,176.90$ 1,034.42$5 394,868.23$ 3,134.29$ 4,176.90$ 1,042.60$6 393,817.37$ 3,126.04$ 4,176.90$ 1,050.86$7 392,758.19$ 3,117.72$ 4,176.90$ 1,059.18$8 391,690.63$ 3,109.34$ 4,176.90$ 1,067.56$9 390,614.61$ 3,100.88$ 4,176.90$ 1,076.01$

10 389,530.08$ 3,092.37$ 4,176.90$ 1,084.53$11 388,436.96$ 3,083.78$ 4,176.90$ 1,093.12$12 187,335.19$ 3,075.13$ 204,176.90$ 201,101.77$13 184,641.36$ 1,483.07$ 4,176.90$ 2,693.83$14 181,926.21$ 1,461.74$ 4,176.90$ 2,715.15$15 179,189.56$ 1,440.25$ 4,176.90$ 2,736.65$16 176,431.24$ 1,418.58$ 4,176.90$ 2,758.31$17 173,651.09$ 1,396.75$ 4,176.90$ 2,780.15$

18 170,848.93$ 1,374.74$ 4,176.90$ 2,802.16$19 168,024.58$ 1,352.55$ 4,176.90$ 2,824.34$20 165,177.88$ 1,330.19$ 4,176.90$ 2,846.70$21 162,308.64$ 1,307.66$ 4,176.90$ 2,869.24$22 159,416.68$ 1,284.94$ 4,176.90$ 2,891.96$23 156,501.83$ 1,262.05$ 4,176.90$ 2,914.85$24 153,563.91$ 1,238.97$ 4,176.90$ 2,937.93$25 150,602.72$ 1,215.71$ 4,176.90$ 2,961.18$26 147,618.10$ 1,192.27$ 4,176.90$ 2,984.63$27 144,609.84$ 1,168.64$ 4,176.90$ 3,008.26$28 141,577.77$ 1,144.83$ 4,176.90$ 3,032.07$29 138,521.70$ 1,120.82$ 4,176.90$ 3,056.07$30 135,441.43$ 1,096.63$ 4,176.90$ 3,080.27$31 132,336.77$ 1,072.24$ 4,176.90$ 3,104.65$32 129,207.54$ 1,047.67$ 4,176.90$ 3,129.23$

33 126,053.53$ 1,022.89$ 4,176.90$ 3,154.01$34 122,874.56$ 997.92$ 4,176.90$ 3,178.97$35 119,670.42$ 972.76$ 4,176.90$ 3,204.14$36 116 440 91$ 947 39$ 4 176 90$ 3 229 51$



Problem 6

50 68,448.95$ 570.44$ 4,176.90$ 3,606.46$51 64,813.94$ 541.89$ 4,176.90$ 3,635.01$52 61,150.15$ 513.11$ 4,176.90$ 3,663.79$53 57,457.36$ 484.11$ 4,176.90$ 3,692.79$54 53,735.33$ 454.87$ 4,176.90$ 3,722.03$

55 49,983.84$ 425.40$ 4,176.90$ 3,751.49$56 46,202.64$ 395.71$ 4,176.90$ 3,781.19$57 42,391.51$ 365.77$ 4,176.90$ 3,811.13$58 38,550.21$ 335.60$ 4,176.90$ 3,841.30$59 34,678.51$ 305.19$ 4,176.90$ 3,871.71$60 30,776.14$ 274.54$ 4,176.90$ 3,902.36$61 26,842.89$ 243.64$ 4,176.90$ 3,933.25$62 22,878.50$ 212.51$ 4,176.90$ 3,964.39$63 18,882.72$ 181.12$ 4,176.90$ 3,995.78$64 14,855.31$ 149.49$ 4,176.90$ 4,027.41$65 10,796.02$ 117.60$ 4,176.90$ 4,059.29$66 6,704.59$ 85.47$ 4,176.90$ 4,091.43$67 2,580.76$ 53.08$ 4,176.90$ 4,123.82$68 -$ 20.43$ 2,601.20$ 2,580.76$

The mortage will be fully paid down at the end of the 68th month (5 yrs & 8 mths).

You can determine this analytically as follows. You need the present value of an annuity that pays $4,176.90 per month. When is this to the amount of the mortgage still outstanding which is $187,335.19? Start by writing the annuity factor as

An = (P/i)*(1-Zn)=Vwhere:An = n-period annuity factorP = monthly payment 4,176.90$V = Amount outstanding at end of period 12 187,335.19$Zn = (1/(1+i))n = Z1nZ1 = one period discount factor 0.9921I = per period interest rate 0.0079

Then solve for Zn:Zn = Z1n = 1-(iV/P) 0.64

Take logs on both sides:n*log(Z1) = log(1-(iV/P) -0.43860563

Divide by log (Z1):n = log(1-(iV/P))/log(Z1) 55.62183085 You can also use EXCEL functions: 55.621830

Take the highest integer exceeding this: 56 5Add the first 12 months 68



Problem 7

Problem 7)

7a

0r1 = 0.05

1f2

= 0.06

Invest @ 1 yr spot and reinvest @ current 1 yr forward:

100*(1.05)*(1.06) = $111.30 at the end of two years

7b

assume that $100*(1+0r2)2=$111.3

So, 0r2=(111.3/100)^(1/2)-1= 5.4988%

Double check your result. Investing $100 at the rate of 5.4988% gives:

100*1.0549882= $111.30

hence, investing at 5.4988% for two years yields the same as investing at 5% now for one year, and investing forward at 6% for the follow



Problem 8)

First, we need to find the monthly payment on the old mortgage. Since the APR is 11%, the monthly effective rate is 11%* 1/1 0.92%We need to solve for the monthly payment C where:

Solving for the monthly payment C gives:

$1,476.10

Second, we need to find the amount of the mortgage we need to pay off after 10 years of payments (i.e., in 1999).The calculation below gives an amount of 143,006.96.

$143,006.96

Including the 5% additional cost of refinancing, the couple will need to take out a new mortgage of $143,006.96* #########

Third, we need to find the monthly payment on this new 20-year mortgage. At 9% monthly compounding the monthly rate is 9 0.75%Using the annuity formula we need to solve for a new monthly payment C:

$1,351.00

Since this payment is less, the couple should refinance.

C*[(1-1.0075-240

)/0.0075] = 150,157.31

Monthly Payment:

C*[(1-1.0092-360

)/0.0092] = 155,000

Monthly Payment:

1,476*[(1-1.0092-240

)/0.0092] = ?

New Amount:



Problem 9)

we are given the following information on zero coupon bonds with $100 face value:

Bond Maturity Price

(in years) (in $)

A 1 95.24B 2 89.85C 3 83.96

First, write down the cash flows generated by the coupon bond:

In one year In two years Maturity$5,000 $5,000 $105,000

The price of the 5% coupon bond with three years to maturity is then just a portfolio of the zero coupon bonds:

Cash Flow Maturity Number of PV

(in years) zero coupon bonds (in $)$5,000 1 50 4762$5,000 2 50 4493

$105,000 3 1050 88158Price: 97413



Problem 10)

Bond

ABC

(a) The forward rate between year 1 and 2 is given by ((1+r2)2 /(1+r1))-1 = (1.0552 /1.05) - 1 = 6.00%

(b) The forward rate between year 2 and 3 is given by ((1+r3)3 /(1+r2)

2)-1 = 1.06

3 /1.055

2- 1 = 7.01%

(c) The forward rate between year 1 and 3 is given by ((1+r3)3 /(1+r1))

1/2- 1 = (1.06

3 /1.05)

1/2- 16.50%

In total, we have the following table:

Bond Maturity Spot Rate(in years) in % per year

A 1 5.00 - -B 2 5.50 6.00C 3 6.00 7.01

Maturity (in years) Spot Rate (in % per year)

1 5.0%

Forward Rate(in % per year)

6.50

2 5.5%3 6.0%



Problem 11)

Option #1: $54,000

5

6.0%1260

0.50%

$1,043.97

Option #2: $58,000

5

5.5%

1260

0.46%

$1,107.87

Decision:

Number of compounding periods denoted by N:The effective preiodic interest rate denoted by i is:

The monthly payment, denote by P, is given by:

Therefore, P =

The cost taking into account the $4,000 discount:# of years of financing which we denote by t:

Financing interest rate which we denote by R:Compounding frequency which we denote by m:

Go with the option 1 and take the $4,000 discount.

Number of compounding periods denoted by N:The effective preiodic interest rate denoted by i is:

The monthly payment, denote by P, is given by:

Therefore, P =

The cost:# of years of financing which we denote by t:

Financing interest rate which we denote by R:

Compounding frequency which we denote by m:

Pú

û

ùêë

é -=

-

005.

)005.1(1000,54

60

Púû

ùêë

é -=

-

00458333.

)00458333.1(1000,58

60



Problem 12)

Step 1: First value the bond at the time of the first coupon:

Face ($ mil) 2

T 6m 2N 12R 6.00%i 3.00%coupon rate c 6% hence, the semi-annual coupon is 0.12 in $ mil.

In three months the present value of all future cashflows (in millions of dollars) will be:

$2.717240

Step 2: Discount this back 3 months to the present:

$2.677376 mil

Discussion:The strategy to take is to find the present value of the bond as of 3 months from today, and then, as a last step,discount back half a period to today. In 3 months, we're going to receive a coupon for 120,000. Six months afterthat we will receive another coupon for 120,000, and so for another 6 years until maturity when we receive ourfinal coupon plus the principal. To value the bond as of 3 months from today, we just need to apply the formula

for valuing a straight coupon bond and take into account that we'll also receive 120,000. That is the first formulaabove (which does everything in millions of dollars). This formula gets us the value of the bond as of three monthsfrom today. The last step is to take this value and discount it back one half a period, or 3 months. That is the secondequation that produces the final result.

=

5.0)03.1(

71724.2

=+úû

ùêë

é -+

-

12

12

)03.1(

212.0

03.

)03.1(112.0



Problem 13)

(a) The bond is selling at par. This means that the coupon rate equals the current yield of the bond.Hence the annually compounded yield is 8%.

(b) The Macaulay duration equals 2.78 (computed according to the table below).

Time Payment Yield PV TxPV(in $) (in % per year)

1 8 8 7.41 7.41

2 8 8 6.86 13.723 108 8 85.73 257.20

100.00 278.33

2.782.58

(c) At R=8%, P=$100. At R=7.99%, P=$100.0258. The DV01 is then given by $100.0258-$100=$0.0258.For a change of 10 basis points, we have a price change of $0.258.

3

Sum

Macaulay Duration (in years)Modified Duration

Period

1

2



Problem 14)

The Macaulay duration of zero-coupon bonds is equal to their maturity (2 years) because allcash-flows occur at the same time. The duration of the coupon bonds is given in the question(12 years). If interest rates change by 50 basis points, the value of the zeros will change by

Duration(2-year bonds)/(1+Yield(2-year bonds)) times the change in the yield, that is, (2/1.06)*(50/100)=0.9434%,

which is $9,434. Similarly, the value of the coupon bonds will change byDuration(Long-term bonds)/(1+Yield(Long-term bonds))times the change in the yield, that is, (12/1.07)*(50/100)=5.6075%, which is $56,075.In both cases the value of the bonds changes in the opposite directions to the interest rates.

In particular, if interest rates increase by 50 basis points, the value of the firms' liabilities decreases by $9,434 andthe value of its assets decreases by $56,075.

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