AS/AK/MATH2580 A AS/AK/MATH2580 A Assignment #4 Answers

1. Method 1: 40000000/(a20|0.03125) = \$2,719,796.28 per half year.

Method 2: Each bond costs 1000+(45-32.50)a20|0.0325 = \$1181.74. Selling the strip bond gives 1000(1.03)-20 = \$553.68. Thus the net cost of each bond (coupons only) is 1181.74-553.68 = \$628.06. Let n be the number that can be purchased. Then 628.06(n) = 40,000,000 so n = 63688.183 or n = 63,688 bonds. Thus the coupons at \$45 each bring in (63688)(45) = \$2,865,960.00 per half year.

Hence method 2 produces over \$146,000 more revenue per half year than method 1.

2. The price P0 on January 1, 2001 is, by Makeham's formula, F0+(0.04/0.03)(F-F0) where F0 = 1000000((1.03)-17+º+(1.03)-29) = 1000000(a29|0.03-a16|0.03) = \$6,627,352.56 and F = \$13,000,000. Hence P0 = \$15,124,215.81. Finally on May 6, 2001 the price P = P0(1+(0.03)(126-1)/(182-1)) = \$15,437,562.82.

3. At 7% the NPV is -1,400,000+200,000a11|0.07-100,000(1.07)-12 =\$55,333.67. Yes, at 7% this is a suitable investment.

At 8% the NPV is -1,400,000+200,000a11|0.08-100,000(1.08)-12 = -\$11,918.52. No, at 8% this is not a suitable investment.

We can estimate an IRR by doing linear interpolation between 7% and 8%.

Thus (i-7)/(8-7) = (0-55333.67)/(-11918.52-55333.67) = 0.823. Thus an IRR for this cash flow is 7.82%

4. KA = 125000 + (125000-18000)/(1.079-1) + 7900/0.07 = \$365,472.18.

KB = 175000 + (175000-25000)/(1.0712-1) + 9200/0.07 = \$426,218.55.

On a per ton basis: KA/39 = \$9371.08 and KB/45 = \$9471.52. Hence the type A truck is more economical.

5. Without the kit, K1 = 21000 + (21000-5000)/(1.069-1) + 1000/0.06 = \$60,872.60.

With a kit costing X, K2 = 21000+X + (21000+X-5000)/(1.0612-1) + 500/0.06 = \$45140.54 + 1.98795049X.

Let the number of units made per year without the kit be U. Then with the kit the units made per year is 1.1U. Thus on a unit basis: K1/U = K2/(1.1U) or 1.1K1 = K2. Thus X = \$10,975.79 is the amount one should be willing to pay for the kit.

1. R5 = (1/8)(35000-7000) = \$3500.00.
2. 35000(1-d)8 = 7000 so 1-d = 0.8177654 and d = 0.182235. Thus R5 = 35000(0.8177654)4(0.182235) = \$2852.43.

6. New car price in 4 years = 36000(1.05) = \$37800. Book value of her car in 4 years = 36000(1-d)4 where 36000(1-d)10 = 4000. Thus the book value in 4 years is 36000(4000/36000)4/10 = \$14948.77. She must save 37800.00 - 14948.77 = \$22851.23 in 4 years (= 48 months). Hence her monthly SF deposit should be 22851.23/s48|0.06/12 = \$422.41.

1. The probability a person aged 18 survives 20 years to reach age 38 is l38/l18. But this ratio depends on the mortality class: female nonsmoker, female smoker, male nonsmoker, male smoker. Using the appropriate table we get the four ratios (9871801/9956686) = 0.991474573, (9828627/9955889) = 0.987217415, (9761201/9938575) = 0.982152975, (9684598/9936586) = 0.974640385. The probability that all 25 students survive 20 years is (0.991474573)5(0.987217415)5(0.982152975)7(0.974640385)6 = 0.678879. Thus the probability that at least one dies by this time is 1 - 0.678879 = 0.3211.
2. The probability a person aged 18 survives 80 years to reach age 98 is l98/l18. As above, there are four ratios for the 4 classes of mortality. They are: 0.068030969, 0.041391984, 0.02035161, 0.009406551. The probabilities of dying before age 98 are one minus these values or: 0.931969031, 0.958608016, 0.97964839, 0.990593449. The probability that all 25 students die before age 98 is the product (0.931969031)5(0.958608016)5(0.97964839)7(0.990593449)6 = 0.4656686. The probability that at least one survives is one minus this or 0.5343.

7. If the payments are each R then the expected values of the payments are R(0.99), R(0.99)2, to R(0.99)48. To find the present value of these payments, introduce an interest rate i such that (1+i) = (1+0.09/12)/0.99. Hence i = 0.017676768. Thus 10000 = Ra48|0.017676768 and R = \$310.80 per month for 4 years.

8. 10000 = XD65/D32 so X = 10000D32/D65.

a. Using the male smoker mortality table with i = 6% we get X = 90188.04. The size of the endowment is \$90,188.04.

b. From a bank he would get 10000(1.06)33 = \$68,405.90.

c. Using the female nonsmoker tables at 6% gives and endowment of \$74,874.14.

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On 27 Oct 2000, 06:02.