final hw
Stat 274 - Hartman Final Exam - Page 1 of 3 15 April 2020 This exam is now a larger homework assignment. It is due 15 April at 2:00 pm. Please email your completed assignment to Mackenzie (
[email protected]) 1. What is the Macaulay duration of a level per- petuity with an annual interest rate of 3.42%. Assume annual end of year payments. [30.2398] A. Less than 25 B. At least 25, but less than 30 C. At least 30, but less than 35 D. At least 35, but less than 40 E. At least 40 2. The bank loans out 3000 each to 1500 individuals. They want to make back an average of 3600 on these loans but they expect that a certain number of these individuals will default. You are given the following: • 16 are expected to default with 45% recov- ery • 11 are expected to default with 23% recov- ery • 2 are expected to default with no recovery What is the adjusted amount the bank should require to account for these defaults? [3646.85] A. Less than 3620 B. At least 3620, but less than 3630 C. At least 3630, but less than 3640 D. At least 3640, but less than 3650 E. At least 3650 3. For an n-year bond the ratio of the per-period ef- fective interest rate to the modified coupon rate is 6/11. The present value of the redemption value is 3000. vn = 0.766416732. What is the price of the bond? [4676.25] A. Less than 4675 B. At least 4675, but less than 4677 C. At least 4677, but less than 4679 D. At least 4679, but less than 4681 E. At least 4681 4. You invest 200 at time 0 into an account which pays a force of interest of t2/100. At what time will the account balance grow to 350? [5.52] A. Less than 5 B. At least 5, but less than 5.2 C. At least 5.2, but less than 5.4 D. At least 5.4, but less than 5.6 E. At least 5.6 5. A perpetuity pays 10 at time 1, (10 · 1.02) + 5 at time 2, (10 · 1.022) + 2 · 5 at time 3, continu- ing in that pattern where the time t payment is 10(1.02t−1) + 5(t − 1). Using i = 0.03, find the present value of this perpetuity. [6555.56] A. Less than 6750 B. At least 6750, but less than 7000 C. At least 7000, but less than 7250 D. At least 7250, but less than 7500 E. At least 7500 6. You give your children a perpetuity which pays X at the beginning of every year. Abby takes the first 15 payments and Sophie gets the rest. The present value of each child’s share is equal. What is the annual effective interest rate? [0.0473] A. Less than 0.02 B. At least 0.02, but less than 0.03 C. At least 0.03, but less than 0.04 D. At least 0.04, but less than 0.05 E. At least 0.05 7. You buy a unique annuity which pays 1 and the end of each month for the first 12 months, 2 at the end of each month for the next 12, 3 for the next 12, increasing arithmetically until it pays 70 at the end of each month for months 829-840. Assuming a nominal interest rate of 0.12 convert- ible monthly, calculate the present value of this annuity. [886.64] A. Less than 875 B. At least 875, but less than 880 C. At least 880, but less than 885 D. At least 885, but less than 890 E. At least 890 8. You take out a loan and will repay it with 17 an- nual payments of X. Immediately before the 7th payment, the loan balance is p · X. The annual effective interest rate is 0.06. Calculate p. [8.36] Stat 274 - Hartman Final Exam - Page 2 of 3 15 April 2020 A. Less than 8.2 B. At least 8.2, but less than 8.3 C. At least 8.3, but less than 8.4 D. At least 8.4, but less than 8.5 E. At least 8.5 9. You borrow 10,000 and repay it with 48 monthly payments of 250. You miss the 12th and 19th payments. Calculate the outstanding loan balance immediately after the 15th payment. [7516.38] A. Less than 7480 B. At least 7480, but less than 7500 C. At least 7500, but less than 7520 D. At least 7520, but less than 7540 E. At least 7540 10. Find the Macaulay duration of a 15-year mort- gage for X at an annual effective interest rate of 0.06. [6.463] A. Less than 6.1 B. At least 6.1, but less than 6.3 C. At least 6.3, but less than 6.5 D. At least 6.5, but less than 6.7 E. At least 6.7 11. Calculate the modified convexity for an asset which pays 500 at time 3 and 625 at time 7. As- sume i = 0.06. [30.163] A. Less than 25 B. At least 25, but less than 27 C. At least 27, but less than 29 D. At least 29, but less than 31 E. At least 31 12. You owe a liability of 1000 at time 6. You have one asset which pays X at time 4 and another as- set which pays Y at time 9. Under this scenario and i = 0.04, your liability is fully immunized. Calculate X − Y . [104.78] A. Less than 60 B. At least 60, but less than 80 C. At least 80, but less than 100 D. At least 100, but less than 120 E. At least 120 13. You enter into a two-year deferred, three-year swap with a notional amount of 1,000,000 and the following yield curve. y1 y2 y3 y4 y5 0.032 0.035 0.039 0.042 0.043 Calculate the swap rate. [0.04837] A. Less than 0.045 B. At least 0.045, but less than 0.05 C. At least 0.05, but less than 0.055 D. At least 0.055, but less than 0.06 E. At least 0.06 14. You purchase an annuity which pays 20 at the end of each quarter for the first year, 19 at the end of each quarter for the second year, and so on until it pays 1 at the end of each quarter for the twentieth year. If the annuity is priced with an annual effective rate of 0.03, what is the fu- ture value of the annuity (at the end of year 20)? [1247.37] A. Less than 1000 B. At least 1000, but less than 1100 C. At least 1100, but less than 1200 D. At least 1200, but less than 1300 E. At least 1300 15. You finance the purchase of a 280,000 home with a 15-year mortgage. Your level monthly pay- ments are 1950. At some time during the lifetime of the loan, you miss two consecutive payments without realizing your mistake. The total pay- ment you must make at the end of the 15th year to pay off the loan is 7452.32. At the end of which month did the 1st missed payment occur? [47] A. Less than 45 B. At least 45, but less than 50 C. At least 50, but less than 55 D. At least 55, but less than 60 E. At least 60 16. A bond with annual coupons and unknown face value is sold for 1089.38. The ratio of the annual effective yield rate to the nominal coupon rate is 1.6. You are given vnj = 0.27. Find the par value of this bond. [1500.01] A. Less than 1450 B. At least 1450, but less than 1490 Stat 274 - Hartman Final Exam - Page 3 of 3 15 April 2020 C. At least 1490, but less than 1530 D. At least 1530, but less than 1570 E. At least 1570 17. A bond with quarterly coupons is sold to yield 6% nominal annual compounded quarterly for 5000. The ratio of the principal to interest portion of the first coupon payment is 23 . How much is each coupon payment? [125] A. Less than 80 B. At least 80, but less than 100 C. At least 100, but less than 120 D. At least 120, but less than 140 E. At least 140 18. You have liabilities of 1000 in 3 years, 800 in 2 years, and 600 in 1 year. You would like to ex- actly match your liabilities using the following three bonds currently available: Bond A is a 3- year bond with 10% annual coupons. Bond B is a 2-year bond with 5% annual coupons. Bond C is a zero-coupon bond that yields 3%. What face value of bond B will you purchase? [675.32] A. Less than 700 B. At least 700, but less than 800 C. At least 800, but less than 900 D. At least 900, but less than 1000 E. At least 1000 19. You decide to invest 1 at times t = 0, 1, 2, 3, 4, and 5. You put this money each year into a fund that earns interest at the force of interest given by: δt = 1 10 − t Determine how much money is in the fund just after the deposit made at time t = 5. [9] A. Less than 8.5 B. At least 8.5, but less than 9.5 C. At least 9.5, but less than 10.5 D. At least 10.5, but less than 11.5 E. At least 11.5 20. A loan is amortized over 8 years with quarterly payments at an annual nominal interest rate of 8% compounded quarterly. The first payment is 500 and is to be paid three months from the date of the loan. Each succeeding quarterly payment will be 15 less than the prior payment. Calculate the outstanding loan balance immedi- ately after the 27th payment. [309.17] A. Less than 200 B. At least 200, but less than 250 C.