Ruckman 1/4.1–4.3
Chris Ruckman and Joe Francis, Chapter 1: "Interest Rates and Factors," Section 2.6: "Equations of Value," Section 4.1: "Non-Annual Interest and Discount Rates," Section 4.2: "Nominal pthly Interest Rates," and Section 4.3: "Nominal pthly Discount Rates," in Financial Mathematics , pp. 1–25, 95–106.
OUTLINE
I.
INTRODUCTION A. Definitions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Accumulated value factor – "accumulated value after t years of a deposit of $1" Annual effective interest rate – compound interest rate compounded annually Biannual a.k.a. semiannual interest rate – interest rate for one-half of a year Biennial interest rate – interest rate for a two-year period Capital – asset on which interest is earned Collateral – "something of value pledged as security against the risk of default" Compound interest – interest "earned on the initial deposit and the interest that has previously accrued" Compounding – process whereby interest is earned on interest Default risk – "risk that the borrower will not be able to repay the loan principal" Discount – interest "paid the beginning of the time period" Discounting – "process of allowing for future interest in determining a present value" Force of interest – "continuously compounded interest rate;" "instantaneous change in the account value, expressed as an annualized percentage of the current value;" "derivative of the accumulated value with respect to time expressed as a percentage of the accumulated value at time t" Future value – value "taking into account any interest that will be earned during the investment period" Interest – "payment by one party (the borrower) for the use of an asset that belongs to another party (the lender) over a period of time" Interest rate – Interest "expressed as a percent of the capital amount" Interest rate convertible pthly – interest rate compounded p times per year Nominal interest a.k.a. convertible rate – annualized rate produced "by multiplying the effective pthly rate of interest by the number of time periods, p" Present value – payment "discounted to a previous point in time, . . . taking into account any interest that will be earned during the investment period" Present value factor – "present value of a payment of $1 to be made in t years" Present value of $X payable in t years – "amount that, if invested now at an annual effective rate i, will accumulate to $X at time t years" Principal amount – monetary capital; "amount provided to the borrower when the loan is originated"
13. 14. 15. 16. 17. 18. 19. 20. 21.
Ruckman 1/4.1–4.3 23. 24. 25. Rate of discount – "amount of discount earned over the year divided by the ending accumulated value" Simple interest – interest "only earned on the initial deposit . . . [and not] on the interest that has previously accrued" Time value of money – principle that "a $1 payment now is worth more than $1 payable in one year's time"
B.
Symbol 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. AVt - accumulated value at time t AVFt - accumulated value factor at time t d - discount rate i - interest rate d(p) - nominal pthly discount rate i(p) - nominal pthly interest rate p, m - number of conversion periods per year PVt - present value at time t PVFt - present value factor at time t t - number of years v - one-year present value factor δ - constant force of interest δt - force of interest at time t
C.
Equations 1. Amount of interest earned from time t to time (t + s) Interest = AVt+s − AVt 2. Interest rate in effect from time t to time (t + 1) i = 3. AVt+1 − AVt AVt
AV of an investment X after t years with simple interest AVt = X(1 + ti)
4.
AV of an investment X after t years with compound interest AVt = X(1 + i)t
5.
Compound interest AV factor AVFt = (1 + i)t
6.
Simple interest AV value factor
Ruckman 1/4.1–4.3 AVFt = 1 + ti
Interest Rates
PAST CAS AND SoA EXAMINATION QUESTIONS A.
A1. A2. A3.
Simple and Compound Interest
A constant rate of simple interest implies a decreasing effective rate of interest. (86–4–1–1) A constant rate of compound interest implies a decreasing effective rate of interest. (86–4–1–1) A loan of $1,000 is made on July 1 and will be repaid with interest two months later on September 1. Interest will be calculated using the banker's rule and a simple interest rate of 20% per annum. What will be the total interest paid? A. < $33.75 E. ≥ $34.50 B. ≥ $33.75 but < $34.00 (87–4–2–1) C. ≥ $34.00 but < $34.25 D. ≥ $34.25 but < 34.50
A4.
Fund A is invested at an effective annual interest rate of 3%. Fund B is invested at an effective annual interest rate of 2.5%. At the end of 20 years, the total in the two funds is $10,000. At the end of 31 years, the amount in fund A is twice the amount in fund B. Calculate the total in the two funds at the end of 10 years. A. $5,732 B. $6,602 C. $7,472 D. $7,569 E. $8,123 (87F–140–18)
A5.
Two funds, A and B, start with the same amount. Fund A grows at an annual interest rate of i > 0 for n years, and at an annual interest rate of j > 0 for the next n years. Fund B grows at an annual interest rate of k > 0 for 2n years. Fund A equals 1.5 times fund B after n years. The amounts in the two funds are equal after 2n years. Which of the following are true? 1. j < k < i A. 1,2 B. 1,3 2. k < (i + j)/2 C. 2,3 D. 1,2,3 3. j = k(2/3)1/n E. None of these answers is correct. (88S–140–3)
A6.
Gertrude deposits $10,000 in a bank. During the first year, the bank credits an annual effective rate of interest (i − 5%). During the second year, it credits an annual effective rate of interest i. At the end of two years, she has $12,093.75 in the bank. What would Gertrude have in the bank at the end of three years if the annual effective rate of interest were (i + 9%) for each of the three years? A. $16,851 B. $17,196 C. $17,499 D. $17,936 E. $18,113 (88F–140–3)
A7.
An investor puts 100 into fund X and 100 into fund Y. Fund Y earns compound interest at the annual rate of j > 0, and fund X earns simple interest at the annual rate of 1.05j. At the end of two years, the amount in fund Y is equal to the amount in fund X. Calculate the amount in fund Y at the end of 5 years. A. 150 B. 153 C. 157 D. 161 E. 165 (90S–140–1)
A8.
At an annual effective interest rate of i, i > 0, the following are all equal: i) ii) iii) The present value of 10,000 at the end of 6 years The sum of the present values of 6,000 at the end of year t and 56,000 at the end of year 2t 5,000 immediately
Interest Rates Calculate the present value of a payment of 8,000 at the end of year (t + 3) using the same annual effective interest rate. A. 1,330 B. 1,415 C. 1,600 D. 1,775 E. 2,000 (90F–140–4)
Interest Rates Solutions are based on Broverman, pp. 1–14; Ruckman, pp. 1–11. A1. A2. A3. T. F – Substitute "constant" for "decreasing." Use the formula for ordinary simple interest : I = + (Principal)(Annual Rate)(Number of Days ) = (1,000)(.20)(3136031 ) 360 = 34.44
Answer: D A4. Solve two simultaneous equations for A and B and calculate the value of the funds at year ten: A(1.03)31/B(1.025)31 = 2 A(1.03)20 + B(1.025)20 = 10,000 B = 2,107.46 A = 3,624.73 Amount in Funds10 = (3,624.73)(1.03)10 + (2,107.46)(1.025)10 = 7,569 Answer: D A5. Set up two equations at t = n and t = 2n: (1 + i)n = (1.5)(1 + k)n 1. 2. 3. (1 + i)n (1 + j)n = (1 + k)2n
T – The first equation indicates that k < i and dividing the second equation by the first demonstrates that j < k. T – The first equation produces: i = (1 + k)(3/2)1/n − 1. Dividing the second equation by the first equation produces: j = (1 + k)(2/3)1/n − 1. Averaging these values produces a value that is greater than k. F – Dividing the second equation by the first equation gives us: j = (1 + k)(2/3)1/n − 1.
Answer: A A6. Solve for i and use to calculate the AV: (1 + i − .05)(1 + i) = 12,093.75/10,000 (1 + i)2 − (.05)(1 + i) − 1.209375 = 0 −(−.05) ± (−.05)2 − (4)(1)(−1.209375) 1+i = = 1.125 (2)(1) (10,000)(1.125 + .09)3 = 17,936 Answer: D A7. Solve for the interest rate and use the accumulation function: 1 + (2)(1.05j) = (1 + j)2 2.1j = 2j + j2 A(5) = A(0) a(5) = (100)(1.1)5 = 161 Answer: D A8. Equate ii) and iii) and using the interest rate to solve for vt. Equate i) and iii) to solve for v6. Then calculate the PV: j=.1
Interest Rates 5,000 = 6,000vt + 56,000v2t 56v2t + 6vt − 5 = 0 vt = .25 6 = .5 t v3 = (8,000)(.25) .5 = 1,414.21 v A(0) = A(t + 3) v Answer: B 5,000 = 10,000v6