Comprehensive Guide to Time Value of Money and Annuity Calculations
Time Value of Money
does not change(i.e. The interest rate is ignored), then the present value of a 10-year annuity-due with annual payments of $200 is: $2000
b) If interest is credited on the initial investment only, simple interest is being used.
c) The five basic quantities involved in an annuity problem are the beginning value, the ending value, the number of payments, the interest rate, & payment amount.
d) The present value of a future payment will decrease if you receive it later than expected. Assume a positive interest rate.
e)(an)/(aduen) = (1+i)-1
2) On July 1, 1984, a person invested 1000 in a fund for which the force of interest at time t is given by delta = (3+2t)/50, where t is the number of years since January 1, 1984. Determine the accumulated value of investment on January 1, 1985.Ans: 1046
3) What is the nominal annual rate of interest equivalent to a nominal annual rate of discount of 4%, both rates convertible quarterly? .0404
4) You will receive three cash flows: (i) $400, today (ii) $1000, 4 years from today (iii) $500, eight years from today. At an annual effect interest rate i, the combined present value of these three cash flows is $1505.80. At the same annual effective rate i, find the accumulated value of $60 after 3 years.
5) Tony just won the actuarial lottery. He is allowed to select one of three payout options in order to collect his winnings: (A) $575,000 payable six months from today. (B) $300,000 today plus an additional $300,000 payable two years from today (C) $20,500 paid at the beginning of each six-month period for 40 years (with the first payment coming six months later). The nominal rate of interest, convertible semiannually, is 8%. Select the option that results in the highest present value for Tony. Evaluate all 3 options and show all work.
6) Find the simple effective interest rate such that $85,000 will accumulate to $100,000 in two periods.
7) Jerry pays $27,506.28 for an annuity-immediate today. This annuity will pay him $1,750 at the end of each month for n months with the first payment coming at the end of the first month. If i(12), find n.
8) Frank, who turns 40 today, will deposit $2,500 in his empty retirement account at the end of every three months, with the first deposit coming 3 months from now and the final deposit coming on his 50th birthday. There are no other withdrawals or deposits. The nominal interest rate convertible quarterly, is 10%. (Hint: find the fund balance on 50th birthday)
9) The present value of 200 paid at the end of n years, plus the present value of 100 paid at the end of 2n years is 200. Determine the annual effective rate of interest.
Test 2 1)An annuity-due has monthly payments of 150, 200, 250, … , 800 at i(12)=.12 starting immediately. Find the present value of annuity.
2) You are given the following values: (i) a20=11.196 (ii) (Ia)20 = 95.360
3) Jerry bought a car with $10,000 loan. He paid off the loan completely in 36-months with $330 payments at the end of each month. What is th nominal rate of interest convertible monthly on this loan?
4) Bob’s home mortgage is amortized over 30 years using level payments at the end of each quarter (3 months) and an annual effective interest rate of 5 %. What percentage of his 27th payment is principal?
5) A family took home improvement loan for $25,000. Interest on the loan at a rate i(2) = 10% must be paid at the end of each six month period. Also at the end of each 6 month period payments are made into the sinking fund that earns j(2) = 6%. At the end of 5 years the balance in the sinking fund is the same as the value of loan. Find the total amount of money that family needs to pay every six months.
6) Evaluate (Iacont)5 = if delta = .06
7) A perpetuity pays 1 at the end of each year plus an additional 1 at the end of every second year. The present value of the perpetuity is K for i >= 0. Find the expression for K in terms of i.
8) Joanna deposits $200 at the beginning of every quarter in the bank account. Five years later the balance in the bank is 4,500. What was the annual effective rate of interest?
Test 3 1) A 1000 par value 10-year bond with coupons at 5% convertible semiannually is selling for $1081.78. The bond salesman’s method is used to approximate the yield rate convertible semiannually. Calculate the difference between the approximation and the exact value.
2) A 26-week T-bill is bought for $9500 at issue and will mature for $10,000. Find the yield rate computed as: a) A discount rate, using the typical method for counting days (actual/36) on a T-bill. b) An annual effective rate of interest, assuming the investment period is exactly half a year.
3) A loan is repaid with payments which start at $200 the first year and increase by $50 per year until a payment of $900 is made, at which time payments cease. If interest is 4% effective, find the amount of interest in the fifth payment.
4) Suppose you purchase a twenty-year bond with 6% semiannual coupons bought to yield 8.16% (annual effective). Assume a face value of $1,000. Further suppose that this bond is callable at $1080, but it may not be called until the end of the third year (after six coupon payments). After the third year, the annual effective interest rate has risen to 10.25%. Find the market value of the bond will be called at this point in time.
5) At the end of the semester, you decide to forego a ritualistic burning of actuarial textbooks. Instead, you sell Kellison’s Theory of Interest on Amazon Marketplace and make a net profit of $60. You use this money (and this money alone) to buy a 5-year zero-coupon bond with par value of $100. Find this bond’s yield-to-maturity.
6) Consider a five-year bond, with a $1,000 face value and a 6% coupon paid annually. The bond was bought to yield 4% annually. Find the market value of the bond using the theoretical method 3 months after the third coupon payment. That is, find Bm3.25.
7) A $700 par value five-year 10% bond with semiannual coupons is purchased for 670.60. The present value of the redemption value of 372.05. Calculate the redemption value.