Annuity and Perpetuity Time Value of Money Annuity and Perpetuity

Q 1) Perpetuity Curly’s Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs $44,000 per year forever. A representative for Curly’s tells you the policy costs $690,000. At what interest rate would this be a fair deal? PV = C / r $690,000 = $44,000 / r   We can now solve for the interest rate as follows:   r = $44,000 / $690,000 r = .0638, or 6.38%

Q2) Present Value An investment will pay you $89,000 in four years. Assume the appropriate discount rate is 8.25 percent compounded daily.   What is the present value? For this problem, we need to find the PV of a lump sum using the equation:   PV = FV / (1 + r)t   It is important to note that compounding occurs on a daily basis. To account for this, we will divide the interest rate by 365 (the number of days in a year, ignoring leap year), and multiply the number of periods by 365. Doing so, we get:   PV = $89,000 / [(1 + .0825 / 365)4(365)] PV = $63,986.60

Q3) Annuity You are to make monthly deposits of $775 into a retirement account that pays an APR of 9.9 percent compounded monthly.   If your first deposit will be made one month from now, how large will your retirement account be in 30 years?  This problem requires us to find the FVA. The equation to find the FVA is:    FVA = C{[(1 + r)t – 1] / r} FVA = $775({[1 + (.099 / 12)]360– 1} / (.099 / 12)) FVA = $1,714,944.94

Q4)Annuity Beginning three months from now, you want to be able to withdraw $4,400 each quarter from your bank account to cover college expenses over the next four years.   If the account pays .84 percent interest per quarter, how much do you need to have in your bank account today to meet your expense needs over the next four years? The cash flows are an annuity with four payments per year for four years, or 16 payments. We can use the PVA equation:    PVA = C({1 – [1 / (1 + r)t]} / r) PVA = $4,400{[1 – (1 / 1.008416)] / .0084} PVA = $65,617.00

Q5)Annuity You want to buy a new sports car from Muscle Motors for $43,100. The contract is in the form of a 72-month annuity due at an APR of 6.35 percent.   What will your monthly payment be? We need to use the PVA due equation, which is:   PVAdue = (1 + r)PVA   Using this equation:   PVAdue = $43,100 = [1 + (.0635 / 12)] × C[{1 – 1 / [1 + (.0635 / 12)]72} / (.0635 / 12)] $42,873.13 = C[{1 – 1 / [1 + (.0635 / 12)]72} / (.0635 / 12)] C = $717.64   Notice, to find the payment for the PVA due, we find the PV of an ordinary annuity, then compound this amount forward one period.

Q6)PV and FV You are saving to buy a $191,000 house. There are two competing banks in your area, both offering certificates of deposit yielding 7.6 percent. How long will it take your initial $108,000 investment to reach the desired level at Second Bank, which compounds interest monthly? FV = PV(1 + r)t $191,000 = $108,000[1 + (.076 / 12)]t t = 90.31 months, or 7.53 years 

Q7)Annuity and Perpetuity Mary is going to receive a 31-year annuity of $8,600. Nancy is going to receive a perpetuity of $8,600.   If the appropriate interest rate is 9 percent, how much more is Nancy’s cash flow worth? Here, we need to find the difference between the present value of an annuity and the present value of a perpetuity. The annuity time line is: PVA = C({1 – [1 / (1 + r)t]} / r) PVA = $8,600({1 – [1 /(1.09)31]} / .09) PVA = $88,948.10 And the present value of the perpetuity is: PV = C / r PV = $8,600 / .09 PV = $95,555.56   So, the difference in the present values is:   Difference = $95,555.56 – 88,948.10 Difference = $6,607.46   There is another common way to answer this question. We need to recognize that the difference in the cash flows is a perpetuity of $8,600 beginning 32 years from now. We can find the present value of this perpetuity and the solution will be the difference in the cash flows. So, we can find the present value of this perpetuity as:   PVP = C / r PVP = $8,600 / .09 PVP = $95,555.56   This is the present value 31 years from now, one period before the first cash flows. We can now find the present value of this lump sum as:   PV = FV / (1 + r)t PV = $95,555.56 / (1 + .09)31 PV = $6,607.46   This is the same answer we calculated before.

Q8) Perpetuity Given an interest rate of 6.85 percent per year, what is the value at Year 7 of a perpetual stream of $3,600 payments that begin at Year 18? Here we need to find the present value of a perpetuity at a date before the perpetuity begins. We will begin by finding the present value of the perpetuity. Doing so, we find:   PV = C / r PV = $3,600 / .0685 PV = $52,554.74   This is the present value of the perpetuity at Year 17, one period before the payments begin. So, using the present value of a lump sum equation to find the value at Year 7, we find:   PV = FV / (1 + r)t PV = $52,554.74 / (1 + .0685)10 PV = $27,093.60