Question

For your job as the business reporter for a local​ newspaper, you are given the task of putting together a series of articles that explain the power of the time value of money to your readers. Your editor would like you to address several specific questions in addition to demonstrating for the readership the use of time value of money techniques by applying them to several problems. What would be your response to the following memorandum from your​ editor?

​To: Business Reporter

​From: Perry​ White, Editor, Daily Planet

​Re: Upcoming Series on the Importance and Power of the Time Value of Money

In your upcoming series on the time value of​ money, I would like to make sure you cover several specific points. In​ addition, before you begin this​ assignment, I want to make sure we are all reading from the same​ script, as accuracy has always been the cornerstone of the Daily

Planet.

In this​ regard, I'd like a response to the following questions before we​ proceed:

a.  What is the relationship between discounting and​ compounding?

b.  What is the relationship between the​ present-value factor and the annuity​ present-value factor?

c. 1. What will $6,700 invested for 26 years at 6 percent compounded annually grow to?

2. How many years will it take $370 to grow to $2,687.44 if it is invested at 11 percent compounded annually?

3. At what rate would $1,800 have to be invested to grow to $16,243.68 in 18 years?

d. Calculate the future sum of 1,200​, given that it will be held in the bank for 15 years and earn 17 percent compounded semiannually.

e. What is an annuity​ due? How does this differ from an ordinary​ annuity?

f. What is the present value of an ordinary annuity of $2,600 per year for 18 years discounted back to the present at 6 percent? What would be the present value if it were an annuity​ due?

g. What is the future value of an ordinary annuity of $2,600 per year for 18 years compounded at 6 percent? what would be the future value if it were an annuity​ due?

h. You have just borrowed $150,000, and you agree to pay it back over the next 30 years in 30 equal end-of-year payments plus 14% compound interest on the unpaid balance. What will be the size of these payments?

i.  What is the present value of a perpetuity of $1,600 per year discounted back to the present at 18%?

j.  What is the present value of an annuity of $1,900 per year for 10 years, with the first payment occuring at the end of year 10 (that is, ten $1,900 payments occurring at the end of year 10 through year 19), given a discount rate of 16%?

k. Given a discount rate of 9%, what is the percent value of a perpetuity of $1,800 per year if the first payment does not begin until the end of year​ 10?

​

​

Answer 1

a). Discounting and compounding are opposites. While discounting is used to bring back a future number to the present, compounding is used to take a present number to the future. In other words, if you take a number and compound it, then discount the compounded amount to the present, you will get the same number back.

b). Present value factor is the discount factor for a single cash flow whereas the annuity present value factor is the discount factor for a series of same cash flows made at regular time periods.

c-1). Compounded amount = initial amount*(1+ annual interest rate)^number of years

= 6,700*(1+6%)^26 = 30,480.87

c-2). Compounded amount = initial amount*(1+ annual interest rate)^number of years

number of years = ln(Compounded amount/initial amount)/ln(1+ annual interest rate)

= ln(2,687.44/370)/ln(1+11%) = 19

It will take 19 years.

c-3). Compounded amount = initial amount*(1+ annual interest rate)^number of years

annual interest rate = [(Compounded amount/initial amount)^(1/number of years)] -1

= [(16,243.68/1,800)^(1/18)] -1 = 13.00%

d). Annual interest rate = 17%; semi-annual rate = 17%/2 = 8.50%

Number of compoundings (n) = number of years*2 = 15*2 = 30

Future value = initial amount*(1+ semi-annual rate)^n

= 1,200*(1+8.5%)^30 = 13,869.90

e). Annuity due is an annuity in which payments start at the beginning of the time period unlike the ordinary annuity where payments are made at the end of the time period.

f). Present value (PV) of an ordinary annuity = P*(1 - (1+r)^-n)/r

where P = 2,600; r = 6%; n = 18

= 2,600*(1 - (1+6%)^-18)/6% = 28,151.77

PV of annuity due = [P*(1 - (1+r)^-n)/r]*(1+r)

= 28,151.77*(1+6%) = 29,840.88

g). Future value (FV) of an ordinary annuity = P*((1+r)^n -1)/r

= 2,600*((1+6%)^18 -1)/6% = 80,354.70

FV of annuity due = [P*((1+r)^n -1)/r ]*(1+r)

= 80,354.70*(1+6%) = 85,175.98