Question

c. What annual interest rate would cause $100 to grow to $119.10 in 3 years? d. If a company's sales are growing at a rate of 10% annually, how long will it take sales to double? e. What's the difference between an ordinary annuity and an annuity due? What type of annuity is shown here? How would you change it to the other type of annuity? 100

Answer 1

(c) Present Value = PV = $100

Future Value = FV = $119.10

Number of Years = n = 3

Interest Rate = r

FV = PV(1+r)ⁿ

119.10 = 100(1+r)³

=> r = (119.10 /100)^1/3 - 1 = 0.06 = 6%

(d) Let Present Value = PV = x

Future Value = FV = 2x

Interest Rate = r = 10%

Let number of years = n

=> FV = PV(1+r)ⁿ

=> n = ln (FV/PV) / ln (1+r) = ln (2/1)/ln(1+0.10) = 7.27 years

(e) In an Ordinary annuity payments are made at end of each period

In an Annuity due payments are made at starting of each period

The given annuity is an ordinary annuity

For Ordinary Annuity, PV = P/(1+r) + P/(1+r)² +....+ P/(1+r)ⁿ = P[1- (1+r)^-n]/r

For Annuity Due, PV = P + P/(1+r) +....+ P/(1+r)^n-1 =P + P[1- (1+r)^-(n-1)]/r

The Present Value of both should be equal

Let the payment in ordinary annuity be P (P = 100)

Let the payment in annuity due be X

Number of years = n = 3

=> P[1- (1+r)^-n]/r = X + X[1- (1+r)^-(n-1)]/r

=> P[1- (1+r)^-n] = Xr + X[1- (1+r)^-(n-1)]

=> X = P[1- (1+r)^-n]/(r + [1- (1+r)^-(n-1)]) = 100[1- (1+r)^-3]/(r + [1- (1+r)^-2])