Ralph buys a perpetuity due paying 500 annually. He deposits the payments into a savings account earning interest at an effective annual rate of 10%. Ten years later, before receiving the eleventh payment, Ralph sells the perpetuity based on an effective annual interest rate of 10%. Using the proceeds from the sale plus the money in the savings account, Ralph purchases an annuity due paying X per year for 20 years at an effective annual rate of 10%. Calculate X.

X = 1523

Explanation

Perpetuity due = (C/r) + C. Where Annual payment C =500, Annual effective interest rate = 10%

Perpetuity due = (500/10%) + 500 = 5500

Value of perpetuity due will remain same after 10 years

Money in saving account can be calculated with FV of an Annuity due formula

FV = C*(1+r) *{(1+r) ^n−1} / r

Where n = 10 years

FV = 500*(1+10%) * {(1+10%)^10 - 1} / 10%

FV = 500*1.10 * [1.10^10 - 1 / 0.10}

FV = 550 * 1.5937424601/0.10

FV = 550 * 15.937424601

FV = 8765.58353055

FV = 8766

Total proceeds = 5500 + 8766 = 14266

Now this proceed is the present value for annual payment of X calculation  . Formula of the present value (PV) of annuity due: PV = X * [1- (1+r) ^-n / r] * (1+r) : Where  PV = 14266, Annuity payment X = ?, Interest rate r = 10%, Period of annuity = 20 years.

1.10^-20

PV = X * [1- (1+r)^-n / r] * (1+r)

14266 = X * (1 - (1+10%)^-20 / 10%) * (1+10%)

14266 = X * [1 - 0.14864362802/0.10]*1.10

14266 = X * [8.5135637198*1.10]

14266 = X * 9.3649

X = 14266 / 9.3649

X = 1523.347820051469

X = 1523

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