Question

II. Find the present value of $10,000 received at the start of every year for 20 years if the interest rate is J1 = 12% p.a. and if the first payment of $10,000 is received at the end of 10 years.

Answer 1

Present Value computation

This represents a constant annuity with payments at the end of the year. To compute the present value (PV), the period before the first payment of $10,000 and the period beginning of year 10 will be considered. Ideally, the computation of the PV represents one which involves computing the PV in different conditions.

PMT = $10,000

n = 20 years

J₁ = 12 % p.a

First payment = $10,000 at end of 10 years

PV = PMT* (1 – 1/ (1+r)ⁿ)/r

PV (beginning of year 10) = ($10,000 / 12%) (1 – (1/(1+12%)²⁰ )

PV = $(10,000 / 12) \times (1 – (1/1.12^{20}))$

PV = $10,000 / 12%)* (1 – 1/9.646293)

PV = $83,333.33)* (1 – 0.103667)

PV =  $83,333.33 *0.896333

PV = $ 74,694.41

PV (year 0) = PV (beginning of year 10)/ (1+r)^n-1

PV = $74,694.41 / (1+0.12)^9$

PV = 74,694.41 / 2.773079

PV = $ 26,935.55

Total present value (PV) = 74,694.41 + 26,935.55 =$ 101,629.96