Question

An insurance annuity offers to pay you $1,000 per quarter for 20 years starting immediately. If you want to earn an effective annual rate of return (EAR) of 6.5 percent, what is the most you are willing to pay as a lump sum today to obtain this annuity?

A) $44,008.24

B) $45,840.95

C) $45,124.60

D) $44,927.59

Answer 1

For quarterly payment, effective annual rate (EAR) = (1 + r/m) ^m – 1

Where,

Effective annual rate (EAR) = 6.50%

Quarterly rate r =?

Period m = 4 quarters

Therefore,

6.50% = (1 + r/4) ^4 - 1

Or r = (1+ 6.5%) ^ (1/4) – 1 = 0.001587 or 1.587% per quarter

Formula of the present value (PV) of annuity due as the payment is beginning of the period

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

Where,

Present value PV =?

Quarterly payments PMT = $1,000

Interest rate r = 1.587% per quarter

Time period of annuity n = 20 years or 20*4 = 80 quarterly payments

Therefore,

PV = $1000 * [1- (1+1.587%) ^-80 / 1.587%] * (1+1.587%)

Or PV= $45,850.95

Therefore correct answer is option: B. $45,850.95