Question

Suppose an annuity will pay $15,000 at the beginning of each year for the next 7 years. How much money is needed to start this annuity if it earns 7.5%, compounded annually? (Round your answer to the nearest cent.) $

Answer 1

present value of annuity due

when the payments are made at beginning of each year then it was called annuity due

and we need to find the present value by using below formula

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

Pv = present value = ?

P = periodic payment = $16000

N = number of payments = 7 years = 7

Rate of interest r = 7.5% = 7.5/100 = 0.075

So substitute all values

Pv = 16000 + 16000[1-(1+0.075)^-(7-1) / 0.075]

Pv = 16000 + 16000[1-(1.075)^-(6) / 0.075]

Pv = 16000 + 16000[1-(0.64796) / 0.075]

Pv = 16000 + 16000[0.35204 / 0.075]

Pv = 16000 + 16000[4.69386]

Pv = 16000 + 75101.76

Pv = $91101.76

We should start the annuity with $91101.76

We can find by using the present value formula for ordinary annuity

PV = pmt x [(1 - 1 / (1 + i)ⁿ)] / i]

PV = present value = $200000

Pmt = Periodic payment = $5500

i = interest rate = 7% = 7/100 = 0.07

and compounding frequency = quarterly so r = 0.07/4 = 0.0175

n = Number of remaining payments = ?

we can use below formula

PV = pmt x [(1 - 1 / (1 + i)ⁿ)] / i

200000 = 5500 x [(1 - 1 / (1 + 0.0175)ⁿ)] / (0.0175)

200000/5500 = [(1 - 1 / (1.0175)ⁿ)] / (0.0175)

36.3636 = [(1 - 1 / (1.0175)ⁿ)] / (0.0175)

36.3636 x 0.0175 = [(1 - 1 / (1.0175)ⁿ)]

0.6363 = [(1 - 1 / (1.0175)ⁿ)]

1 / (1.0175)ⁿ) = [(1 -0.6363)]

1 / (1.0175)ⁿ) = [0.3637]

1 /0.3637 = (1.0175)ⁿ

2.7495 = (1.0175)ⁿ

Now apply ln both sides

Ln(2.7495) = ln(1.0175)ⁿ

1.01141 = n ln(1.0175)

1.01141 = n (0.01734)

1.01141 / 0.01734 = n

58.3281 ~ 53 = n

So number of payments is 53