Question

An annuity-immediate pays 40 per year for 10 years, then decreases by 2 per year for 9 years. At an annual effective interest rate of 6%, the present value is equal to X. Calculate X.

Answer 1

ANSWER:

X = present value of annuity-immediate paying 40 per year for 10 years + present value of (present value of annuity at end of 9 years from now)

present value of annuity-immediate paying 40 per year for 10 years

PV of annuity-immediate = P (1 + r) [1 - (1 + r)-n] / r,

where P = periodic payment. This is 40.

r = interest rate per period. This is 6%.

n = number of periods. This is 10.

PV of annuity-immediate = P (1 + r) [1 - (1 + r)-n] / r

PV of annuity-immediate = 40 (1 + 6%) [1 - (1 + 6%)-10] / 6%

PV of annuity-immediate = 312.07

present value of (present value of annuity at end of 9 years from now)

Present value of growing annuity = P * [1 - ((1 + g) / (1 + r))n] / (r - g),

where P = first payment. This is 40 * (1 - 2%) = 39.20

r = rate per period. This is 6%.

g = growth rate. This is -2%.

n = number of periods. This is 9

present value of annuity at end of 9 years from now = P * [1 - ((1 + g) / (1 + r))n] / (r - g)

present value of annuity at end of 9 years from now = 39.20 * [1 - ((1 + (-2%)) / (1 + 6%))9] / (6% - (-2%))

present value of annuity at end of 9 years from now = 248.19

present value of (present value of annuity at end of 9 years from now) = present value of annuity at end of 9 years from now / (1 + r)n

present value of (present value of annuity at end of 9 years from now) = 248.19 / (1 + 6%)9

present value of (present value of annuity at end of 9 years from now) = 248.19 / (1 + 6%)9 = 146.90

X = present value of annuity-immediate paying 40 per year for 10 years + present value of (present value of annuity at end of 9 years from now)

X = 312.07 +146.90

X = 458.97

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