Question

Marian Kirk wishes to select the better of two 7 ​-year annuities. Annuity 1 is an ordinary annuity of ​$2 comma 890 per year for 7 years. Annuity 2 is an annuity due of ​$2 comma 670 per year for 7 years. a.  Find the future value of both annuities at the end of year 7 ​, assuming that Marian can earn​ (1) 8 ​% annual interest and​ (2) 16 ​% annual interest. b.  Use your findings in part a to indicate which annuity has the greater future value at the end of year 7 for both the​ (1) 8 % and​ (2) 16 ​% interest rates.. c.  Find the present value of both​ annuities, assuming that Marian can earn ​(1) 8 ​% annual interest and​ (2) 16 ​% annual interest. d.  Use your findings in part c to indicate which annuity has the greater present value for both the​ (1) 8 ​% and​ (2) 16 ​% interest rates. e. Briefly​ compare, contrast, and explain any differences between your findings using the 8 ​% and  16 ​% interest rates in parts b and d.

Answer 1

a.

(1)

FV of Ordinary annuity at 8% = 2890*(1+8%)^6 + 2890*(1+8%)^5 + 2890*(1+8%)^4 + 2890*(1+8%)^3 + 2890*(1+8%)^2 + 2890*(1+8%)^1 + 2890*(1+8%)^0

FV of Ordinary annuity = $25,786.90

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FV of annuity due = 2670*(1+8%)^7 + 2670*(1+8%)^6 + 2670*(1+8%)^5 + 2670*(1+8%)^4 + 2670*(1+8%)^3 + 2670*(1+8%)^2 + 2670*(1+8%)^1

FV of annuity due = $27,849.85

(2)

FV of Ordinary annuity at 16% = 2890*(1+16%)^6 + 2890*(1+16%)^5 + 2890*(1+16%)^4 + 2890*(1+16%)^3 + 2890*(1+16%)^2 + 2890*(1+16%)^1 + 2890*(1+16%)^0

FV of Ordinary annuity = $32,986.09

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FV of annuity due = 2670*(1+16%)^7 + 2670*(1+16%)^6 + 2670*(1+16%)^5 + 2670*(1+16%)^4 + 2670*(1+16%)^3 + 2670*(1+16%)^2 + 2670*(1+16%)^1

FV of annuity due = $35,351.05

b.

Future value of annuity due is higher in both cases because the annuity due interest rate starts from beginning of the year and for ordinary annuity first payment is after a year.

c.

(1)

PV of Ordinary annuity at 8% = 2890/(1+8%)^1 + 2890/(1+8%)^2 + 2890/(1+8%)^3 + 2890/(1+8%)^4 + 2890/(1+8%)^5 + 2890/(1+8%)^6 + 2890/(1+8%)^7

PV of Ordinary annuity = $15,046.41

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PV of annuity due at 8% = 2670/(1+8%)^0 + 2670/(1+8%)^1 + 2670/(1+8%)^2 + 2670/(1+8%)^3 + 2670/(1+8%)^4 + 2670/(1+8%)^5 + 2670/(1+8%)^6

PV of annuity due = $15,013.09

(2)

PV of Ordinary annuity at 16% = 2890/(1+16%)^1 + 2890/(1+16%)^2 + 2890/(1+16%)^3 + 2890/(1+16%)^4 + 2890/(1+16%)^5 + 2890/(1+16%)^6 + 2890/(1+16%)^7

PV of Ordinary annuity = $11,671.45

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PV of annuity due at 16% = 2670/(1+16%)^0 + 2670/(1+16%)^1 + 2670/(1+16%)^2 + 2670/(1+16%)^3 + 2670/(1+16%)^4 + 2670/(1+16%)^5 + 2670/(1+16%)^6

PV of annuity due = $12,508.24

d.

Present value of ordinary annuity is higher in case (1) and in case (2) present value of annuity due is higher because of higher discount rate.