Answer to Question #135787 in Finance for abdullah

Question

Marian Kirk wishes to select the better of two 10-year annuities, C and D. Annuity C is an ordinary annuity of $2,500 per year for 10 years. Annuity D is an annuity due of $2,200 per year for 10 years.
a. Find the future value of both annuities at the end of year 10, assuming that Marian can earn (1) 10% annual interest and (2) 20% annual interest.
b. Use your findings in part a to indicate which annuity has the greater future value at the end of year 10 for both the (1) 10% and (2) 20% interest rates.
c. Find the present value of both annuities, assuming that Marian can earn (1) 10% annual interest and (2) 20% annual interest.
d. Use your findings in part c to indicate which annuity has the greater present value for both (1) 10% and (2) 20% interest rates.
e. Briefly compare, contrast, and explain any differences between your findings using the 10% and 20% interest rates in parts b and d.

Answer 1

solution

Annuity C:

ordinary annuity

$Payment,p=2500$

$Period,n=10 \ years$

Annuity D:

Annuity due

$Payment,p=2200$

$Period,n=10$

part a) future value

i) when interest $i=10\%$

Annuity C

$$FV=p*\frac{(1+i)^n-1}{i}$$

$$=2500*\frac{(1.10)^{10}-1}{0.10}=39843.5615$$

answer: the future value is $39,843.56

Annuity D

$$FV=p+p*\frac{(1+i)^{n-1}-1}{i}$$

$$=2200+2200*\frac{(1.1)^{9}-1}{0.1}=32074.8492$$

answer: the future value is $32,074.85

ii) when interest $i=20\%$

Annuity C

$$=2500*\frac{(1.20)^{10}-1}{0.20}=64896.7053$$

answer: the future value is $64896.71

Annuity D

$$=2200+2200*\frac{(1.2)^{9}-1}{0.2}=47957.5839$$

answer: the future value is $47,957.58

part b)

answer: the ordinary annuity has a greater present value than the annuity due under both interest rates

part c) present value

i) when interest $i=10\%$

Annuity C

$$PV=p*\frac{1-(1+i)^{-n}}{i}$$

$$=2500*\frac{1-(1.1)^{-10}}{0.1}=15361.4178$$

answer: the present value is $15,361.42

Annuity D

$$PV=p+p*\frac{1-(1+i)^{-{(n-1)}}}{i}$$

$$=2200+2200*\frac{1-(1.1)^{-{9}}}{0.1}=14869.8524$$

answer: the present value is$14,869.85

ii) when interest $i=20\%$

Annuity C

$$=2500*\frac{1-(1.2)^{-10}}{0.2}=10481.1802$$

answer: the present value is$10,481.18

Annuity D

$$=2200+2200*\frac{1-(1.2)^{-{9}}}{0.2}=11068.1263$$

answer: the present value is $11,068.13

part d)

Answer: the present value of the annuity due is greater than that of the ordinary annuity when interest rate is 20%. However, when interest rate is 10%, the present value of the ordinary annuity is greater than that of the annuity due.

part e) the present value of the annuities decreased when the interest rate was increased for both annuities. This is because the cash flows are discounted at a higher rate.

For both annuities, the future value increased when the interest rate increased from 10% to 20%. This is because cash flows earn higher interest on 20%

At 20%, annuity D has a greater present value than annuity C. However, at 10%, annuity C has a greater present value than annuity D