Question

A is saving for her retirement and contributes $1000 to his account at the end of every year for 40 years. B is also saving for his retirement and contributes $950 to his account at the beginning of every year for 40 years.

If they have the same amount of money after 40 years, what is the annual effective interest rate?

Answer 1

Let the annual interest rate be r

A contributes $1,000 annually for 40 years. Hence,after 40 years, the value of this deposit can be calculated as under:

Future value = Annuity x CVAF(_{r%, 40})

where, CVAF(_{r%, 40}) = Compound value annuity factor at r% for 40 years

Hence, future value = 1,000 x CVAF(_{r%, 40}) Equation (i)

B contributes $950 annually for 40 years at the begining of every year. Hence,after 40 years, the value of this deposit can be calculated as under:

Future value = Annuity x CVAF(_{r%, 40}) x (1 + r)

= 950 x x CVAF(_{r%, 40}) x (1 + r) Equation (ii)

Since, future value of both the deposits is same, hence by equation (i) and equation (ii)

1,000 x CVAF(_{r%, 40}) = 950 x CVAF(_{r%, 40}) x (1 + r)  

1,000 = 950 x (1 + r)

1 + r = 1,000/950

1+ r = 1.0526

r = 1.0526 - 1

= 0.0526

= 5.26%

Hence, annual interest rate is 5.26%

Kindly give a positive rating if you are satisfied with the answer. Feel free to ask if you have any doubts. Thanks.