Question

Anurag receives an annuity that pays $1,000 at the end of each month. He wishes to replace it with an annuity that has the same term and has only one payment each year, and that payment should be at the beginning of the year. How much should the payments be if the exchange is based on a nominal discount rate of 3% payable quarterly?

Answer 1

Answer:

$11,804.97

Explanation:

Data provided in the question:

Amount paid at the end of each month, P = $1,000

Nominal discount rate,  i = 3% = 0.03

n  = 4 for quarterly payable

thus,

Effective annual discount rate = $(1+\frac{i}{n})^n-1$

or

= $(1+\frac{0.03}{4})^4-1$

= 0.03034

thus,

monthly interest rate, r = $\frac{\textup{Annual rate}}{\textup{12}}$

= $\frac{0.03034}{12}$

= 0.00253

Now,

the annuity is given as:

Annuity = $\frac{P(1-\frac{1}{(1+r)^{12}})}{r}$

or

Annuity = $\frac{\ 1,000(1-\frac{1}{(1+0.00253)^{12}})}{0.00253}$

or

Annuity = $11,804.97