Assume that each deposit is made at the end of every year.

Question

You plan to retire in 30 years and decide to save $10,000 per year. If the interest rate is 6% compounded monthly, how much will you have in 30 years? Assume that each deposit is made at the end of every year.

Answer 1

 

   A = P ( 1 + 0.06/12)^(12*30) = 10,000*(1.005)³⁶⁰ = 60225.75

Answer 2

This problem asks for the future value of an annuity with monthly deposits.

 

The standard formula for this amount is

 

FV =R*((1+i)ⁿ-1)/i,

 

where R is the payment per compounding period, i = the interest rate per compounding period, and n the number of compounding periods. This formula assumes that one payment is made each compounding period, which in your problem is not the case: the compounding period is a month (n=30*12), while a payment R=10,000 is made once year.

 

We can get an approximation if we assume instead that a payment of R=$10,000/12 is made every month.

With i=0.06/12 we get

 

FV =(10000/12) *((1+(0.06/12))^30*12-1)/(0.06/12)

=$ 837,100

 

This is an overestimate, because it assumes monthly payments with monthly compounding.

 

We can get an underestimate if we assume annual compounding with annual payments. In this case n=30, i=0.06 and

 

FV=10000 * ((1+0.06)³⁰-1)/0.06

=$ 790,580

 

The actual answer will lie between these two estimates.

 

 

Answer 3

Since the first deposit will be made by the end of the first year, there will be 29 deposits.

 

The first deposit will grow to be 10,000*(1+0.06/12)^12*29; the second deposit grows to be 10,000*(1+0.06/12)^12*28 and so on. The last deposit will grow to 10,000*(1+0.06/12)¹²≈$10616.78.

 

We need to find the sum 10,000*(1+0.06/12)¹²+10,000*(1+0.06/12)^12*2+…+10,000*(1+0.06/12)^12*29=

=10000*[(1+0.06/12)¹²+((1+0.06/12)¹²)²+…+((1+0.06/12)¹²)²⁹]

 

So we need to sum geometrical progression with the first term (1+0.06/12)¹²≈1.061678 and the base 1.061678, the total number of terms being 29. 

 

S=q*(q^N-1-1)/(q-1), where q is the base of geometrical progression, N is the number of terms.

 

Plug in the number to get:

 

S≈80.432448

 

Final answer is S*10000=804324.48