Question

Suppose someone wants to accumulate $120,000 for retirement in 30 years. The person has two choices. Plan A is a single deposit into an account with annual compounding and an APR of 6%. Plan B is a single deposit into an account with continuous compounding and an APR of 5.8%. How much does the person need to deposit in each account in order to reach the goal?The person must deposit $______ into the account for Plan A to reach the goal of $.The person must deposit $______ into the account for Plan B to reach the goal of $.(Round to the nearest cent as needed.)

Answer 1

We want to calculate the amount needed as an initial investment to have 120000 after 30 years.

Recall that the formula of annual compounding is given by the formula

$S\text{ =}P\text{ \lparen1+r\rparen}^t$

where P is the principal, r is the interest rate and t is the time in years. When compounded continously the formula is

$S=Pe^{rt}$

where the variables have the same meaning. In both cases we want to find P sucht that

$S=120000$

when t=30 and r is the interest rate that we are given.

So we have the following equation in the first case

$120000=P\text{ \lparen1+}\frac{6}{100})^{30}$

so if we divide both sides by (1+6/100)^30 we get

$P=\frac{120000}{(1+\frac{6}{100})^{30}}\approx20893.22$

so for Plan A 20893.22 is needed to have 120000 after 30 years.

now, we want to do the same with the second plan. We have

$120000=Pe^{\frac{5.8}{100}30}$

so we divide both sides by exp(5.8*30/100). So we get

$P=\frac{120000}{e^{\frac{5.8}{100}\cdot30}}\approx21062.45$

so for Plan B 21062.45 is needed to have 120000 after 30 years