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+}(6)/(100))^(30)`

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

`P=(120000)/((1+(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^{(5.8)/(100)30}`

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

`P=\frac{120000}{e^{(5.8)/(100)\cdot30}}\approx21062.45`

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