Bruce and Sharon are the same age and both had pension plans with their place of employment. Sharon started contributing $2500 per year at age 25. Bruce did not start making regular contributions until age 40, but at that point contributed $6500 per year. Sharon’s plan earned an annual rate of 4.55% compounded quarterly whereas Bruce’s plan paid an annual rate of 4.85% compounded semi-annually.

a. What was the value of each plan when each is 65 years old?

Let's calculate the future value (FV) of each pension plan using the formula for compound interest with regular contributions. For compound interest with regular contributions, the future value is given by:

\[
FV = P \times \frac{(1 + r/n)^{nt} - 1}{r/n} \times (1 + r/n)
\]

Where:
- $ P $ = annual contribution
- $ r $ = annual interest rate (as a decimal)
- $ n $ = number of compounding periods per year
- $ t $ = number of years the contributions are made

### For Sharon:

- Annual contribution ($ P $) = \$2500
- Annual interest rate ($ r $) = 4.55% = 0.0455
- Compounding quarterly ($ n $) = 4 times a year
- She contributes from age 25 to age 65, so $ t = 65 - 25 = 40 $ years

Using the formula:

\[
FV_{\text{Sharon}} = 2500 \times \frac{(1 + 0.0455/4)^{4 \times 40} - 1}{0.0455/4} \times (1 + 0.0455/4)
\]

### For Bruce:

- Annual contribution ($ P $) = \$6500
- Annual interest rate ($ r $) = 4.85% = 0.0485
- Compounding semi-annually ($ n $) = 2 times a year
- He contributes from age 40 to age 65, so $ t = 65 - 40 = 25 $ years

Using the formula:

\[
FV_{\text{Bruce}} = 6500 \times \frac{(1 + 0.0485/2)^{2 \times 25} - 1}{0.0485/2} \times (1 + 0.0485/2)
\]

Let's calculate both future values.The value of each pension plan when Bruce and Sharon are 65 years old is as follows:

- Sharon's pension plan value: \$1,135,580.07
- Bruce's pension plan value: \$635,175.57

Sharon's early start and consistent contributions at a lower rate of interest allowed her to accumulate a significantly larger retirement fund than Bruce, even though Bruce contributed more annually but started later.

Would you like any additional details on these calculations, or do you have any questions?

Here are 5 related questions for further exploration:
1. What would Bruce's pension be if he started contributing at age 30?
2. How much more would Sharon have if her interest rate was 5% compounded annually?
3. How much does the compounding frequency affect their final amounts?
4. What if Sharon stopped contributing after 30 years—how would that affect her final amount?
5. How would inflation affect the real value of their retirement funds?

**Tip:** Starting contributions earlier, even with a smaller amount, often results in a larger future value due to the power of compound interest.

Bruce and Sharon are the same age and both had pension plans with their place of employment. Sharon started contributing $2500 per year at age 25. Bruce did not start making regular contributions until age 40, but at that point contributed $6500 per year. Sharon’s plan earned an annual rate of 4.55% compounded quarterly whereas Bruce’s plan paid an annual rate of 4.85% compounded semi-annually. What was the value of each plan when each is 65 years old?

Calculate the future value of Sharon and Bruce's pension plans with detailed solutions. Sharon contributes $2500 per year at 4.55% compounded quarterly, while Bruce contributes $6500 per year at 4.85% compounded semi-annually. Explore how starting early affects retirement savings and see the final amounts when they are 65 years old.

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