Skill: Evaluate long-term financial plans

To evaluate a long-term savings plan:

1. Put all plans on the same ending date.
2. Match the interest rate to the payment period.
3. Account for fees by finding the amount actually invested.
4. Use the future value formula for regular deposits.
5. Compare the final values and the total amount paid.

For deposits made at the end of each month, use the ordinary annuity formula:

$$FV=P\left(\frac{(1+i)^n-1}{i}\right)$$

where:

• $P$ is the amount invested each period,
• $i$ is the interest rate per period,
• $n$ is the number of deposits,
• $FV$ is the future value.

Worked example

You are comparing two 10-year savings plans.

• Plan A: Pay $200 per month. A$2 monthly fee is taken from this amount, so $198 is invested. The account earns$6%$ per year, compounded monthly.
• Plan B: Invest $180 per month with no fee. The account earns$6.5%$ per year, compounded monthly.

Which plan has more money after 10 years?

Step 1: Find the monthly rate and number of deposits

Both plans use monthly deposits, so convert the annual rates to monthly rates.

For Plan A:

$$i_A=\frac{0.06}{12}=0.005$$

For Plan B:

$$i_B=\frac{0.065}{12}=0.0054167$$

There are:

$$n=10(12)=120$$

monthly deposits.

Step 2: Calculate Plan A

The amount actually invested each month is:

$$P_A=200-2=198$$

Substitute into the future value formula:

$$FV_A=198\left(\frac{(1.005)^{120}-1}{0.005}\right)$$ $$FV_A\approx \$32,\!448$$

The investor pays $200 each month for 120 months:

$$200(120)=\$24,\!000$$

The $2 monthly fee is already included in this amount.

Step 3: Calculate Plan B

For Plan B:

$$P_B=180$$

Substitute into the formula:

$$FV_B=180\left(\frac{(1.0054167)^{120}-1}{0.0054167}\right)$$ $$FV_B\approx \$30,\!312$$

The total amount paid is:

$$180(120)=\$21,\!600$$

Step 4: Compare the plans

At the end of 10 years:

\vert Plan \vert Total paid \vert Final value \vert \vert --- \vert ---: \vert ---: \vert \vert A \vert $24,000 \vert$32,448 \vert \vert B \vert $21,600 \vert$30,312 \vert

Plan A produces about:

$$32,\!448-30,\!312=\$2,\!136$$

more than Plan B.

Therefore, Plan A is the better plan if the goal is to have the larger balance after 10 years. However, it also requires $2,400 more in total payments, so the best choice depends on both the final value and how much the investor can afford to pay.