Evaluate long-term financial plans

Evaluating long-term financial plans involves modeling deposits, withdrawals, repayments, fees, and interest through the time value of money, including compound interest, present and future value, annuities, and amortization. Plans are compared on a common time basis by aligning interest rates with compounding periods and considering specified inflation or taxes; tables, timelines, formulas, and graphs reveal how term, contribution, rate, and payment frequency affect outcomes, while stochastic investment modeling and advanced optimization are beyond this scope.

Detailed Explanation: Evaluate long-term financial plans

To evaluate a long-term savings plan:

Put all plans on the same ending date.
Match the interest rate to the payment period.
Account for fees by finding the amount actually invested.
Use the future value formula for regular deposits.
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.

Learn by doing: Evaluate long-term financial plans

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