Solving Compound Interest Problems

6.5 Solving Compound Interest Problems

PLEASE NOTE: This book is currently in draft form; material is not final.

Learning Objectives

Create timelines to solve single cash flow problems.
Calculate the FV of a cash flow.
Calculate the PV of a cash flow.
Calculate the interest rate implied by two cash flows.
Use a calculator or spreadsheet to calculate FV, PV, r, or n, given the other 3 inputs.

When solving compound interest problems, the first crucial step is to visualize the problem as a timeline. In creating the timeline, there are two important conventions to follow. First, cash inflows should be represented coming toward the reader (pointing down), whereas cash outflows are away (pointing up). Second, the PV is always earlier in time (to the left) of FV.

Figure 6.2 Basic Timeline

We will demonstrate how to solve these problems by using the equation and by using a financial calculator and spreadsheet software. Please note that, when using a calculator or spreadsheet, cash inflows will be represented using positive values, while outflows will be negative. Also, while the inputs are fairly standardized among brands, please see the appendix to this chapter for differences among the popular calculator and spreadsheet brands.

Solving for FV

Susan deposits $2,000 into a savings account. If the interest rate is 2% compounded annually, what is her expected balance in five years?

Since Susan is giving her money to the bank, that is a cash outflow. In five years, she could potentially withdraw the money, causing an inflow.

Figure 6.3 FV Example Timeline

Note that if we solve the problem from the bank’s point of view, the only difference is the direction of the cash flows. The numbers will necessarily have to be the same!

Figure 6.4 FV Example Timeline (Bank’s View)

We can solve this problem without resorting to formulas or technology by reasoning through the balances at the end of each of the years. After the first, her $2,000 will grow by 2%: $2,000 + ($2,000 × 2%) = $2,000 + $40 = $2,040. For year two, the entire balance will grow by another 2%: $2,040 + ($2,040 × 2%) = $2,040 + $40.80 = $2,080.80. And so on until we reach a year five balance of $2,208.16.

Figure 6.5 Ending balances

Solving with the formula yields the result faster, but it is very important to conceptualize why the formula works! Using a PV = $2,000, n = 5 years, r = 2% per year, results in: $$ 2, 000 \times {(1 + .02)}^{5} = $ 2, 208.16$ . Note that we must use the decimal representation of the interest rate (.02 instead of 2%) for the formula to work properly.

Sanity Check

Since the difference between simple interest and compound interest are relatively small, we can quickly use simple interest to “sanity check” our work for errors. 2% of $2,000 is $40, so with simple interest we would get $40 per year, or 5 × $40 = $200 total interest. Compounding should yield slightly more than simple interest, which we can verify only adds $8.16 more. Of course, the higher the interest rate or the more periods there are will cause the difference to increase, but it’s a useful benchmark to test that we are in the ballpark.

Solving with a financial calculator is fairly straightforward, once you’ve identified the proper inputs and account for a few minor differences. The first is that cash inflows are represented as positive numbers and outflows as negative numbers. Thus, to represent our problem accurately, the PV should be entered as a negative number (since Susan is depositing the money in the bank, which is a cash outflow. Of course, if we solve this problem from the perspective of the bank, with a positive PV, we will get the same answer for FV, just with the sign flipped!). The other important difference for financial calculators is that they expect the interest rate as a percentage, not decimal. For now, you can ignore the pmt key.

should show the proper solution of 2,208.16.

Using a spreadsheet function is very similar to a financial calculator. pmt will probably be a required input, so for now just enter 0; we’ll explain what this is used for in the next chapter.

=FV(rate, nper, pmt, pv) =FV(2%, 5, 0, −2000) 2,208.16

Also, when determining the inputs for a problem, no matter what method you use, it is a good habit to write the time units for n and r, as they must match the compounding period. Traditionally, interest rates are quoted as an Annual Percentage Rate (APR)A rate quote determined by multiplying the compounding period’s interest rate by the number of periods in a year., or the compounding period’s interest rate multiplied by the number of periods in a year. Since the compounding period in our sample problem was yearly, we didn’t need to take this into account, but most real-world examples tend to have shorter compounding periods (semi-annually, monthly, or daily). Unless specifically stated otherwise, all subsequent interest rates in this book should be assumed to be quoted as APRs.

Equation 6.7 Periodic Interest Rate from APR

APR / # of periods in a year = periodic interest rate

How would our answer change if the interest was compounded monthly? PV would still be $2,000, but r would need to be 2% / (12 months per year) = 0.1667% per month, and n would need to be (5 years × 12 months per year) = 60 months. Now that r and n both match the compounding period, we can use our equation: $$ 2, 000 \times {(1 + .001667)}^{60} = $ 2, 210.20.$ Compounding more frequently will get Susan about $2 more in interest; this might seem like a small difference, but for higher interest rates or longer duration, more frequent compounding can have a larger impact.

Common Formula Mistakes

Pay attention to the details when using the formula, as there are ample ways to make mistakes. The two biggest are using percentage when decimal is necessary, and not matching the compounding period. Since the numbers we are dealing with for interest rates can be small, a good rule of thumb is to keep four digits (not counting leading zeros!) of the interest rate before rounding. A 2% APR compounded daily has 0.005479% of interest per day. Converting to decimal (another potential source of error—count those decimal places!) gives r = 0.00005479.

Another common source of confusion can come from the labels “present” and “future”, since present day does not always line up with PV. Consider the following example:

Five years ago, Susan deposited $2,000 into a savings account. If the interest rate is 2% compounded annually, what is her balance today?

This example is exactly equivalent with the first in this section, despite present day now being the FV! If one were to always assume “present value” must be “present day”, the set-up to the problem would go all wrong. Drawing the timeline, however, safely puts the labels where they properly should be. This is why, by the way, we will almost exclusively use the labels PV and FV instead of the full words.

Figure 6.6 Graph of FV over Time for Different Interest Rates

Solving for PV

Susan has a $3,312.24 balance in her savings account. If the interest rate has been 2% compounded annually, how much did she deposit five years ago when she opened the account?

Figure 6.7 PV Example Timeline

It is very important to note that, even though her current balance is 3,312.24, this is our FV, as we need to solve for the amount earlier on the timeline (which must be the PV). A rephrasing of the same problem would be:

Susan wants to have $3,312.24 balance in her savings account in five years. If the interest rate is 2% compounded annually, how much must she deposit now?

PV \times {(1 + .02)}^{5} = 3, 312.24

Solving for PV yields the solution of $3,000. Because solving for PV is such a frequent task in finance, it is worthwhile to rearrange the equation.

Equation 6.8 Compound Interest for n Periods (rearranged)

PV \times (1 + r) ⁿ = FV

PV = \frac{FV}{{(1 + r)}^{n}}

Using a financial calculator:

should show the proper solution of −3,000.

Spreadsheet:

=PV(rate, nper, pmt, fv) =PV(2%, 5, 0, 3312.24) −3,000.00

Figure 6.8 Graph of PV over Time for Different Interest Rates

Solving for the Interest Rate

David currently has $100 thousand and would like to double this money, through investing, by his planned retirement in 10 years. At what interest rate must he invest to achieve this goal?

Since David has $100 thousand and wants to double his money, he must want to receive $200 thousand in 10 years.

Figure 6.9 Rate Example Timeline

So Many Zeros….

Sometimes in finance we deal with very large sums of cash: thousands, millions, or even billions! Rather than type all of those zeros each time (and possibly leave one off in error), we can still use these formulas just fine by treating all of our cash numbers in their respective units. For example, this problem can still be solved properly with PV=100 and FV=200, since our units are now thousands of dollars. Just remember to be consistent!

$100, 000 \times {(1 + r)}^{10} = 200, 000$ .Solving for r (you’ll need to take the 10th root) yields the solution of 0.07177 or 7.177%.

Using a financial calculator:

should show the proper solution of 7.177%. Note that one of the cash flows must be negative and one positive, or the calculator will produce an error.

Spreadsheet:

=RATE(nper, pmt, pv, fv) =RATE(10, 0, −100000, 200000) 0.07177

Growth Rate

Note that we can use this formula to solve for any rate of growth, for example, revenue growth or growth in the dividends of a stock!

Solving for the Number of Periods

David currently has $100 thousand and would like to double this money, through investing, by his retirement. If the best interest rate he can find is 6%, how long will he have to wait to retire?

Figure 6.10 Number of Periods Example Timeline

$100, 000 \times {(1 + .06)}^{n} = 200, 000$ . Solving for n is a little tricky: we’d need to use logarithms. Most professionals will just use a financial calculator or spreadsheet for these types of problems.

Using a financial calculator:

<CLR TVM> −100,000 <PV> 6 <I> 200,000 <FV> <CPT> <N> should show the proper solution of 11.90 years. Again, one of the cash flows must be negative and one positive, or the calculator will produce an error.

Spreadsheet:

=NPER(rate, pmt, pv, fv) =NPER(6%, 0, −100000, 200000) 11.90

Rule of 72

Finding time-to-double is so common that there is a quick trick to get the approximate answer. Simply divide 72 by the interest rate, and that’s your time (in matching periods, of course). So if my business is growing at 9% a month, it will double in about (72 / 9) = 8 months. It also works to find the interest rate given number of periods: if I want my money to double by my retirement in 20 years, then I need a (72 / 20) = 3.6% rate.

Key Takeaways

Always make a timeline. PV is to the left of FV.
When using a financial calculator or a spreadsheet function, cash inflows are positive and outflows are negative. There must always be at least one of each type if solving for r or n.

Exercises

If $300 is invested at a 4% APR, how much interest will be earned in 10 years compounded annually? Compounded monthly? Compounded daily?
If a loan shark offers to give you $5,000 today in return for $8,000 in three weeks, what is the implied weekly interest rate? What is this rate quoted as an APR?
A bank account currently has a balance of $7,400 dollars. If the initial deposit was made four years ago, and the interest rate is 5% compounded annually, what was the initial deposit? If the interest rate is instead a 5% APR compounded monthly, what was the initial deposit?