Simple vs. Compound Interest: Fundamentals for Actuaries
Compare simple and compound interest and their formulas for Exam FM.
Simple Interest
Under simple interest, the accumulation function is a(t) = 1 + it, where i is the annual interest rate and t is time measured in years. Interest is earned only on the original principal, not on any accumulated interest. The growth is linear: the same dollar amount of interest is added each period regardless of how much has already accumulated.
For example, $1000 invested at 5% simple interest for 3 years accumulates to 1000(1 + 0.05 * 3) = 1000(1.15) = $1150. The interest earned is $50 per year, totaling $150 over three years. Notice that the interest in year 2 is still $50 (5% of the original $1000), not 5% of the $1050 balance at the end of year 1.
Simple interest is rarely used in practice for periods longer than one year. However, it commonly applies for fractional periods, such as calculating interest for days within a year. Two conventions exist for counting days: exact interest uses a 365-day year, while ordinary interest (the "banker's rule") uses a 360-day year. On Exam FM, the problem will specify which convention to use.
An important property of simple interest is that the effective rate per period is not constant. The effective rate for the n-th year is i / (1 + i(n-1)), which decreases as n increases. This declining effective rate is economically unreasonable for long periods, which is one reason compound interest is the standard for multi-year calculations.
Compound Interest
Under compound interest, the accumulation function is a(t) = (1 + i)^t. Interest earned in one period itself earns interest in subsequent periods. This "interest on interest" effect causes exponential growth rather than linear growth.
For $1000 at 5% compound interest for 3 years: 1000(1.05)^3 = $1157.63. Compare this to the $1150 from simple interest. The extra $7.63 comes from compounding. In year 1, the interest is $50. In year 2, it is 5% of $1050 = $52.50. In year 3, it is 5% of $1102.50 = $55.13. The total is $157.63.
Compound interest is the default assumption on Exam FM. Unless a problem explicitly states simple interest, you should use compound interest. The effective interest rate in each period is constant at i, which is a defining and desirable property. This constancy means that $1 invested at any point in time grows by the same factor (1 + i) over the next period.
The present value under compound interest is PV = FV / (1 + i)^t = FV * v^t, where v = 1/(1+i) is the discount factor. This formula works for any time t, including non-integer values. For instance, the present value of $5000 due in 2.5 years at 6% compound interest is 5000 / (1.06)^2.5 = 5000 / 1.15369 = $4333.85.
Comparison of Simple and Compound Interest
The relationship between simple and compound interest depends on the time horizon. For t less than 1, simple interest produces a larger accumulation than compound interest: 1 + it > (1+i)^t. At t = 1, they are exactly equal: both give 1 + i. For t greater than 1, compound interest produces a larger accumulation: (1+i)^t > 1 + it.
This crossover can be understood graphically. The simple interest accumulation function is a straight line through (0, 1) with slope i. The compound interest accumulation function is an exponential curve through (0, 1) that is tangent to the line at t = 1. Before t = 1, the line is above the curve; after t = 1, the curve rises above the line.
Numerically, consider i = 10%. At t = 0.5: simple gives 1.05, compound gives (1.10)^0.5 = 1.04881. At t = 2: simple gives 1.20, compound gives (1.10)^2 = 1.21. At t = 10: simple gives 2.00, compound gives (1.10)^10 = 2.5937. The gap widens dramatically over long horizons.
Mixed Calculations: Simple and Compound Together
Some Exam FM problems use compound interest for whole years and simple interest for fractional years. This is called the "practical method" or "mixed method." For example, to accumulate $1000 for 3 years and 4 months at 8% annual compound interest with simple interest for the fractional year: first compound for 3 years to get 1000(1.08)^3 = $1259.71, then apply simple interest for 4/12 of a year to get 1259.71(1 + 0.08 * 4/12) = 1259.71(1.02667) = $1293.29.
Under pure compound interest, the answer would be 1000(1.08)^(10/3) = $1293.07. The mixed method gives a slightly higher result, which is consistent with the fact that simple interest exceeds compound interest for fractional periods.
Another common setup involves a bank that credits compound interest at the end of each year and simple interest between anniversary dates. A deposit made on March 15 and withdrawn on November 20 of the same year would earn simple interest for the number of days between those dates.
The Rule of 72
A useful approximation for compound interest is the Rule of 72: the number of years for money to double is approximately 72 / (100i), where i is expressed as a percentage. At 6%, money doubles in about 72/6 = 12 years. The exact answer is ln(2)/ln(1.06) = 11.90 years, so the approximation is quite good.
The Rule of 72 works best for interest rates between 4% and 12%. For very low or very high rates, the approximation becomes less accurate. While this rule is not directly tested on Exam FM, it provides a quick sanity check for your calculations.
Study Tips
For Exam FM, make sure you can quickly distinguish whether a problem calls for simple or compound interest. Watch for keywords like "simple interest," "exact interest," or "ordinary interest" that signal the simple interest framework. In the absence of such keywords, default to compound interest. Practice problems that mix both methods, especially those involving fractional periods. Remember the crossover property (simple beats compound for t less than 1) because it appears occasionally in true/false or comparison questions. Finally, be comfortable with the accumulation and present value formulas for both types, as speed with these basic calculations frees up time for harder problems on exam day.