Force of Interest: Continuous Compounding for Actuaries
Understand the force of interest and continuous compounding for Exam FM.
Definition of the Force of Interest
The force of interest delta(t) = a'(t)/a(t) is the instantaneous rate of growth of the accumulation function at time t. It measures how fast money is growing at a specific instant, expressed as a proportion of the current balance. In calculus terms, it is the logarithmic derivative of the accumulation function: delta(t) = d/dt [ln(a(t))].
For constant force of interest delta, integrating gives ln(a(t)) = delta * t, so a(t) = e^(delta * t). This is continuous compounding: money grows exponentially at a constant instantaneous rate. The force of interest is the limit of the nominal interest rate as the compounding frequency approaches infinity: delta = lim(m to infinity) i^(m) = ln(1+i).
Conversely, i = e^delta - 1. For example, if delta = 0.05, then i = e^0.05 - 1 = 0.05127, or about 5.13%. The force of interest is always slightly less than the effective rate because delta = ln(1+i) and ln(1+x) is less than x for x > 0.
Relationships to Other Rate Measures
The force of interest connects to every other rate measure on Exam FM. Here is the complete set of conversions you should memorize: delta = ln(1+i) = -ln(v) = -ln(1-d). From these, i = e^delta - 1, v = e^(-delta), and d = 1 - e^(-delta).
For nominal rates: i^(m) = m * (e^(delta/m) - 1) and d^(m) = m * (1 - e^(-delta/m)). As m increases, both i^(m) and d^(m) converge to delta from above and below, respectively. At the limit, continuous compounding erases the distinction between interest and discount.
A useful numerical example: at delta = 0.06, the annual effective rate is i = e^0.06 - 1 = 0.06184, the discount rate is d = 1 - e^(-0.06) = 0.05824, and the discount factor is v = e^(-0.06) = 0.94176. These values satisfy all the standard identities: d = iv = 0.06184 * 0.94176 = 0.05824, and (1+i)(1-d) = 1.06184 * 0.94176 = 1.
Variable Force of Interest
When the force of interest varies with time, the accumulation function from time 0 to time t is a(t) = exp(integral from 0 to t of delta(s) ds). This follows from integrating the differential equation a'(t) = delta(t) * a(t) with the initial condition a(0) = 1.
The present value of 1 due at time t is 1/a(t) = exp(-integral from 0 to t of delta(s) ds). The accumulation from time t1 to time t2 is exp(integral from t1 to t2 of delta(s) ds). This last formula is particularly important because it handles accumulation over any sub-interval, not just from time 0.
Example: if delta(t) = 0.01 * t, the accumulation from 0 to 5 is exp(integral from 0 to 5 of 0.01t dt) = exp(0.01 * t^2/2 evaluated from 0 to 5) = exp(0.01 * 12.5) = exp(0.125) = 1.1331. The present value of $1000 due at time 5 is 1000/1.1331 = $882.50.
For the accumulation from time 2 to time 5 with the same force: exp(integral from 2 to 5 of 0.01t dt) = exp(0.01 * (25/2 - 4/2)) = exp(0.01 * 10.5) = exp(0.105) = 1.1107.
Common Forms on Exam FM
Exam FM frequently uses three forms of variable force of interest. The first is linear: delta(t) = a + bt. The integral is at + bt^2/2, and the accumulation is exp(at + bt^2/2). This is the most common variable force on the exam.
The second is reciprocal: delta(t) = c/(d + t). The integral is c * ln(d + t), and the accumulation from 0 to n is exp(c * ln((d + n)/d)) = ((d + n)/d)^c. This form produces a power function accumulation, which simplifies many calculations.
The third is constant: delta(t) = delta, giving a(t) = e^(delta * t). While this is the simplest case, problems may present it in disguise by giving the force of interest and asking for the effective rate or vice versa.
Example with reciprocal force: if delta(t) = 2/(5 + t), find the present value of $500 due at time 3. The accumulation from 0 to 3 is ((5+3)/5)^2 = (8/5)^2 = 2.56. The present value is 500/2.56 = $195.31.
Continuous Payment Streams
The force of interest naturally pairs with continuous payment streams. If payments flow at a continuous rate of r(t) per year, the present value of payments from time 0 to time n is: PV = integral from 0 to n of r(t) * exp(-integral from 0 to t of delta(s) ds) dt. For a constant payment rate r and constant force delta, this simplifies to PV = r * (1 - e^(-delta * n)) / delta = r * a-bar-angle-n, where a-bar-angle-n is the present value of a continuous annuity.
The continuous annuity formulas are: a-bar-angle-n = (1 - v^n) / delta and s-bar-angle-n = ((1+i)^n - 1) / delta. These are analogous to the discrete annuity formulas but use delta in place of i or d. They appear regularly on Exam FM.
Study Tips
For Exam FM, practice evaluating integrals of common force of interest functions until the process is routine. The most time-consuming part of these problems is usually the integration, so speed with basic calculus pays off significantly. Memorize the relationships delta = ln(1+i), v = e^(-delta), and the continuous annuity formulas. When given a variable force of interest, always write out the integral explicitly before evaluating. Check your work by verifying that the accumulation function equals 1 at time 0 and that the present value of a positive future payment is less than the payment amount. Work through at least 15 variable force problems, as these are among the most algebra-intensive on the exam.