Rate of Return

Study guide: Exam FM-2

The rate of return on an investment is the interest rate at which the money put in and the money taken out are equivalent — the yield rate or internal rate of return (IRR) of its cash flows.

$\sum_{t} C_t\, v^t = 0$

$C_t$ is the net cash flow at time $t$ (outflows negative) and $v = 1/(1+i)$ ; the IRR is the $i$ that makes the net present value zero. For a bond it is the yield to maturity.
It is unique when the cash flows change sign only once — all outlays before all receipts; with several sign changes there can be more than one root.
On Exam FM, "yield rate/rate of return" is a term of Topic 5, General Cash Flows, Portfolios, and Asset Liability Management, studied alongside Duration and Immunization.

IRR of an Investment An insurer pays $10,000 today for a note that returns $6,000 in one year and $5,500 in two years. Find the rate of return.
Answer Set the net present value to zero and solve the quadratic in $v$ :
 $\begin{align*} -10{,}000 + 6{,}000v + 5{,}500v^2 &= 0 \\ v &= \frac{-6{,}000 + \sqrt{6{,}000^2 + 4(5{,}500)(10{,}000)}}{2(5{,}500)} \\ &= \frac{-6{,}000 + 16{,}000}{11{,}000} \\ &= 0.90909 \end{align*}$ 
So $i = 1/0.90909 - 1 = 10\%$ . There is one sign change, so this is the only rate of return.

Dollar-weighted and time-weighted returns

Not on the December 2026 Exam FM syllabus. Its learning outcomes name only "yield rate/rate of return"; the textbook sections devoted to these measures are left out of its assigned readings (Broverman §5.2, Dollar-weighted and Time-weighted Rate of Return; Vaaler §2.6, Dollar-weighted yield rates; Brown & Kopp §§7.3–7.4, since only §§7.1–7.2 of that chapter are assigned); and SOA's sample questions, pruned to the syllabus since October 2022, include none. They are kept here for reference.

For a fund with deposits and withdrawals during the year, the rate of return is measured two ways: dollar-weighted, which reflects the investor's own timing, and time-weighted, which removes it.

$i_{DW} \approx \frac{I}{A + \sum_t C_t\,(1-t)}$

$1 + i_{TW} = \prod_{k=1}^{m} \frac{B_k}{B_{k-1} + C_{k-1}}$

Dollar-weighted. Over one year, $A$ is the fund at the start, $B$ at the end, $C_t$ the net deposit at time $t$ (withdrawals negative), $C = \sum C_t$ , and $I = B - A - C$ the interest earned. The formula is the simple-interest approximation to the exact equation $A(1+i) + \sum_t C_t (1+i)^{1-t} = B$ .
Time-weighted. Split the year at every deposit or withdrawal. $B_0 = A$ , $C_0 = 0$ , $B_k$ is the fund value just before cash flow $C_k$ , and $B_m = B$ . Each ratio is one sub-period's growth; chaining them gives a return that does not depend on how much money was in the fund when.
Which to use. The time-weighted rate measures the fund manager (who does not control the deposits); the dollar-weighted rate measures the investor's actual experience. With no cash flows during the year, the two coincide.

Dollar-Weighted Versus Time-Weighted Return An insurer's investment fund is worth $100,000 on January 1. On July 1 it is worth $120,000, and the insurer immediately deposits another $50,000. On December 31 the fund is worth $160,000. Find the dollar-weighted and time-weighted rates of return.
Answer Dollar-weighted, with $I = 160{,}000 - 100{,}000 - 50{,}000 = 10{,}000$ :
 $\begin{align*} i_{DW} &\approx \frac{10{,}000}{100{,}000 + 50{,}000(0.5)} \\ &= \frac{10{,}000}{125{,}000} \\ &= 8.00\% \end{align*}$ 
(Solving the exact equation $100{,}000(1+i) + 50{,}000(1+i)^{0.5} = 160{,}000$ gives $8.03\%$ .)

Time-weighted, splitting the year at July 1:
 $\begin{align*} 1 + i_{TW} &= \frac{120{,}000}{100{,}000} \times \frac{160{,}000}{120{,}000 + 50{,}000} \\ &= 1.2 \times 0.94118 \\ &= 1.12941 \end{align*}$ 
So $i_{TW} = 12.94\%$ . The manager earned $20\%$ in the first half and lost $5.9\%$ in the second; the dollar-weighted rate is lower because the insurer added money just before the losing half.