Investment Income on Loss Reserves

Why Payment Timing Matters

A loss reserve represents funds associated with claims that have not yet been paid. While those funds remain invested, they can earn investment income. The amount of income depends not only on the reserve amount and the assumed yield, but also on how long each portion of the reserve remains invested. A reserve paid out near the beginning of a period supports less investment income than the same reserve paid out at the end.

This tutorial estimates income attributable to funds supporting unpaid claims. It treats expected claim payments and the investment yield as inputs; it does not estimate the claim reserve itself. The calculation is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: For an investment-income calculation, the loss-reserve balance at time $t$, written $R(t)$, is the amount of claim funds assumed to remain invested and available to support unpaid claims at that time. An expected payment reduces the balance when it is made. The assumed yield is the rate earned on those invested funds over the stated period.

A reserve figure must be interpreted in context before applying a yield. It may be a balance at a particular date, an average balance over a period, or an estimate derived from future payment amounts. These are not interchangeable without assumptions about payment timing. The examples below state which interpretation is used.

Two Calculation Approaches

When reserve balances are available at the start of each investment period, calculate income period by period. With a simple annual yield $j$, a balance $R_k$ held for one full year earns $jR_k$. If claims are paid during that year, a beginning-of-year balance alone does not describe the actual amount held throughout the year; use payment timing or a suitable average-balance estimate.

I \approx j\sum_{k=1}^{n} R_k \Delta t_k

Here, $I$ is investment income, $R_k$ is the reserve balance assumed to be invested during period $k$, and $\Delta t_k$ is the period length in years. This is a simple-yield calculation: it does not add earned income back to the claim-reserve balance to earn further income. If the problem specifies a different reinvestment or compounding convention, use that convention instead.

If expected payments and their dates are known, each expected payment amount can instead be matched to its assumed holding time. Under simple proportional accrual at annual yield $j$, an amount $C_m$ paid at time $t_m$ contributes $jC_m t_m$ of income. This calculation assumes the funds backing that payment are invested from time zero until payment, and that income is credited separately rather than reinvested.

I = j\sum_m C_m t_m

The expression is a direct way to calculate the area under the reserve-balance curve. The amount $C_m$ contributes to the reserve balance for $t_m$ years; adding those amount-times-time contributions across payments gives total reserve dollar-years. Multiplying by the annual yield gives investment income under the stated simple-accrual convention.

Worked Calculations

Worked Example: Annual Payments from a Loss Reserve

A hypothetical insurer has \$100,000 set aside at time zero for expected claim payments. It expects to pay \$40,000 at the end of year 1 and \$60,000 at the end of year 2. The funds earn a simple annual yield of 4.0%. Assume investment income is credited separately and is not added to the claim-reserve balance. Estimate the investment income.

Method. Track the unpaid claim funds at the beginning of each year. This avoids treating the entire initial balance as invested for both years.

Year 1. The beginning reserve balance is \$100,000, so income for the year is $0.04 \times \$100{,}000 = \$4{,}000$. At year-end, the \$40,000 payment reduces the claim-reserve balance to $\$100{,}000-\$40{,}000=\$60{,}000$.

Year 2. The beginning balance is \$60,000, so income is $0.04 \times \$60{,}000 = \$2{,}400$. The year-end \$60,000 payment exhausts the claim reserve.

Total and check. Total investment income is $\$4{,}000+\$2{,}400=\$6{,}400$. The payment amounts sum to the initial reserve, $\$40{,}000+\$60{,}000=\$100{,}000$, and each amount earns income only until its payment date. The common mistake is to apply 4% to \$100,000 in both years, giving \$8,000; that wrongly leaves the first payment invested during year 2.

Worked Example: Payments at Different Times Within a Year

A hypothetical liability reserve of \$240,000 supports three expected payments: \$90,000 after 3 months, \$80,000 after 8 months, and \$70,000 after 12 months. Assume a simple annual yield of 5.0%, prorated by time, and assume investment income is credited separately. Estimate income and the payment-weighted average holding time.

Method. Convert each payment date to years: 3 months is $0.25$ years, 8 months is $8/12=2/3$ years, and 12 months is $1$ year. Apply $jC_m t_m$ to each payment.

The income contributions are:

\begin{aligned} \$90{,}000(0.05)(0.25) &= \$1{,}125.00,\\ \$80{,}000(0.05)(2/3) &= \$2{,}666.67,\\ \$70{,}000(0.05)(1) &= \$3{,}500.00. \end{aligned}

Using unrounded values, the total is $\$1{,}125+\$2{,}666.666\ldots+\$3{,}500=\$7{,}291.666\ldots$, or \$7,291.67 to the nearest cent.

The payment-weighted average holding time is total reserve dollar-years divided by total payments:

\frac{\$90{,}000(0.25)+\$80{,}000(2/3)+\$70{,}000(1)} {\$240{,}000} = \frac{145{,}833.333\ldots}{240{,}000} = 0.607638\ldots\text{ years}.

Check. Applying the yield to the total dollar-years gives $0.05 \times \$145{,}833.333\ldots = \$7{,}291.67$, the same result. The common mistake is to treat all \$240,000 as invested for a full year, producing \$12,000. The early payments are not invested for the entire year.

Why the Weighted-Time Formula Holds

Suppose the reserve initially supports expected payments $C_1,\ldots,C_m$, with each payment made at time $t_1,\ldots,t_m$. Before the first payment, the invested claim funds include every amount still unpaid. Each payment amount $C_m$ is part of that balance from time zero until its payment date, so it contributes $C_m t_m$ reserve dollar-years. Adding the contributions gives the total area under the reserve-balance curve.

\int_0^T R(t)\,dt = \sum_{m=1}^{M} C_m t_m \qquad\Longrightarrow\qquad I=j\int_0^T R(t)\,dt =j\sum_{m=1}^{M}C_m t_m

The equality assumes the reserve starts at time zero with the funds backing these expected payments, the reserve decreases by each payment when paid, and the yield is simple and constant over the period. It also assumes no additional claim funding enters and no investment income is reinvested into the claim reserve. If the reserve includes other cash flows, or if yields vary over time, the balance-by-period approach must reflect those features. The formula is not a substitute for defining the reserve balance and timing convention.

Worked Example: Income from a Multi-Year Reserve Schedule

A hypothetical insurer begins year 1 with a \$420,000 claim reserve. Expected payments at each year-end are \$120,000 in year 1, \$150,000 in year 2, and \$150,000 in year 3. The reserve earns a simple annual yield of 4.0%; income is credited separately. Calculate total income over the three years.

Method. Calculate the opening reserve for each year by subtracting prior year-end payments. Each opening balance is invested for the full year because the specified payments occur at year-end.

Year 1: opening reserve $=\$420{,}000$; income $=0.04(\$420{,}000)=\$16{,}800$. After the \$120,000 payment, the reserve is $\$300{,}000$.

Year 2: opening reserve $=\$300{,}000$; income $=0.04(\$300{,}000)=\$12{,}000$. After the \$150,000 payment, the reserve is $\$150{,}000$.

Year 3: opening reserve $=\$150{,}000$; income $=0.04(\$150{,}000)=\$6{,}000$. The final \$150,000 payment reduces the reserve to zero.

Total and check. Total investment income is $\$16{,}800+\$12{,}000+\$6{,}000=\$34{,}800$. Total expected payments are $\$120{,}000+\$150{,}000+\$150{,}000=\$420{,}000$, equal to the starting reserve. Applying 4% to the initial balance for all three years would give \$50,400 and would incorrectly ignore the reduction in funds after each payment.

Choosing and Checking the Timing Convention

A problem may provide dated expected payments, annual opening reserve balances, or an average reserve. Use the information in the form supplied, and avoid mixing conventions. A beginning balance is appropriate for a period when the amount is assumed to remain invested for the full period. If payments are spread through the period, a beginning balance alone typically overstates the funds invested.

When only beginning and ending balances are available, the arithmetic average can be used as an estimate of the balance during the period if the balance changes reasonably steadily:

\bar{R}\approx\frac{R_{\text{begin}}+R_{\text{end}}}{2}, \qquad I\approx j\bar{R}\Delta t.

This is an approximation, not an identity. It can be misleading when payments are concentrated early or late in the period, or when large payments cause abrupt reserve changes. If the timing of individual payments is known, use it rather than an average-balance shortcut.

Investment income is distinct from the expected amount of claim payments: it arises because funds are held before payment. A useful reasonableness check is therefore to ask whether earlier payments produce less income than later payments of the same amount. Also check the time unit: a payment at 3 months has a holding time of 0.25 years, not 3 years.

Key takeaway: Estimate loss-reserve investment income by matching each reserve amount to the time it remains invested. State the reserve balance, yield convention, payment timing, and treatment of earned income before calculating.

Check Your Understanding

For each problem, use the stated simple-yield convention and assume investment income is credited separately unless specified otherwise.

A hypothetical reserve of \$80,000 supports payments of \$30,000 at the end of year 1 and \$50,000 at the end of year 2. At a 3.0% annual yield, calculate total income. (Answer: \$3,900)
A \$160,000 reserve supports a \$100,000 payment after 6 months and a \$60,000 payment after 1 year. At a 6.0% simple annual yield prorated by time, calculate income. (Answer: \$6,600)
A reserve begins at \$420,000. Payments at the ends of years 1, 2, and 3 are \$120,000, \$150,000, and \$150,000. At a 4.0% annual yield, find the total income. (Answer: \$34,800)
A \$100,000 reserve supports payments of \$25,000 after 3 months and \$75,000 after 9 months. At a 4.0% simple annual yield prorated by time, find the investment income. (Answer: \$2,500)
A hypothetical \$100,000 reserve supports expected payments of \$20,000 after 6 months, \$50,000 after 1 year, and \$30,000 after 2 years. At a 5.0% simple annual yield, calculate the payment-weighted average holding time and total investment income. (Answer: 1.2 years; \$6,000)

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