Why Investment Income Credits Need Careful Inputs
A premium dollar may be received before the claims and expenses it ultimately supports are paid. While that money is held, it can earn investment income. In a rate indication, an investment income credit recognizes this contribution, but its calculation depends on more than the stated yield: the amount invested changes over time, the holding period matters, and the denominator must match the requested measure.
The central discipline is to keep the cash-flow timeline explicit. A quoted annual yield is not automatically earned for a full year, a reserve estimate is not automatically the balance invested, and an investment income amount does not have a unique percentage until the premium denominator is specified. The examples use hypothetical cash flows. Unless stated otherwise, all calculations use a simple annual yield and assume cash flows occur at the stated times.
This simplified method represents income on the balances and timing specified in the problem. It is not a universal model of an insurer’s assets, statutory reserves, or actual investment portfolio. A reserve balance should be included only when the problem says it is available for investment and provides an appropriate timing and yield basis. The method also does not account for taxes, investment expenses, default risk, or changing yields unless those are explicitly incorporated.
Calculate Income From Balances and Periods
Divide the year at each relevant receipt or payment. Within each interval, the invested balance is constant under the stated assumptions. Multiply that balance by the interval’s fraction of a year to obtain balance-years, then multiply total balance-years by the simple annual yield.
Here \(B_j\) is the balance invested during interval \(j\), and \(t_j\) is the interval length in years. For example, three months is \(3/12\) of a year. If a payment reduces the balance, use the balance before that payment for the interval leading up to it, and the reduced balance for the following interval. This is a discrete cash-flow calculation: the balance changes at the specified dates.
Determine whether the stated annual yield is simple or effective. Do not interchange them.
Split the year whenever a receipt or payment changes the invested balance.
For each interval, record the amount invested and the interval length in years.
Sum interval income, then divide by the premium measure explicitly requested.
Worked Calculations
Worked Example: Updating the Balance After Each Payment
A hypothetical cohort receives $800 in premium at the start of a year. It pays $200 of expenses after three months and $300 of claims after six months. Assume the remaining balance stays invested through year-end at a 4% simple annual yield. Find the investment income and its credit as a proportion of the $800 written premium.
Method and timeline. Split the year at the three-month and six-month payment dates. The balance is $800 for the first three months. After the $200 expense payment, it is $600 for the next three months. After the $300 claim payment, it is $300 for the final six months.
At a 4% simple annual yield, income is \(0.04(500)=\$20.00\). The written-premium credit ratio is \(\$20/\$800=0.025\), or 2.50%.
Check. The balance is nonnegative and falls by exactly the stated payments: \(800-200=600\), then \(600-300=300\). The income is less than 4% of the original premium because the entire $800 is not invested for the full year. The common mistake is to use an incorrect balance or holding period and report $22.00 and 2.75%; the balance-years calculation gives $500, so the corrected results are $20.00 and 2.50%.
Worked Example: Converting Months to Fractions of a Year
A hypothetical policy cohort produces $1,500 available for investment at the start of a year. Claims of $500 are paid after four months, and expenses of $200 are paid after eight months. Assume the remaining balance is invested at a 6% simple annual yield. Calculate the investment income and the credit as a percentage of the $1,500 premium.
Method. There are three intervals, each four months long. The invested balances are $1,500, then \(1,500-500=\$1,000\), then \(1,000-200=\$800\). Each interval lasts \(4/12=1/3\) of a year.
Income is \(0.06(1{,}100)=\$66.00\). The credit ratio is \(\$66/\$1{,}500=0.044\), or 4.4%.
Check. The annual yield is applied to balance-years, not to the sum of the three balances as if each were invested for a full year. That incorrect calculation would produce \(0.06(1,500+1,000+800)=\$198\), three times the correct income because each actual interval is one-third of a year.
Worked Example: Matching the Credit to the Premium Denominator
A hypothetical insurer invests a $2,500 premium-related balance, with no intervening cash flows, for six months. The yield is 4% effective annually. The earned premium for the relevant period is $2,000. Find the investment income and the credit ratio using both written and earned premium.
Method. An effective annual rate includes compounding over a full year. For a fraction \(t\) of a year, the accumulation factor is \((1+i)^t\), so income on a balance left invested throughout that interval is \(B[(1+i)^t-1]\). Here \(B=\$2,500\), \(i=0.04\), and \(t=0.5\).
Using written premium, the credit ratio is \(49.51/2,500=0.019804\), or 1.980%. Using earned premium, it is \(49.51/2,000=0.024755\), or 2.476%. Ratios are rounded to three decimal places as percentages; income is rounded to the nearest cent.
Check. The two ratios differ because the denominators differ, not because the investment income changes. Dividing by written premium and labeling the result an earned-premium credit would use the wrong base. Also, treating a 4% effective annual rate as a 2% half-year rate would give $50.00, not the correct $49.51.
Why the Balance-Time Formula Works
For a simple annual yield \(r\), a constant balance \(B\) held for a fraction \(t\) of a year earns \(rBt\). This follows from the definition of a simple annual yield: income is proportional both to the principal and to the fraction of the annual investment period. If the balance changes at specified times, apply that result separately to each interval.
The last equality uses the same yield \(r\) for every interval. If the yield varies by interval, the appropriate expression is \(\sum_j r_jB_jt_j\), provided each \(r_j\) is a simple annual yield for its interval. If the problem instead states an effective annual rate and an unchanged balance held for \(t\) years, use \(B[(1+i)^t-1]\). For multiple periods with changing balances, apply the stated effective-rate accumulation convention to each period and follow the problem’s treatment of interim investment earnings. Do not substitute one convention for another without justification.
Common Errors and Their Corrections
| Error | Why it fails | Correction |
|---|---|---|
| Applying the annual yield as if every balance were invested for a full year | It ignores the actual interval lengths and overstates income when balances are held for less than a year. | Multiply each balance by its time in years before applying a simple annual yield. |
| Keeping the original balance after claims or expenses are paid | The paid amount is no longer part of the balance available for investment after the payment date. | Update the balance at each cash-flow date and use the new balance for the next interval. |
| Using an unpaid reserve estimate as though it were automatically invested cash | A reserve measure and an invested asset balance are not necessarily identical; timing and availability matter. | Use only balances the problem identifies as available for investment, with their specified timing. |
| Treating a stated effective annual rate as a simple annual yield | Fractional-year accumulation differs under the two conventions. | For an unchanged balance over \(t\) years at effective annual rate \(i\), use \(B[(1+i)^t-1]\). |
| Dividing by a convenient but unstated premium amount | The same income produces different ratios with written and earned premium. | Name the denominator in the answer and use the one requested. |
A useful final check is to ask whether the magnitude is plausible. Under a positive yield, positive invested balances, and no fees or losses, income should be positive. For a simple annual yield over no more than one year, income cannot exceed the yield times the largest balance for a full year. A credit ratio is not itself the investment yield: its denominator may be larger or smaller than the average invested balance.
Careful treatment of these inputs keeps the investment income credit internally consistent with the assumptions used in the rate calculation. The distinctions among period, balance, yield convention, and premium base are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
Check Your Understanding
For each problem, identify the relevant timing or denominator before calculating. Round dollar amounts to cents and percentages to three decimal places where needed.
- A hypothetical cohort has $1,000 invested at the start of the year. It pays $250 after three months and $350 after nine months. At a 5% simple annual yield, find investment income and the credit as a percentage of the $1,000 written premium. (Answer: $36.25; 3.625%)
- A hypothetical $1,600 balance is invested at the start of a year. Claims of $600 are paid after six months, and expenses of $200 are paid after nine months. At a 4.5% simple annual yield, find income and the credit as a percentage of written premium. (Answer: $56.25; 3.516%)
- A $1,200 balance remains unchanged for nine months at an 8% effective annual yield. Find investment income. (Answer: $71.30)
- A hypothetical rate calculation has $45 of investment income, $1,500 of earned premium, and $1,800 of written premium. Find the credit ratio using each premium base. (Answer: 3.000% of earned premium; 2.500% of written premium)
- A hypothetical $2,000 balance earns a 3% simple annual yield. It is $2,000 for two months, $1,500 for the next five months, and $900 for the final five months. Find income and the credit as a percentage of $2,000 written premium. (Answer: $40.00; 2.000%)