Investment Income per Premium Dollar

Why Express Investment Income per Premium Dollar?

A rate indication is often expressed relative to premium: losses, expenses, and profit provisions are each compared with the premium they support. Investment income can be put on the same basis. Expressing it as a proportion of premium makes the scale of the credit easier to interpret and compare across books of different sizes.

For this calculation, investment income has two sources. Premium collected in advance can remain invested while it is unearned. Separately, funds associated with incurred losses can remain invested until claims are paid. The first component is based on an unearned-premium investment amount; the second depends on expected losses and how long the corresponding funds remain invested.

Definition: Investment income per premium dollar is the investment income attributable to a defined period divided by the premium for that same period. It is a dimensionless ratio: for example, an investment income of $ \$120{,}000 $ on $ \$6{,}000{,}000 $ of premium is $0.02$, or 2.0% of premium.

The calculation below uses a simplified ratemaking model. Let $P$ be the premium for the period, $U$ the one-year-equivalent amount of unearned premium funds invested, $L$ the expected losses for the period, $T$ the average time in years that those loss funds remain invested, and $i$ the annualized investment yield. Apply the yield linearly over fractional-year durations, as a simple-interest approximation. This is not an exact compound-interest accumulation model.

Model: When the premium investment component is represented by $U$, and the loss investment component is represented by expected losses $L$ held for average duration $T$, estimated investment income per premium dollar is

\frac{I}{P} = i\left(\frac{U}{P}+\frac{L}{P}T\right).

Here $I$ is the modeled investment income. The first term assumes the premium investment base earns for one year under this convention; the second converts expected losses into invested dollar-years using $T$. All quantities must refer to consistent periods and premium bases.

The model assumes the stated yield applies to both investment components, that the balances and expected losses are appropriate for the period being analyzed, and that the duration $T$ summarizes the loss-fund holding time. It excludes taxes, investment expenses, compounding within the year, and any other income or financing effects. No mortality model or probability distribution is required: this is a deterministic calculation using stated expected values and durations.

How the Two Components Fit Together

The premium component is $iU$. The loss component is $iLT$, because $L$ dollars of expected losses held for $T$ years represent $LT$ dollar-years of invested funds under the linear approximation. Adding them gives estimated income $I=iU+iLT$. Dividing each side by $P$ gives the per-premium-dollar formula.

Sometimes the loss investment input is supplied not as expected losses and duration, but as an average unpaid-loss reserve balance $R$. If $R$ is the average amount invested throughout the modeled year, the corresponding income estimate is $iR$. In that case, the combined calculation is:

\frac{I}{P}=i\left(\frac{U}{P}+\frac{R}{P}\right).

Use either the reserve-balance form or the expected-losses-and-duration form, depending on the data provided. Under a steady-state approximation, the reserve balance may be represented by $R \approx LT$; if that relationship is already embodied in a supplied average reserve, multiplying that reserve by another duration would count holding time twice.

Fix the basis.
Confirm that premium, unearned premium, expected losses, and reserves refer to the same book, coverage, and period.

Choose the loss-fund representation.
Use $R$ for a supplied average reserve balance, or use $L$ and $T$ when expected losses and their investment duration are supplied.

Calculate income before the ratio.
Find the premium income component and the loss income component in dollars, then add them.

Divide by premium and interpret.
Report $I/P$ as a decimal or percentage, and check that the dollar income divided by premium gives the same result.

Worked Calculations

Worked Example: Combining Premium and Loss-Duration Components

A hypothetical specialty insurer has $ \$10{,}000{,}000 $ of annual premium, $ \$800{,}000 $ of one-year-equivalent unearned-premium investment funds, and expected annual losses of $ \$6{,}500{,}000 $. The average loss-fund investment duration is 0.40 years. Assume an annualized yield of 4.0%, prorated linearly over the duration. There is no mortality assumption or probability distribution in this deterministic model. Calculate investment income in dollars and per premium dollar.

Method: calculate each component separately. First, the premium component is:

\frac{U}{P} = \frac{\$800{,}000}{\$10{,}000{,}000} = 0.08.

Its income is $0.04 \times \$800{,}000=\$32{,}000$. Next, the expected-loss ratio is $ \$6{,}500{,}000/\$10{,}000{,}000=0.65$. Applying the 0.40-year duration gives a loss investment factor of $0.65 \times 0.40=0.26$, or, in dollar-years, $ \$6{,}500{,}000 \times 0.40=\$2{,}600{,}000$. The loss component therefore earns $0.04 \times \$2{,}600{,}000=\$104{,}000$.

Total modeled investment income is $ \$32{,}000+\$104{,}000=\$136{,}000$. Dividing by premium gives:

\frac{I}{P} = \frac{\$136{,}000}{\$10{,}000{,}000} = 0.0136 = 1.36\%.

Check: The ratio formula gives $0.04(0.08+0.26)=0.04(0.34)=0.0136$, matching the dollar calculation. The common mistake avoided is omitting $T$ from the expected-loss component: expected losses are not all assumed to remain invested for a full year.

Worked Example: Using an Average Reserve Balance

A hypothetical commercial property program has $ \$2{,}500{,}000 $ of annual premium, $ \$150{,}000 $ of one-year-equivalent unearned-premium investment funds, and an average unpaid-loss reserve balance of $ \$1{,}200{,}000 $. Assume a 5.0% annualized yield, applied to each balance for the modeled year. The reserve is already an average balance, so no additional duration is supplied or applied. No mortality model or probability distribution is used. Find the investment-income credit per premium dollar.

Method: use the reserve-balance form. The unearned-premium component earns $0.05 \times \$150{,}000=\$7{,}500$. The reserve component earns $0.05 \times \$1{,}200{,}000=\$60{,}000$. Total modeled investment income is $ \$7{,}500+\$60{,}000=\$67{,}500$.

The two balances as proportions of premium are $ \$150{,}000/\$2{,}500{,}000=0.06$ and $ \$1{,}200{,}000/\$2{,}500{,}000=0.48$. Thus:

\frac{I}{P} = 0.05(0.06+0.48) = 0.05(0.54) = 0.027 = 2.7\%.

Check: $ \$67{,}500/\$2{,}500{,}000=0.027$, also 2.7%. The common mistake avoided is using a duration multiplier on the supplied average reserve balance without evidence that it is needed. The balance itself is already the investment base for this calculation.

Worked Example: Solving for the Implied Loss Duration

A hypothetical equipment insurer has annual premium of $ \$6{,}000{,}000 $, one-year-equivalent unearned-premium funds of $ \$300{,}000 $, and expected annual losses of $ \$4{,}500{,}000 $. The annualized yield is 5.0%, applied linearly over the loss duration. The target investment-income credit is 3.0% of premium. Find the loss-fund duration $T$. No mortality assumption or probability distribution is used.

Method: solve the per-premium-dollar equation. The two ratios are $U/P=\$300{,}000/\$6{,}000{,}000=0.05$ and $L/P=\$4{,}500{,}000/\$6{,}000{,}000=0.75$. Set the modeled credit equal to the target:

$$ 0.03=0.05(0.05+0.75T). $$

Divide by 0.05 to obtain $0.60=0.05+0.75T$. Subtract $0.05$: $0.55=0.75T$. Therefore:

T=\frac{0.55}{0.75}=0.733333\ldots\text{ years}\approx 0.7333\text{ years}.

Check in dollars: Target investment income is $0.03 \times \$6{,}000{,}000=\$180{,}000$. The premium component contributes $0.05 \times \$300{,}000=\$15{,}000$, leaving $ \$165{,}000$ for loss-fund income. The duration that produces this amount is $ \$165{,}000/(0.05 \times \$4{,}500{,}000)=0.733333\ldots$ years, as above.

A tempting but incorrect answer is $0.70$ years. Substitution shows why it fails: $0.05(0.05+0.75(0.70))=0.05(0.575)=0.02875$, or 2.875%, not 3.0%. The correction is to solve the equation and then substitute the result to verify it.

Using the Credit Carefully

The result is an investment-income estimate stated relative to premium. For example, a value of 0.027 means modeled investment income of 2.7 cents per premium dollar under the stated assumptions. It is not, by itself, a complete rate indication or a universal reduction to a selected rate. A rate calculation must keep investment income on a basis consistent with its other components and with the intended profit and expense treatment.

The calculation is most sensitive to inputs that change either the size of the invested base or its holding time. A higher $U/P$, $L/P$, $R/P$, yield, or loss duration raises the modeled credit, all else equal. A shorter duration lowers the loss component even when expected losses are unchanged. The ratio also depends on the premium denominator: mixing written premium in one component with earned premium in another can make the resulting percentage internally inconsistent.

Do not count duration twice. Use an average reserve balance directly when that balance is the investment base; use expected losses multiplied by duration when those are the supplied inputs.
Keep units aligned. A duration in months must be converted to years when the yield and other durations are annual. For example, 9 months is $9/12=0.75$ years.
Distinguish a ratio from a dollar amount. $I/P=0.02$ is a 2.0% credit; $I=\$200{,}000$ is the corresponding dollar income only when the premium denominator is $ \$10{,}000{,}000$.
State the approximation. Linear prorating is appropriate for this simplified calculation. If the required model uses compounding, variable yields, or changing investment balances, use the cash-flow timing and accumulation assumptions of that model instead.

Key takeaway: Put both investment components on the same premium basis: $iU/P$ for one-year-equivalent unearned-premium funds, plus $iLT/P$ for expected losses held for average duration $T$. If an average reserve balance is supplied instead, use it as the loss-fund investment base without adding another duration factor.

Check Your Understanding

For each problem, use the stated simple annualized-yield convention and report the final percentage to three decimal places where needed.

A hypothetical marine portfolio has annual premium of $ \$4{,}000{,}000 $, one-year-equivalent unearned-premium funds of $ \$200{,}000 $, expected losses of $ \$2{,}000{,}000 $, and loss duration of 0.50 years. At a 6.0% yield, calculate investment income in dollars and per premium dollar. (Answer: $ \$72{,}000$; 1.800%.)
A hypothetical warranty program has premium of $ \$8{,}000{,}000 $, unearned-premium funds of $ \$400{,}000 $, and average unpaid-loss reserves of $ \$1{,}600{,}000 $. At a 4.5% yield, calculate the credit. Treat the reserve as an average balance, not as expected losses requiring a further duration adjustment. (Answer: $ \$90{,}000$; 1.125%.)
A hypothetical small-business book has premium of $ \$5{,}000{,}000 $, unearned-premium funds of $ \$250{,}000 $, expected losses of $ \$3{,}000{,}000 $, and average loss duration of 0.60 years. At a 5.0% yield, calculate the separate income components and total credit. (Answer: premium component $ \$12{,}500$; loss component $ \$90{,}000$; total $ \$102{,}500$; 2.050%.)
A hypothetical insurer targets an investment-income credit of 2.4% on premium of $ \$10{,}000{,}000 $. Unearned-premium funds are $ \$500{,}000 $, expected losses are $ \$8{,}000{,}000 $, and the yield is 4.0%. What loss duration is required? (Answer: 0.6875 years.)
A hypothetical program has premium of $ \$3{,}000{,}000 $, unearned-premium funds of $ \$90{,}000 $, and expected losses of $ \$1{,}500{,}000 $. Its average unpaid-loss reserve is $ \$900{,}000 $, consistent with a 0.60-year duration. At a 5.0% yield, calculate the credit using the reserve-balance method and check it using the loss-duration method. (Answer: both methods give $ \$49{,}500$ of income, or 1.650% of premium.)

What You'll Learn