Sensitivity of Profit Provisions to Investment Yield

Why Yield Sensitivity Matters

An investment-income-adjusted profit provision depends on more than the target profit and the expected investment yield. It also depends on how much money is available to invest, and for how long. If the assumed yield changes while the premium, expected losses, expenses, and payment timing remain fixed, the investment-income credit changes. The underwriting profit provision must then move in the opposite direction if the target total operating profit is to remain unchanged.

This sensitivity is useful when comparing plausible yield assumptions or explaining a change in an indication. It also clarifies which cash flows matter: a dollar paid later reduces the invested balance later, leaving more funds invested for longer. The calculations below use a deliberately stated simple-interest approximation so that the effect of yield and timing can be isolated.

Definition: Let $P$ be premium received at the start of a one-year period, $L$ the expected loss amount, and $E$ the expected expense amount. Write $l=L/P$ and $e=E/P$. If losses are paid at time $t_L$ and expenses at time $t_E$, measured in years after premium receipt, define the investment factor per premium dollar as $$ K=1-l(1-t_L)-e(1-t_E). $$ Under the simple-interest approximation, with annual yield $y$, the investment-income credit as a proportion of premium is $yK$. If the target total operating profit is $T$ as a proportion of premium, the required underwriting profit provision is $$ U=T-yK. $$ The model assumes premium is received at time zero; the expected loss and expense amounts are paid at their stated times; the invested balance earns the same annual simple yield throughout; and there are no taxes, other cash flows, or compounding effects in this calculation. Use the formula only when those assumptions are appropriate to the stated approximation.

How the Cash-Flow Factor Works

The factor $K$ is the average invested balance over the year, measured in premium-dollar-years, per dollar of premium. Before any payments, the full premium is available to invest. A loss payment reduces the balance only after its payment date, as does an expense payment. Thus, an expected outflow $lP$ paid at time $t_L$ reduces invested funds for $1-t_L$ years, not for $t_L$ years. The expense term has the same interpretation.

This gives a practical way to read the factor: later payments increase $K$, because the funds remain invested longer. Increasing an expected payment amount generally decreases $K$, because more funds leave the invested balance. The factor is not a forecast of investment income by itself; it must be multiplied by a yield expressed on the same annual basis.

Fix the premium and cash-flow assumptions.
Express expected losses and expenses as proportions of premium, and record when each is paid.

Calculate the investment factor.
Subtract from one each expected outflow proportion multiplied by the fraction of the year that outflow is no longer invested.

Apply each yield assumption.
Multiply the factor by the annual yield to obtain investment income per premium dollar.

Recalculate the underwriting provision.
Subtract the investment-income credit from the fixed target total operating profit ratio.

Deriving the Yield Sensitivity

The sensitivity follows directly from the provision equation. Hold the cash flows, and therefore $K$, fixed. For two yields $y_1$ and $y_2$, the corresponding underwriting provisions are $U_1=T-y_1K$ and $U_2=T-y_2K$. Subtracting the first equation from the second gives the change in provision:

$$ U_2-U_1=(T-y_2K)-(T-y_1K)=-K(y_2-y_1). $$

Therefore, for a positive investment factor, a higher yield reduces the underwriting profit provision. The magnitude of the change is the yield change multiplied by $K$. This is a linear result under the stated model, not a general guarantee that real investment returns or cash flows behave linearly. If payment timing, investment strategy, taxes, or other assumptions also change, the difference cannot be attributed to yield alone.

In differential form, the same result is $\frac{\partial U}{\partial y}=-K$. This derivative is a sensitivity per unit change in yield. For example, a factor of $0.50$ means a one-percentage-point increase in yield reduces the provision by $0.50$ percentage points of premium, provided all other inputs remain fixed.

Worked Example: Recalculating at Two Yields

A hypothetical one-year policy receives $P=\$1{,}000$ at inception. Expected losses are $\$600$, paid at the midpoint of the year, and expected expenses are $\$250$, paid one-quarter of the way through the year. The target total operating profit is $8\%$ of premium. Calculate the underwriting profit provision at annual yields of $3\%$ and $5\%$, and check the change.

Method and assumptions. Use the simple-interest model above. The loss and expense ratios are $l=\$600/\$1{,}000=0.60$ and $e=\$250/\$1{,}000=0.25$. The payment times are $t_L=0.50$ and $t_E=0.25$.

Investment factor. The loss payment leaves funds invested for $1-0.50=0.50$ years. The expense payment leaves funds invested for $1-0.25=0.75$ years. Thus

$$ K=1-(0.60)(0.50)-(0.25)(0.75) =1-0.300-0.1875 =0.5125. $$

At a $3\%$ yield. The investment-income credit is $0.03(0.5125)=0.015375$, or $1.5375\%$ of premium. The underwriting provision is

U_1=0.080000-0.015375=0.064625=6.4625\%.

At a $5\%$ yield. The credit is $0.05(0.5125)=0.025625$, or $2.5625\%$ of premium. The underwriting provision is

U_2=0.080000-0.025625=0.054375=5.4375\%.

Check. The yield rises by $0.02$, so the provision should fall by $K(0.02)=0.5125(0.02)=0.01025$, or $1.025$ percentage points. Direct subtraction gives $6.4625\%-5.4375\%=1.0250$ percentage points, as required. On $\$1{,}000$ of premium, the provisions are $\$64.63$ and $\$54.38$, respectively, rounded to the nearest cent. The common mistake avoided here is treating the time already elapsed before payment as the period the money is unavailable; the relevant period is the time remaining after payment.

Separating Yield Changes from Timing Changes

A comparison can involve more than one changed assumption. To explain the result, change one input at a time. First compare yields with the cash-flow factor fixed. Then, if payment timing differs, recalculate the factor and measure that effect separately. This decomposition identifies whether the change is driven by the yield assumption, the amount of cash remaining invested, or both.

Worked Example: Distinguishing Yield and Claim-Timing Effects

Two hypothetical portfolios each have premium of $\$10{,}000$, expected losses of $70\%$ of premium, expenses of $20\%$ of premium paid at $t_E=0.25$, and a target total operating profit of $6.5\%$. In Portfolio A, losses are paid at $t_L=0.50$; in Portfolio B, they are paid at $t_L=0.75$. Compare Portfolio A at a $3\%$ yield with Portfolio B at a $5\%$ yield. Separate the yield effect from the claim-timing effect by moving first from Portfolio A at $3\%$ to Portfolio A at $5\%$, then from Portfolio A at $5\%$ to Portfolio B at $5\%$.

Portfolio A factor. Here $l=0.70$, $e=0.20$, $t_L=0.50$, and $t_E=0.25$. Therefore

$$ K_A=1-(0.70)(0.50)-(0.20)(0.75) =1-0.35-0.15 =0.50. $$

At $3\%$, Portfolio A receives an investment credit of $0.03(0.50)=1.50\%$, so its underwriting provision is $6.50\%-1.50\%=5.00\%$, or $\$500.00$. At $5\%$, its credit is $0.05(0.50)=2.50\%$, and its provision is $4.00\%$, or $\$400.00$. The yield increase reduces the provision by $1.00$ percentage point.

Portfolio B factor. The only factor change is the later loss payment. Losses are now paid at $t_L=0.75$, so they reduce the invested balance for $1-0.75=0.25$ years:

$$ K_B=1-(0.70)(0.25)-(0.20)(0.75) =1-0.175-0.15 =0.675. $$

At $5\%$, the credit is $0.05(0.675)=3.375\%$, and the provision is $6.50\%-3.375\%=3.125\%$, or $\$312.50$. Moving from Portfolio A at $5\%$ to Portfolio B at $5\%$ increases the factor by $0.675-0.50=0.175$. The additional credit is $0.05(0.175)=0.00875$, or $0.875$ percentage points, so the provision falls by the same amount.

Check the total comparison. The provision moves from $5.00\%$ to $3.125\%$, a decrease of $1.875$ percentage points. The two sequential effects add to $1.00+0.875=1.875$ percentage points. The later claim payment increases the investment credit because more of the premium remains invested for longer; it does not change the target total operating profit in this model.

Reading the Factor by Cash-Flow Component

The expression for $K$ makes the drivers visible. The loss contribution to the reduction in invested funds is $l(1-t_L)$, and the expense contribution is $e(1-t_E)$. A higher loss ratio or expense ratio lowers the factor, all else equal. A later loss or expense payment raises it, because the outflow reduces the balance for fewer years. These are statements about the investment-income factor under the model, not statements that a later payment changes the expected amount of the loss or expense.

When multiple yields are being considered, report both the fixed-factor yield sensitivity and any changes to the factor. If $K$ changes along with $y$, the exact provision comparison is

$$ U_2-U_1=(T_2-y_2K_2)-(T_1-y_1K_1). $$

Only when the target $T$ and factor $K$ are unchanged does this reduce to $-K(y_2-y_1)$. This distinction prevents attributing a cash-flow change to yield alone.

Worked Example: Finding the Yield Needed for a Selected Provision

A hypothetical book has premium of $\$5{,}000$, expected losses equal to $60\%$ of premium paid at $t_L=0.40$, and expenses equal to $25\%$ of premium paid at $t_E=0.80$. The target total operating profit is $7.2\%$ of premium. What annual yield makes the underwriting profit provision $4.5\%$ of premium? Check the result using dollars.

Method. First calculate the factor from the specified payment dates. Then solve $U=T-yK$ for $y$. The loss and expense are paid before the end of the year, so their respective remaining investment periods are $0.60$ and $0.20$ years.

$$ K=1-(0.60)(0.60)-(0.25)(0.20) =1-0.36-0.05 =0.59. $$

Rearranging the provision equation gives $y=(T-U)/K$. Substituting $T=0.072$, $U=0.045$, and $K=0.59$,

y=\frac{0.072-0.045}{0.59} =\frac{0.027}{0.59} =0.0457627\ldots \approx 4.576\%.

Dollar check. The target profit is $0.072(\$5{,}000)=\$360.00$, and the selected underwriting provision is $0.045(\$5{,}000)=\$225.00$. The required investment income is therefore $\$135.00$. At the unrounded yield, the investment credit is $0.0457627\ldots(0.59)(\$5{,}000)=\$135.00$, confirming the result. The yield is reported to three decimal places as $4.576\%$; the intermediate calculation is not rounded.

Common Mistake: Using the Wrong Payment-Time Fraction

A frequent error is to subtract $lt_L$ and $et_E$ from one when calculating the investment factor. That calculation treats an outflow as though it reduces the balance for the time elapsed before payment. In fact, the cash remains invested until the outflow occurs, and it is the remaining time after payment that determines how long the balance is reduced.

For the first worked example, the incorrect calculation would be

K_{\text{wrong}}=1-(0.60)(0.50)-(0.25)(0.25)=0.6375.

This is wrong because the expense paid at $t_E=0.25$ is treated as unavailable for only $0.25$ years, rather than reducing the invested balance for the remaining $0.75$ years. The correct factor is $0.5125$. At a $5\%$ yield, the incorrect factor would imply a $3.1875\%$ credit rather than the correct $2.5625\%$ credit. The resulting underwriting provision would be understated by $0.625$ percentage points. Always distinguish the payment date from the time remaining after that date.

Key takeaway: With fixed cash flows and a fixed target total operating profit, the underwriting provision changes by $-K$ times the yield change. Calculate $K$ from the time each premium dollar remains invested, and change one assumption at a time when explaining a sensitivity result.

Check Your Understanding

For each problem, use the stated simple-interest model and keep the target total operating profit fixed unless the problem says otherwise.

A hypothetical policy has loss ratio $0.58$, with losses paid at $t_L=0.40$, and expense ratio $0.30$, with expenses paid at $t_E=0.50$. Find $K$, then find the underwriting provision if the target is $7\%$ and the yield is $4\%$. (Answer: $K=0.502$; underwriting provision $=4.992\%$.)
A book has fixed $K=0.648$. The yield assumption rises from $3\%$ to $5\%$. By how many percentage points does the underwriting provision change? (Answer: It decreases by $1.296$ percentage points.)
A hypothetical portfolio has premium $\$8{,}000$, loss ratio $0.64$ with losses paid at $t_L=0.75$, and expense ratio $0.24$ with expenses paid at $t_E=0.20$. If the target is $7\%$ and yield is $4\%$, calculate the factor, investment credit in dollars, and underwriting provision in dollars. (Answer: $K=0.648$; credit $=\$207.36$; underwriting provision $=\$352.64$.)
For a fixed target and yield of $5\%$, a later claims-payment assumption increases $K$ from $0.48$ to $0.60$. By how many percentage points does the underwriting provision change, and in which direction? (Answer: It decreases by $0.60$ percentage points.)
A hypothetical book has $K=0.55$, target total operating profit $6.8\%$, and selected underwriting provision $4.6\%$. What annual yield is implied? (Answer: $4.000\%$.)

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