Worked Investment Income and Tax Calculation

Why Cash-Flow Timing Matters

A premium received at the start of a policy period can earn investment income before claims and expenses are paid. A claim paid halfway through the period does not earn a full year of investment income for the insurer, while a claim paid at the end earns none during that period. Consequently, a profit calculation that ignores payment dates can misstate the investment contribution and the premium needed to meet a profit target.

This tutorial uses a one-period model with specified cash-flow dates, an effective annual investment yield, and a simplified tax assumption. The calculations are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. Actual tax and accounting treatment can be more involved; the model should not be used when the problem specifies different tax timing or taxable-income rules.

Cash-Flow Model and Definitions

Take the policy period to run from time 0 to time 1. Treat receipts as positive cash flows and payments as negative cash flows. Let $C_j$ be a cash flow at time $t_j$, where $0\leq t_j\leq1$. Assume a constant effective annual yield $i$, with investment accumulated continuously over the period according to the equivalent fractional-year accumulation factor $(1+i)^{1-t_j}$. This assumes the stated yield applies to all funds while held and that cash flows are invested or funded immediately.

Definition: The accumulated pretax profit, denoted $B$, is the signed total of all policy cash flows accumulated to time 1. The nominal underwriting profit, denoted $U$, is the signed total of those same cash flows without accumulation. The investment income, denoted $I$, is the difference $B-U$. Under the simplified tax assumption used here, the tax rate $\tau$ applies to $B$ at the end of the period, so after-tax profit is $(1-\tau)B$.

B=\sum_j C_j(1+i)^{1-t_j}, \qquad U=\sum_j C_j, \qquad I=B-U, \qquad \text{after-tax profit}=(1-\tau)B.

For example, a premium received at time 0 is multiplied by $1+i$, a payment at midyear by $(1+i)^{1/2}$, and a payment at time 1 by 1. In these examples, the tax assumption means that taxes are based on the accumulated pretax profit $B$, not on investment income alone. It also assumes no tax loss carryforwards, separate tax treatment of individual cash-flow items, or tax payments at an earlier date. If any of those conditions matter, the tax cash flows must be modeled separately.

Why Investment Income Is the Difference

The relationship $I=B-U$ follows by comparing each cash flow at its original amount with its accumulated amount at time 1. The amount $C_j$ is already included in nominal underwriting profit. Its additional contribution from being held or funded until time 1 is $C_j[(1+i)^{1-t_j}-1]$. Summing these differences gives the total investment contribution.

\begin{aligned} I&=B-U\\ &=\sum_j C_j(1+i)^{1-t_j}-\sum_j C_j\\ &=\sum_j C_j\left[(1+i)^{1-t_j}-1\right]. \end{aligned}

This derivation also shows why the sign of a cash flow matters. A premium received early generally contributes positive investment income when $i>0$. An expense or claim paid early has a negative signed cash flow; accumulating that payment makes the final profit lower than simply subtracting its original amount. Later payments receive less accumulation, so their investment effect is smaller.

Worked Calculations

Worked Example: Separating Investment Income from Underwriting Profit

A hypothetical one-year policy collects a premium of $1,500 at time 0. It pays $50 in expenses at time 0, a $250 claim at midyear, and a $300 claim at year-end. The effective annual investment yield is 10.25%, and the tax rate is 20%. Calculate $B$, $U$, $I$, and after-tax profit. Retain full precision until the final amounts, then round money to cents.

Method and assumptions. Use signed cash flows, accumulate each to time 1, and apply the stated tax rate to $B$. Since $1.1025^{1/2}=1.05$, the midyear accumulation factor is 1.05.

Accumulated pretax profit. The premium and time-0 expense each earn the full-period yield; the midyear claim is accumulated for half a year; the year-end claim is not accumulated:

\begin{aligned} B &=1{,}500(1.1025)-50(1.1025)-250(1.05)-300\\ &=1{,}653.75-55.125-262.50-300\\ &=1{,}036.125. \end{aligned}

Nominal underwriting profit. Add the cash flows without accumulation:

$$ U=1{,}500-50-250-300=900. $$

Investment income and tax. The investment contribution is $I=B-U=1{,}036.125-900=136.125$. After-tax profit is $0.80B=0.80(1{,}036.125)=828.90$. Rounded to cents, $B=\$1{,}036.13$, $U=\$900.00$, $I=\$136.13$, and after-tax profit is $\$828.90$.

Check. Before rounding, $U+I=900+136.125=1{,}036.125=B$, and $0.80B=828.90$. The common mistake avoided here is treating the midyear claim as if it were paid at year-end or accumulated for a full year.

Worked Example: Solving for Premium Under an After-Tax Target

A hypothetical policy collects premium $P$ at time 0, pays expenses equal to $15\%P$ at time 0, and pays a $300 claim at midyear. The effective annual yield is 10.25%, the tax rate is 20%, and the required after-tax profit is $8\%P$. Find the required premium, rounding only the final premium to cents.

Method and assumptions. The premium and percentage expense are accumulated for a full year, while the claim uses the midyear factor $1.05$. Taxes are $20\%$ of accumulated pretax profit, so the retained fraction is $1-0.20=0.80$.

The accumulated pretax profit is

\begin{aligned} B &=1.1025P-0.15P(1.1025)-300(1.05)\\ &=1.1025(0.85)P-315\\ &=0.937125P-315. \end{aligned}

Set after-tax profit equal to the target and solve:

\begin{aligned} 0.80B&=0.08P\\ 0.80(0.937125P-315)&=0.08P\\ 0.7497P-252&=0.08P\\ 0.6697P&=252\\ P&=\frac{252}{0.6697}=376.28789\ldots \end{aligned}

The required premium is therefore $\$376.29$. As a check at the unrounded premium, $B=0.10P$, because the target requires $B=(0.08P)/0.80=0.10P$. Thus after-tax profit is $0.80(0.10P)=0.08P$, exactly the specified target. The common mistake avoided is using $0.08P$ as the required pretax profit; the tax gross-up makes the required pretax amount $0.10P$.

Worked Example: Combining Percentage and Fixed Expenses

A hypothetical policy receives premium $P$ at time 0. It pays an acquisition expense equal to $10\%P$ at time 0, a fixed $40 servicing expense at midyear, and a $200 claim at year-end. The effective annual yield is 21%, the tax rate is 25%, and the required after-tax profit is $90. Find $P$. Round the final monetary results to cents.

Method and assumptions. Since $1.21^{1/2}=1.10$, the midyear expense accumulates by 1.10. The time-0 premium and acquisition expense accumulate by 1.21; the year-end claim has factor 1.

The accumulated pretax profit is

\begin{aligned} B &=1.21P-0.10P(1.21)-40(1.10)-200\\ &=1.089P-244. \end{aligned}

The after-tax target is $0.75B=90$, so the required pretax profit is $B=90/0.75=120$. Substitute this value into the expression for $B$:

\begin{aligned} 1.089P-244&=120\\ 1.089P&=364\\ P&=\frac{364}{1.089}=334.251607\ldots \end{aligned}

Thus $P=\$334.25$. At the unrounded premium, nominal underwriting profit is

U=P-0.10P-40-200=0.90P-240=60.826446\ldots,

and investment income is $I=B-U=120-60.826446\ldots=59.173554\ldots$. Rounded to cents, $U=\$60.83$, $I=\$59.17$, $B=\$120.00$, and after-tax profit is $0.75(120)=\$90.00$. The check $U+I=B$ holds before rounding. This example avoids the mistake of accumulating the fixed midyear expense for a full year or treating the percentage expense as a fixed dollar amount.

Common Timing and Tax Traps

The accumulation period is the time from the cash-flow date to the valuation date, not automatically one year. For the first example, using a full-year factor for the $250 midyear claim would produce the incorrect calculation

$$ 1{,}500(1.1025)-50(1.1025)-250(1.1025)-300=1{,}023.00, $$

instead of the correct $B=\$1{,}036.125$. The claim is paid halfway through the year, so the correct factor is 1.05, not 1.1025. Also, a negative cash flow remains negative when accumulated: expenses and claims are subtracted from the accumulated premium.

Tax treatment is another frequent source of errors. Under the stated model, first calculate pretax profit $B$, then multiply by $1-\tau$. Do not apply the tax rate to $I$ alone, and do not subtract the tax rate as a dollar amount. When solving for premium against an after-tax target $T$, the corresponding pretax target is $T/(1-\tau)$, provided the simplified tax assumption applies. If the stated tax base or tax timing differs, this shortcut does not apply.

Key takeaway: Put every cash flow on the same valuation date before calculating profit. Keep the nominal result $U$, investment income $I$, accumulated pretax result $B$, and after-tax result distinct; check both $B=U+I$ and the stated after-tax target.

Check Your Understanding

For each problem, use the stated one-year accumulation and tax assumptions. Retain full precision in intermediate calculations and round final monetary answers to cents.

A hypothetical policy collects $900 at time 0, pays $60 in expenses at time 0, and pays a $400 claim at year-end. The effective annual yield is 10% and the tax rate is 30%. Find $B$, $U$, $I$, and after-tax profit. (Answer: $B=\$524.00$, $U=\$440.00$, $I=\$84.00$, after-tax profit $=\$366.80$)
A hypothetical policy collects $1,000 at time 0, pays $80 in expenses at time 0, and pays a $300 claim at midyear. The effective annual yield is 21% and the tax rate is 25%. Find $B$, $U$, $I$, and after-tax profit. (Answer: $B=\$783.20$, $U=\$620.00$, $I=\$163.20$, after-tax profit $=\$587.40$)
A hypothetical policy collects $P$ at time 0, pays expenses of $20\%P$ at time 0, and pays a $420 claim at year-end. The effective annual yield is 10%, the tax rate is 20%, and after-tax profit must be $100. Solve for $P$. (Answer: $P=\$619.32$)
A hypothetical policy collects $P$ at time 0, pays expenses of $10\%P$ at time 0 and a fixed $100 at midyear, and pays a $300 claim at year-end. The effective annual yield is 21%, the tax rate is 25%, and after-tax profit must be $120. Solve for $P$, then state $B$. (Answer: $P=\$523.42$, $B=\$160.00$)
A hypothetical policy has an accumulated pretax profit of $240 under the specified cash-flow dates. Its tax rate is 30%. What is its after-tax profit, and what pretax profit would be required for an after-tax target of $140 under the same tax assumption? (Answer: after-tax profit $=\$168.00$; required pretax profit $=\$200.00$)

What You'll Learn