Why Tax Bases and Timing Matter
Taxes reduce the funds available to support a policy, but “tax” is not a single ledger item with one universal base or payment date. A premium tax may be assessed on all premiums received, or only on a specified portion. An investment tax may be assessed on investment gains rather than on the fund itself. In an asset-share projection, applying the correct rate to the wrong base can materially misstate the amount that remains invested.
The calculation therefore begins with the terms of the hypothetical tax arrangement: what amount is taxable, when the tax is paid, and whether a loss produces a tax credit. The timing convention must also identify whether a premium, expense, or benefit is paid at the beginning or end of the year. The examples below assume annual effective investment returns and state the cash-flow timing explicitly.
Let \(AS_k\) be the asset share at the start of year \(k\). Suppose a premium \(P_k\) is received at the start of the year, and the premium-tax base is \(B_k^P\). At premium-tax rate \(\tau_P\), the tax is \(\tau_P B_k^P\). If \(E_k^0\) is an expense paid at the start of the year, the amount available to invest is
If all of \(F_k\) is invested for the full year at effective annual rate \(i_k\), its investment gain is \(G_k=i_kF_k\). When an investment tax at rate \(\tau_I\) applies to positive gains only, with no credit for a loss, the tax is \(T_k^I=\tau_I\max(G_k,0)\). If a benefit \(C_k^1\) is paid at year-end, the end-of-year asset share is
This recurrence is specific to the stated timing: start-of-year cash flows are invested for the full year, while the benefit and investment tax are paid at year-end. If a cash flow occurs partway through the year, it earns investment return only for the period it is held. Likewise, if a tax is paid earlier than year-end, it cannot remain invested after its payment date.
Deriving the Net Accumulation Factor
The formula for an investment tax on gains follows directly from the ledger. Assume \(F\) is invested for one year at effective annual rate \(i\), and the positive investment gain is taxed at rate \(\tau_I\) at year-end. The gain is \(iF\), so the investment tax is \(\tau_I iF\). The amount remaining after investment and tax, before any other year-end cash flow, is
Thus the net accumulation factor is \(1+i(1-\tau_I)\), not \((1+i)(1-\tau_I)\). The latter expression taxes the original invested fund as well as its gain, which is a different tax basis. This result assumes that the whole fund is invested for the year, the gain is positive, and tax is charged only on that gain.
Identify the premium amount subject to premium tax and the investment amount or gain subject to investment tax.
Record when premiums, expenses, taxes, benefits, and withdrawals are paid. This determines which amounts earn investment return.
Apply the stated return to the funds actually invested for the stated period.
Subtract each tax from the asset share at its payment date, then reconcile the opening and closing balances.
Worked Projections
Worked Example: Premium Tax and Investment Tax in One Year
A hypothetical policy has an opening asset share of $0. At the beginning of the year, the insurer receives a $1,000 premium, pays a premium tax of 2% of that premium, and incurs a $100 expense. The remaining funds earn an effective annual return of 5%. At year-end, an investment tax of 10% is paid on positive investment gain, followed by a $200 claim payment. Assume no other cash flows or taxes, and that the policy remains in force until the claim is paid. Find the closing asset share.
Method. Build the ledger in time order. The premium-tax base is the full $1,000 premium, and the investment-tax base is the gain earned on the funds left after the beginning-of-year outflows.
The premium tax is \(0.02(1{,}000)=\$20.00\). The amount invested at the start of the year is
The 5% investment gain is \(0.05(880)=\$44.00\), so the investment tax is \(0.10(44)=\$4.40\). The end-of-year asset share, after the tax and claim, is
Check. Starting from $0, the net change is the $1,000 premium less the $20 premium tax, $100 expense, $4.40 investment tax, and $200 claim, plus $44 investment gain. This also gives $719.60. A common mistake is to calculate investment gain on the full premium before subtracting the premium tax and expense; only $880 was invested for the year.
Worked Example: A Taxable Premium Subset
A hypothetical policy has an opening asset share of $5,000. At the beginning of the year it receives a $1,200 basic premium and a $300 rider premium. The premium tax is 2.5% of the basic premium only. A $120 expense is paid at the same time. The resulting funds earn 4% for the year. At year-end, 15% of the positive investment gain is paid as investment tax, and a $500 benefit is paid after the tax. Assume no other cash flows. Find the closing asset share.
Method. Apply the premium tax to the specified base, not automatically to every premium received. Both premiums enter the fund, but only the basic premium is taxable under the stated terms.
The premium-tax base is $1,200, so the tax is \(0.025(1{,}200)=\$30.00\). The start-of-year invested amount is
Investment gain is \(0.04(6{,}350)=\$254.00\). Investment tax is \(0.15(254)=\$38.10\). Therefore,
Check and trap. If the premium tax were incorrectly charged on both premiums, the tax would be \(0.025(1{,}500)=\$37.50\), an overstatement of $7.50 at the beginning of the year. Because the reduced funds would also earn less, the resulting closing asset share would be $7.50 multiplied by the net accumulation factor \(1+0.04(1-0.15)=1.034\), or $7.755 lower. The incorrect closing value would be $6,058.15 to the nearest cent. The stated tax base produces $6,065.90, which is the correct result.
Worked Example: Repeating the Tax Calculation Across Two Years
A hypothetical policy begins with no asset share. At the start of each of two years, it receives a $2,000 premium and pays a $100 expense. Premium tax is 1.5% of each full premium and is paid immediately. There are no benefits or other cash flows. Investment returns are 3% in year 1 and 5% in year 2. At each year-end, investment tax of 20% is paid on positive investment gain, with no tax credit for losses. Find the asset share at the end of year 2. Round only the final answer to cents.
Method. Calculate each year from its opening asset share. Premium tax and expense reduce the invested amount before that year’s return; investment tax is then deducted from that year’s gain.
In each year, premium tax is \(0.015(2{,}000)=\$30.00\). In year 1, the amount invested is \(0+2{,}000-30-100=\$1{,}870.00\). The investment gain is \(0.03(1{,}870)=\$56.10\), and investment tax is \(0.20(56.10)=\$11.22\). Thus,
In year 2, the opening asset share is $1,914.88. The amount invested after the new premium tax and expense is
The year-2 gain is \(0.05(3{,}784.88)=\$189.244\), and the tax is \(0.20(189.244)=\$37.8488\). Before rounding,
Answer and check. Rounded only at the final step, the year-2 asset share is $3,936.28. Both years reconcile as invested funds plus gain less investment tax; premium taxes and expenses have already been deducted in calculating each year’s invested funds.
Investment Losses and Other Tax Conventions
An investment tax does not always imply that a negative investment result creates a tax benefit. The tax convention must say whether losses are deductible, carried forward, or ignored for the projection. Under the positive-gain-only convention used in the recurrence above, an investment loss produces no investment tax and no tax credit. If a different convention is specified, replace \(\tau_I\max(G_k,0)\) with the tax amount defined by that convention.
For example, suppose $5,900 is invested for a year at an effective return of \(-4\%\). The investment loss is \(-\$236\). If the tax arrangement gives no credit for a loss, the investment tax is $0, not a negative tax of \(-\$47.20\) at a 20% rate. Any separate year-end withdrawal must still be deducted from the fund. The tax treatment of the loss and the investment performance are distinct parts of the projection.
Tax bases can also be defined using an amount other than the actual investment gain on the policy’s own funds. If a problem specifies a tax on credited interest, net investment income, or a stated share of gains, use that specified amount. Do not substitute the fund balance, the gross premium, or an internally calculated return simply because it is available in the ledger.
Common Mistakes and Checks
- Taxing the wrong premium amount. A tax on the basic premium alone is not a tax on basic and rider premiums combined. Write the taxable base before multiplying by the rate.
- Applying investment tax to the fund. If tax is charged on positive investment gains, apply the rate to the gain, not to the opening or closing asset share.
- Investing money after it has been paid out. A start-of-year tax or expense reduces the amount that earns a full year’s return. A year-end benefit does not reduce the gain earned during the year under the timing assumed here.
- Giving an automatic tax credit for a loss. A negative investment result does not establish a tax refund unless the stated tax convention provides one.
- Subtracting an outflow twice. If the premium tax has already been deducted in calculating funds invested, do not subtract it again from the closing asset share.
A useful final check is to reconcile the ledger from opening to closing asset share: add cash inflows, add investment gain or loss, and subtract each expense, tax, benefit, and withdrawal once at its stated time. This topic is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus because it tests careful cash-flow classification and timing in a long-term asset-share projection.
Check Your Understanding
For each hypothetical projection, use the stated tax bases and timing. Unless stated otherwise, round the final asset share to cents.
- A policy has an opening asset share of $4,000. At the start of the year it receives a $800 premium, pays 2% tax on the full premium, and pays an $84 expense. The invested amount earns 4%; 25% of positive investment gain is taxed at year-end, followed by a $300 benefit. Find the closing asset share. (Answer: $4,541.00)
- A policy has an opening asset share of $2,500 and receives $1,000 of basic premium plus $500 of rider premium at the start of the year. Premium tax is 3% of the basic premium only. A $100 expense is paid immediately. The remainder earns 5%, with 10% tax on positive investment gain at year-end, followed by a $500 benefit. Find the closing asset share. (Answer: $3,544.15)
- A policy starts with an asset share of $0. At the start of each of two years it receives a $2,000 premium and pays a $100 expense. Premium tax is 1.5% of the full premium. Investment returns are 3% and 5% in years 1 and 2, respectively; 20% of each positive gain is taxed at that year-end. There are no benefits or loss credits. Find the closing asset share after two years. (Answer: $3,936.28)
- A policy has an opening asset share of $5,000. It receives a $1,000 premium, pays 2% tax on that premium, and pays an $80 expense at the start of the year. The invested amount earns \(-4\%\). The investment-tax terms give no tax credit for a loss. A $100 withdrawal is made at year-end. Find the closing asset share. (Answer: $5,564.00)
- A policy has an opening asset share of $8,000. It receives a $1,200 premium, but only 80% of that premium is subject to a 2.5% premium tax. A $176 expense is paid at the start of the year. The remainder earns 6%, and 15% of positive investment gain is taxed at year-end. A $1,000 benefit is then paid. Find the closing asset share. (Answer: $8,459.00)