Investment Income On Reserve Funds

Introduction

In annual profit testing, investment income can arise from more than one source. Funds held to support policies at the beginning of a policy year may earn a return during that year. Premiums received during the year may also earn a return, but only for the time they are invested. Keeping these sources separate is essential: an opening reserve is already held at the start of the year, whereas a premium may arrive later.

This tutorial focuses on investment income on opening reserve funds. It establishes the annual calculation, shows how to apply it to individual policies and groups, and contrasts it with income on premiums without repeating the full treatment of policy-year cash-flow timing.

Definition: The opening reserve for a policy year is the reserve held at the start of that year for a policy in force at that time. Write ${}_tV_x$ for the reserve at time $t$ for a policy issued at age $x$. If an annual effective investment yield $i$ is earned on the opening reserve for the entire year, the investment income per policy is $i\,{}_tV_x$. The corresponding year-end fund, before any other cash flows, is $(1+i){}_tV_x$.

Why the Opening Reserve Earns a Full Year’s Return

The full-year calculation depends on a timing assumption: the opening reserve is invested at the start of the policy year and remains invested for the whole year. The annual effective yield $i$ means that one dollar invested for a full year grows to $1+i$ dollars. Therefore, an opening fund of ${}_tV_x$ grows to $(1+i){}_tV_x$. Its increase, which is the investment income, is the year-end amount less the opening amount.

\text{Investment income on opening reserve} =(1+i){}_tV_x-{}_tV_x =i\,{}_tV_x.

For a group of $N_t$ identical policies in force at the start of the year, the aggregate opening reserve fund is $N_t{}_tV_x$, so its investment income is $iN_t{}_tV_x$. If policies have different opening reserves, add the reserves first and apply the yield to that aggregate fund.

Assumptions for the annual formula: The opening reserve amount is known; the annual effective yield is $i$; and the opening fund is invested for the full policy year without withdrawals or changes to the invested amount during that year. Under these assumptions, income is $i$ times the opening fund. If funds are added, released, or invested for only part of the year, their investment income must reflect those amounts and timings rather than automatically using the full-year formula.

This calculation is distinct from investment income on premiums. A premium received at the start of the year may earn a full year’s return under an annual effective yield; one received at the end earns none during that year. The opening reserve is treated separately because it is present before any premiums for the year are received. The earlier material on investment income on policy-year cash flows addresses how to handle premium receipt timing.

Worked Example: One Policy with Opening Reserve and Timed Premiums

A hypothetical policy has an opening reserve of $12,500 at the start of a policy year. The annual effective investment yield is 4.8%. A premium of $2,000 is received at the start of the year, and a second premium of $1,000 is received at the end. Assume each amount remains invested after receipt and that there are no other fund movements. Calculate the investment income, separating the reserve from the premiums.

Method. Calculate the full-year return on the opening reserve first. Then calculate each premium’s return using its own time invested. This avoids treating all cash as if it were present at the start of the year.

Opening reserve. The reserve is invested for the entire year:

$$ 12{,}500(0.048)=600. $$

Thus, investment income on the opening reserve is $600. The reserve fund grows to $12{,}500+600=\$13{,}100$, before considering the premiums.

Premium received at the start. Under the stated timing assumption, this $2,000 is invested for the full year:

$$ 2{,}000(0.048)=96. $$

Premium received at the end. It is received at the end of the year, so it earns no investment income during this policy year: $1{,}000(0)=\$0$.

The total investment income from these specified funds is $600+96+0=\$696$. As a check, the income is $600 from the reserve and $96 from the beginning-of-year premium; the end-of-year premium adds no return in the year it arrives.

Common mistake avoided: Applying 4.8% to the opening reserve and both premiums would give $0.048(12{,}500+2{,}000+1{,}000)=\$744$. That is incorrect because the $1,000 premium is not invested for the full year.

Aggregating Opening Reserves

For a portfolio, calculate the total opening reserve from the policies in force at the start of the year. If policy $j$ has opening reserve $V_j$, the aggregate fund is $\sum_j V_j$, and its investment income under a common full-year yield is $i\sum_j V_j$. Equivalently, calculate $iV_j$ policy by policy and sum the results. These methods agree because multiplication by a common rate distributes over addition.

When opening reserves differ across groups, a simple average of the group reserve amounts can be misleading. The correct portfolio total weights each group’s reserve by its number of policies. Applying the yield to an unweighted average of group values does not in general give the right aggregate income.

Worked Example: Aggregating Two Reserve Groups

A hypothetical insurer has 60 policies in one group, each with an opening reserve of $10,000, and 40 policies in another group, each with an opening reserve of $25,000. All are in force at the start of the year. The annual effective yield is 4.25%, and the opening funds are invested for the full year. Find the aggregate investment income on these reserves.

Method. Find the opening reserve total for each group, add the group totals, and apply the annual yield once to the aggregate.

For the first group, the opening reserve total is

$$ 60(10{,}000)=600{,}000. $$

For the second group, it is

$$ 40(25{,}000)=1{,}000{,}000. $$

The aggregate opening reserve is $600{,}000+1{,}000{,}000=\$1{,}600{,}000$. Therefore, investment income is

$$ 1{,}600{,}000(0.0425)=68{,}000. $$

The result is $68,000. As a check by group, the first group earns $600{,}000(0.0425)=\$25{,}500$, and the second earns $1{,}000{,}000(0.0425)=\$42{,}500$. Their sum is $68,000.

Common mistake avoided: Averaging the two per-policy reserves without weighting gives $(10{,}000+25{,}000)/2=\$17{,}500$. Multiplying that by 100 policies would give a supposed fund of $1,750,000, which is not the actual aggregate. The groups contain different numbers of policies; their reserves must be weighted by their respective counts.

Separating Reserve Income from Premium Income

In an annual profit calculation, it can be useful to report investment income by source before combining it with other items. This makes the timing assumptions transparent and helps prevent a premium receipt from being counted as though it were part of the opening reserve. The following example combines a portfolio reserve calculation with premium receipts at two specified times.

Worked Example: Portfolio Reserve Return and Premium Receipts

A hypothetical block has 75 identical policies in force at the beginning of the year, each with an opening reserve of $4,000. The annual effective yield is 3%. During the year, the block receives $90,000 of premiums at the start of the year and $60,000 at the end. Assume the opening reserves and beginning-of-year premiums are invested for the full year, and the end-of-year premiums earn no return during the year. There are no other fund movements. Calculate the investment income by source and in total.

Method. First calculate the aggregate opening reserve and its full-year return. Then calculate the returns on the two premium receipts according to their specified timing.

The opening reserve fund is

$$ 75(4{,}000)=300{,}000. $$

Its investment income is

$$ 300{,}000(0.03)=9{,}000. $$

The $90,000 received at the start of the year earns $90{,}000(0.03)=\$2{,}700$. The $60,000 received at the end earns $60{,}000(0)=\$0$ during this year. Hence total investment income from the specified sources is

$$ 9{,}000+2{,}700+0=11{,}700. $$

The total is $11,700, consisting of $9,000 on opening reserves and $2,700 on premiums. As a check, the opening reserve return is 3% of $300,000, while the premium return is 3% of only the $90,000 present at the start of the year.

Common mistake avoided: Applying 3% to the sum of opening reserves and all premiums would give $0.03(300{,}000+90{,}000+60{,}000)=\$13{,}500$. This overstates income by $1,800 because it assumes that the end-of-year premiums were invested for a full year.

When the Full-Year Calculation Does Not Apply

The formula $i\,{}_tV_x$ is appropriate only when the opening reserve fund is invested for the entire year at the stated annual effective yield. It is not automatically the investment income in every policy-year model. For example, if a reserve fund is released during the year, the amount invested may fall before year-end. If a separate cash flow is added partway through the year, that cash flow does not earn a full year’s return merely because it is included in a year-end total.

The calculation also assumes the stated rate applies to the full year and to the relevant fund. If investment returns are represented using a different convention, such as a rate for a period other than one year, convert or apply the rate consistently with the actual investment period. Do not use an annual effective yield as though it were a monthly rate.

Key takeaway: Under the full-year annual assumption, investment income on an opening reserve is the annual yield multiplied by the reserve held at the start of the year. Keep this amount separate from income on premiums received during the year, because those premiums are invested for different periods.

Common Exam Traps

Using the closing reserve instead of the opening reserve. The calculation in this tutorial uses the fund present at the start of the year. A closing reserve is a different amount and cannot be substituted without a model-specific reason.
Applying the annual yield to every cash flow in the year. A cash flow received after the start of the year is not automatically invested for a full year.
Confusing per-policy and aggregate amounts. Multiply a per-policy income by the relevant number of policies, or apply the yield directly to the aggregate reserve. Do not do both.
Ignoring the investment period. The full-year formula relies on a full-year investment assumption. If the amount or timing differs, the formula must be adjusted to reflect that difference.
Mixing annual and subannual rates. Confirm that the yield and period use consistent time units before multiplying.

Check Your Understanding

In each problem, separate the opening-reserve return from any premium return, and use only the timing assumptions stated.

A hypothetical policy has an opening reserve of $18,000, invested for the full year at an annual effective yield of 4.25%. What is the investment income on the opening reserve? (Answer: $765.00)
A group has 45 policies with opening reserves of $6,000 each and 25 policies with opening reserves of $14,000 each. At an annual effective yield of 3.75%, all funds are invested for the full year. Find aggregate investment income on the opening reserves. (Answer: $23,250.00)
A policy has an opening reserve of $9,500 and earns a full-year annual effective yield of 2.8%. It also receives a $1,500 premium at the start of the year and a $2,500 premium at the end. Under the stated full-year and zero-year investment assumptions for those premium dates, find total investment income. (Answer: $308.00)
A portfolio has 80 policies with opening reserves of $7,500 each and 20 policies with opening reserves of $12,500 each. The annual effective yield is 4%, and all opening funds remain invested for the full year. Calculate the income by group and in total. (Answer: $24,000.00 and $10,000.00 by group; $34,000.00 total)
A policy has an opening reserve of $42,000 invested for a full year at an annual effective yield of 5%. It also receives $3,000 in premium at the start and $4,000 at the end of the year. Assuming the start-of-year premium earns a full year’s return and the end-of-year premium earns none, what is total investment income? (Answer: $2,250.00)

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