Investment Gains and Losses

Investment Experience and the Comparison Basis

A participating insurer invests assets to support policy benefits and other obligations. Investment experience is favorable or unfavorable only relative to a comparison basis: the return allowed for in pricing, valuation, or a projection. A positive investment gain means the assets accumulated to a greater value than they would have under that allowed return, given the same opening assets and cash flows. A negative gain means they accumulated to less.

This comparison measures an asset-side difference. It does not, by itself, measure a change in the value of policy liabilities. A liability value may depend on a valuation interest assumption, future benefits, and other contract features. Actual asset returns do not automatically change a liability measured on a fixed valuation basis.

Definition: For a specified period, an investment experience gain or loss is the difference between the accumulated value of assets under the actual investment return and the accumulated value of the same opening assets and cash flows under the allowed return. A positive difference is a gain; a negative difference is a loss.

For a valid comparison, the actual and allowed projections must use the same opening asset amount, cash-flow amounts, and cash-flow timing. The returns must also use consistent units and treatment of investment expenses. For example, if actual returns are net of asset-management charges, the allowed return should be interpreted on a comparable basis or the charges should be shown separately.

The examples below use hypothetical asset pools, deterministic effective returns, and no taxes, defaults, transaction costs, or fees. Cash flows specified at a period-end are assumed to occur after that period's investment return has been earned. These assumptions make the comparison transparent; a different cash-flow timing convention requires a different accumulation period.

Accumulating Assets and Cash Flows

Suppose an asset pool starts with $A_0$, and the effective return per period is $r$. If $n$ periods pass with no intervening cash flows, the ending asset value is $A_0(1+r)^n$. A contribution made at the end of period $k$ earns returns only in periods $k+1$ through $n$, so its ending value is $C_k(1+r)^{n-k}$. A withdrawal is represented by a negative cash flow.

A_n(r)=A_0(1+r)^n+\sum_{k=1}^{n} C_k(1+r)^{n-k}, \qquad G_n=A_n(r_{\mathrm{actual}})-A_n(r_{\mathrm{allowed}}).

Here $C_k$ is the net cash flow at the end of period $k$, positive for a contribution and negative for a withdrawal. The formula assumes the return is constant within each projection and that each cash flow occurs at the stated period-end. If returns vary by period, each asset amount must instead be accumulated using the sequence of returns it actually experiences.

Set the comparison.
Specify the projection period, actual return, allowed return, and whether the returns are effective per year, quarter, or another period.

List cash flows with dates.
Record the opening assets and each contribution or withdrawal. Do not assume a cash flow earns a return before it occurs.

Project both asset values.
Accumulate the same opening assets and cash flows once at the actual return and once at the allowed return.

Subtract allowed from actual.
The resulting difference is the investment experience gain or loss on the stated asset basis.

Worked Comparisons

Worked Example: A Midyear Contribution in a Quarterly Projection

A hypothetical participating fund begins the year with $100,000. At the end of the second quarter, it receives a $20,000 contribution. The actual effective quarterly return is 2%, while the allowed effective quarterly return is 1%. There are no other cash flows. Calculate the investment gain at year-end. Use unrounded amounts during the calculation and round the final answer to cents.

Method. Accumulate the opening balance for all four quarters, but accumulate the contribution for only the two quarters after it is received.

At the actual return, the opening balance accumulates to:

\$100{,}000(1.02)^4 =\$100{,}000(1.08243216) =\$108{,}243.216.

The contribution accumulates for two quarters:

\$20{,}000(1.02)^2 =\$20{,}000(1.0404) =\$20{,}808.

Thus actual ending assets are $\$108{,}243.216+\$20{,}808=\$129{,}051.216$.

At the allowed return, the opening balance accumulates to:

\$100{,}000(1.01)^4 =\$100{,}000(1.04060401) =\$104{,}060.401.

The contribution accumulates to $\$20{,}000(1.01)^2=\$20{,}402$. Allowed ending assets are therefore $\$104{,}060.401+\$20{,}402=\$124{,}462.401$.

The gain is actual ending assets less allowed ending assets:

\$129{,}051.216-\$124{,}462.401 =\$4{,}588.815 \approx \boxed{\$4{,}588.82\text{ gain}}.

Check. The actual return exceeds the allowed return in each quarter, and all cash flows are identical, so a positive gain is expected. The contribution's gain component is $\$20{,}808-\$20{,}402=\$406$; the opening-balance component is $\$108{,}243.216-\$104{,}060.401=\$4{,}182.815$. Together they give $\$4{,}588.815$.

Common mistake avoided: Applying four quarters of return to the contribution would treat it as if it were present at the start of the year. It was received at the end of quarter two, so it earns only two quarters of return.

Worked Example: A Withdrawal Reduces the Gain

A hypothetical account starts with $200,000 and earns a constant actual quarterly return of 1.2%; the allowed quarterly return is 1.0%. At the end of quarter one, the account pays a $20,000 withdrawal. There are no other cash flows, and the comparison is made at the end of quarter three. Calculate the investment gain to cents.

Method. Accumulate the opening assets for three quarters and treat the end-of-quarter-one withdrawal as a negative cash flow that accumulates for two quarters.

The actual ending value is:

\$200{,}000(1.012)^3-\$20{,}000(1.012)^2 =\$207{,}286.7456-\$20{,}482.88 =\$186{,}803.8656.

The allowed ending value is:

\$200{,}000(1.01)^3-\$20{,}000(1.01)^2 =\$206{,}060.20-\$20{,}402 =\$185{,}658.20.

Therefore:

\$186{,}803.8656-\$185{,}658.20 =\$1{,}145.6656 \approx \boxed{\$1{,}145.67\text{ gain}}.

Check. The opening-asset gain is $\$200{,}000[(1.012)^3-(1.01)^3]=\$1{,}226.5456$. The withdrawal reduces the gain by $\$20{,}000[(1.012)^2-(1.01)^2]=\$80.88$, because the larger actual return would also have earned more on the amount withdrawn if it had remained invested. The net is $\$1{,}226.5456-\$80.88=\$1{,}145.6656$.

Common mistake avoided: Treating the withdrawal as though it were made at the end of quarter three would give it no effect on the projection-period return. It occurs at the end of quarter one and changes the invested amount for the next two quarters.

Why the Cash-Flow Formula Works

The accumulation formula follows by tracking each dollar separately. The opening assets are invested for $n$ periods, so their accumulated value is $A_0(1+r)^n$. A cash flow $C_k$ received at the end of period $k$ is invested for $n-k$ periods; its value at the end of the projection is $C_k(1+r)^{n-k}$. Since investment accumulation is linear in the initial amounts, adding the accumulated values of the opening balance and all cash flows gives the total ending assets.

Applying this reasoning at both returns and subtracting yields a useful expression for the gain:

G_n =A_0\left[(1+r_{\mathrm{actual}})^n-(1+r_{\mathrm{allowed}})^n\right] +\sum_{k=1}^{n} C_k \left[(1+r_{\mathrm{actual}})^{n-k}-(1+r_{\mathrm{allowed}})^{n-k}\right].

A positive contribution generally adds to the gain when the actual return exceeds the allowed return, because the contribution benefits from the higher actual accumulation. A withdrawal is negative, so it generally reduces the measured gain. These directions depend on the actual and allowed returns and on the cash-flow timing; they should be confirmed from the formula rather than assumed.

Asset Performance Is Not Liability Remeasurement

Worked Example: Investment Gain Versus a Liability Value Change

A hypothetical insurer begins a year with $100,000 of assets. The allowed annual effective return is 4%, and the actual annual effective return is 6%. It pays $30,000 of claims at year-end; assume the claims are the same under both projections. Separately, a remaining policy benefit of $100,000 is due in two years. First calculate the asset investment gain at year-end. Then compare the benefit's present value using a 4% valuation interest rate with its present value using a revised 5% rate. Round money to cents.

Asset comparison. Actual assets before the year-end claim payment are $\$100{,}000(1.06)=\$106{,}000$. After the $\$30{,}000$ payment, actual assets are $\$76{,}000$. Under the allowed return, assets are $\$100{,}000(1.04)=\$104{,}000$ before the payment and $\$74{,}000$ afterward. The investment gain is:

\$76{,}000-\$74{,}000=\boxed{\$2{,}000\text{ gain}}.

The identical claim payment cancels in the comparison. The gain is also $\$100{,}000(0.06-0.04)=\$2{,}000$, the excess investment income on the opening assets.

Liability comparison. On the 4% valuation basis, the present value of the benefit due in two years is:

\frac{\$100{,}000}{(1.04)^2} =\frac{\$100{,}000}{1.0816} =\$92{,}455.62.

On a revised 5% valuation basis, it is:

\frac{\$100{,}000}{(1.05)^2} =\frac{\$100{,}000}{1.1025} =\$90{,}702.95.

The liability value decreases by $\$92{,}455.62-\$90{,}702.95=\$1{,}752.67$ when the valuation rate changes from 4% to 5%. This is a liability remeasurement caused by changing the discount assumption; it is not part of the $2,000 asset investment gain. If the liability continues to be measured at 4%, the actual 6% asset return alone does not change that valuation.

Check. The asset comparison uses the actual and allowed accumulation of the same assets and claim payment. The liability comparison discounts a future benefit under two different valuation assumptions. They answer different questions and should not be added together without a defined surplus or valuation framework.

Common mistake avoided: Treating the increase in actual asset return as though it automatically reduces the policy liability. Asset performance and liability valuation are distinct calculations, even when interest assumptions influence both in a broader valuation analysis.

Interpretation and Common Traps

Investment experience is a comparison, not an absolute judgment about whether a return is “good.” A positive result means the selected assets performed better than the specified allowed-return projection, after the stated cash flows and conventions. It does not establish that all policy liabilities are adequately funded or that the same result will continue in future periods.

Mixing time units. A quarterly return must be applied for quarters, not years. Do not substitute an annual rate for an effective quarterly rate without converting it consistently.
Giving a cash flow the wrong number of periods. A contribution at the end of period $k$ earns $n-k$ periods of return, not $n$. A withdrawal has the same timing rule but enters with a negative sign.
Comparing different cash flows. If actual and allowed projections use different contributions or claims, their difference includes more than investment performance. Isolate investment experience by holding the cash flows constant.
Changing the basis silently. State whether returns are gross or net of expenses and whether the rates are effective per period. Inconsistent bases can create a misleading gain or loss.
Confusing assets with liabilities. Actual earned returns determine asset accumulation. A liability value changes only according to its valuation basis and the specified liability cash flows and assumptions.

These distinctions are useful when identifying the investment component of participating-policy experience. The same disciplined comparison is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus: specify the basis, respect timing, and interpret the result as the quantity actually calculated.

Check Your Understanding

For each problem, use the stated timing convention and round money to cents only at the end.

A hypothetical account starts with $50,000 and has no cash flows for one year. Its actual annual effective return is 5%, and its allowed annual effective return is 3%. Find the investment gain. (Answer: $1,000.00 gain)
An account starts with $40,000. It earns actual annual effective returns of 4% in each of two years and allowed returns of 3% in each year. A $5,000 contribution is made at the end of year one. Find the end-of-year-two gain. (Answer: $878.00 gain)
A quarterly account starts with $200,000. For three quarters, its actual effective return is 1.2% per quarter and its allowed return is 1.0% per quarter. A $20,000 withdrawal occurs at the end of quarter one. Find the end-of-quarter-three investment gain. (Answer: $1,145.67 gain)
A future benefit of $10,000 is due in one year. Find the change in its present value when the valuation interest rate increases from 4% to 5%. State whether the value increases or decreases. (Answer: decreases by $91.58, from $9,615.38 to $9,523.81)
An account starts with $50,000, and a $10,000 contribution is made at the end of quarter two. For a four-quarter projection, the actual effective quarterly return is 2% and the allowed return is 1%. Find the investment gain. (Answer: $2,294.41 gain)

What You'll Learn