Investment Return Shocks to Accumulation and Reinvestment

Investment Returns in an Asset Projection

A return assumption affects more than the investment income credited in the current year. If income is retained, it becomes part of the asset base earning returns in later years. A return shock can therefore change both the value accumulated to a future date and the investment income earned along the way.

The key is to compare scenarios using the same cash flows, dates and projection basis, changing only the return assumption being tested. This isolates the effect of the return shock. The methods here use deterministic annual projections; they do not, by themselves, model the market value of assets or revalue liabilities under a changed discount basis.

Definition: Let $A_t$ be assets at the start of year $t$, and let $j_t$ be the effective annual earned return for that year. Let $C_t$ be a net cash flow paid into the fund at the start of the year and $D_t$ a net cash flow paid at the end of the year. A positive cash flow adds to the fund; a negative one is an outgo. Assuming the beginning cash flow earns the full year's return and the ending cash flow earns none of it, the year-end assets are $A_{t+1}=(A_t+C_t)(1+j_t)+D_t$. Unless otherwise stated, returns are net earned returns and are greater than $-1$.

This convention makes cash-flow timing explicit. If a contribution is made at year-end, include it in $D_t$, not $C_t$. If investment income is withdrawn rather than retained, it should be represented as an outgo instead of being allowed to accumulate in the fund.

Why Return Timing and Reinvestment Matter

The roll-forward follows directly from the definition of an effective annual return. The fund has $A_t+C_t$ available at the start of the year. Earning return $j_t$ adds investment income of $j_t(A_t+C_t)$, so the amount before the end-of-year cash flow is $A_t+C_t+j_t(A_t+C_t)$. Adding $D_t$ gives the stated recurrence.

A_{t+1}=(A_t+C_t)+j_t(A_t+C_t)+D_t =(A_t+C_t)(1+j_t)+D_t

Repeated application shows how each year's return carries forward. For a projection of $n$ years, with years indexed from $0$ to $n-1$, the ending assets are

A_n=A_0\prod_{s=0}^{n-1}(1+j_s) +\sum_{t=0}^{n-1}C_t\prod_{s=t}^{n-1}(1+j_s) +\sum_{t=0}^{n-1}D_t\prod_{s=t+1}^{n-1}(1+j_s)

An empty product, such as the product in the final term when $t=n-1$, equals 1. Thus, an end-of-year cash flow is included in assets at that date but earns no return during that year. A beginning-of-year cash flow earns the return for its year and all later projected years. This is the reinvestment effect: retained income and earlier cash flows form part of the base on which later returns are earned.

Result: If only the return in year $k$ changes, and cash flows and returns in all other years remain unchanged, then the change in assets at the end of year $n$, for $n>k$, is $(A_k+C_k)(j'_k-j_k)\prod_{s=k+1}^{n-1}(1+j_s)$. The shock first changes year-$k$ investment income by the return difference times the amount exposed to that year's return. The difference then accumulates at the unchanged subsequent returns.

This expression assumes a single changed year and the same later returns in both scenarios. If a stress changes returns over several years, project each scenario year by year; do not apply the single-year expression as though there were only one shock.

Worked Example: A Persistent Return Reduction

A hypothetical fund begins with assets of $500,000. It receives a $50,000 contribution at the end of year 1, with no other cash flows. Compare a baseline earned return of 6% in each of two years with a stressed return of 3% in each year. At the end of year 2, the required reserve is fixed at $600,000 in both scenarios. Calculate year-end assets, year-2 investment income, and the surplus or deficit relative to that reserve.

Method. Apply the annual roll-forward with the contribution as an end-of-year cash flow in year 1. The contribution earns no year-1 return, but it is part of the asset base earning year-2 return. The fixed reserve is a comparison amount, not a return assumption.

Baseline, end of year 1. Investment income is $500{,}000(0.06)=30{,}000$. Assets after the year-end contribution are $500{,}000+30{,}000+50{,}000=580{,}000$.

Stressed scenario, end of year 1. Investment income is $500{,}000(0.03)=15{,}000$. Assets after the contribution are $500{,}000+15{,}000+50{,}000=565{,}000$.

Year 2. Baseline investment income is $580{,}000(0.06)=34{,}800$, giving year-end assets of $580{,}000+34{,}800=614{,}800$. Stressed investment income is $565{,}000(0.03)=16{,}950$, giving year-end assets of $565{,}000+16{,}950=581{,}950$.

The difference in year-2 income is $34{,}800-16{,}950=17{,}850$. The year-2 ending-asset difference is $614{,}800-581{,}950=32{,}850$. Relative to the fixed $600,000 reserve, baseline surplus is $614{,}800-600{,}000=$14{,}800$, while the stressed result is $581{,}950-600{,}000=-$18{,}050$, a deficit. The change in surplus is $-$32{,}850$, equal to the change in ending assets because the reserve is unchanged.

Check and common mistake. The year-2 income difference includes both the lower stressed rate and the smaller stressed opening asset base. A common error is to apply the year-2 return to the original $500,000 again, or to omit the $50,000 contribution from the year-2 base. The correct bases are $580,000 and $565,000, as confirmed by the year-1 roll-forwards.

Comparing Return Sequences

Even when two scenarios contain the same set of annual returns, their ending assets need not match if cash flows enter between years. The asset base exposed to a return depends on what happened earlier. Consequently, comparing only an arithmetic average return can miss the effect of sequence and reinvestment.

Worked Example: The Same Returns in a Different Order

A hypothetical account starts with $100,000 and receives a $20,000 contribution at the end of each of two years. Compare two return paths: 10% in year 1 followed by $-10$% in year 2, and $-10$% in year 1 followed by 10% in year 2. Assume no other cash flows. Calculate the ending assets and explain the difference.

Method. Roll the account forward in chronological order. Each end-of-year contribution earns no return in its contribution year, but the year-1 contribution is invested during year 2.

10%, then $-10$%. Year-1 investment income is $100{,}000(0.10)=$10{,}000$. After the contribution, year-1 ending assets are $100{,}000+10{,}000+20{,}000=$130{,}000$. Year-2 investment income is $130{,}000(-0.10)=-$13{,}000$. After the second contribution, ending assets are $130{,}000-13{,}000+20{,}000=$137{,}000$.

$-10$%, then 10%. Year-1 investment income is $100{,}000(-0.10)=-$10{,}000$. After the contribution, year-1 ending assets are $100{,}000-10{,}000+20{,}000=$110{,}000$. Year-2 investment income is $110{,}000(0.10)=$11{,}000$. After the second contribution, ending assets are $110{,}000+11{,}000+20{,}000=$141{,}000$.

The second path ends with $141{,}000-137{,}000=$4{,}000$ more. In both paths, the original $100,000 is multiplied by $1.10(0.90)=0.99$; the difference comes from the first $20,000 contribution being exposed to a different year-2 return. The total investment income is $-$3,000$ in the first path and $1,000 in the second, consistent with ending assets of $100,000 plus $40,000 of contributions and the respective investment income.

Check and common mistake. The two rates have the same arithmetic average, but that does not make the account values equal. Treating the contributions as though both were invested for both years would misstate their exposure. The year-1 contribution earns only the year-2 return.

Investment Income, Reserves and Projected Surplus

An asset projection can be compared with a reserve requirement to show whether projected assets cover that requirement. In the example below, the reserve is deliberately held fixed across scenarios. This isolates the asset and investment-income effect of the return shock; it does not estimate a change in the reserve basis. The resulting difference in assets relative to the reserve is a projected surplus change under this stated comparison, not necessarily an accounting profit measure.

Worked Example: A Return Shock and a Fixed Reserve

A hypothetical block begins the year with $2,000,000 of assets. A $100,000 net inflow is received at the beginning of the year. A $250,000 benefit and $30,000 of expenses are paid at year-end. Compare a baseline annual earned return of 5% with a stressed return of 2%. The required year-end reserve is $1,900,000 in either scenario. Calculate investment income, ending assets and surplus or deficit.

Method. Combine opening assets and the beginning inflow before applying the annual return. Combine the benefit and expense as a year-end outgo of $250{,}000+30{,}000=$280{,}000$. Apply the same cash flows in both scenarios.

The amount exposed to return in each scenario is $2{,}000{,}000+100{,}000=$2{,}100{,}000$. Baseline investment income is $2{,}100{,}000(0.05)=$105{,}000$. Baseline ending assets are $2{,}100{,}000+105{,}000-280{,}000=$1{,}925{,}000$, giving surplus of $1{,}925{,}000-1{,}900{,}000=$25{,}000$.

Stressed investment income is $2{,}100{,}000(0.02)=$42{,}000$. Stressed ending assets are $2{,}100{,}000+42{,}000-280{,}000=$1{,}862{,}000$, giving a deficit of $1{,}862{,}000-1{,}900{,}000=-$38{,}000$.

The return reduction is 3 percentage points, so the investment-income and ending-asset reduction is $2{,}100{,}000(0.05-0.02)=$63{,}000$. The surplus changes from $25,000 to $-$38,000$, also a reduction of $63,000. This agrees with the ending-asset comparison because the reserve and all cash flows are the same in both scenarios.

Check and common mistake. Applying the return only to the opening $2,000,000 would omit the beginning inflow that is invested for the full year. Conversely, applying a return to the year-end outgo would incorrectly give the benefit and expenses a full year of investment exposure. The timing convention identifies the correct return base as $2,100,000.

Practical Checks and Limits of the Comparison

A sound return-stress calculation is a controlled scenario projection. Before comparing results, verify that the return period, cash-flow timing and meaning of the return are consistent. An annual effective earned return applies over one year; a monthly return, a rate quoted on a different basis, or an asset return before investment expenses cannot be substituted without adjustment.

Keep the comparison controlled. If the purpose is to isolate a return shock, hold benefits, expenses, contributions and other assumptions fixed.
Apply each return to the correct exposure. Check whether each cash flow occurs before or after the return is earned.
Reinvest retained amounts consistently. If returns remain in the fund, include them in later opening assets. If they are distributed, record the distribution as a cash outflow.
Recalculate the whole path for persistent shocks. A return difference in an early year changes later opening assets, so later investment income may also differ even if later return assumptions are identical.
State what is held fixed. A fixed-reserve comparison measures how projected asset changes affect coverage of that reserve. It does not show how a changed valuation basis would alter the liability.

Key takeaway: Investment-return shocks affect the assets already invested and, through reinvestment, the returns earned on future asset bases. Make the cash-flow timing explicit, project each scenario period by period, and separate an asset accumulation comparison from any change in the reserve valuation basis.

Check Your Understanding

For each question, use the stated timing convention and compare scenarios only where requested.

A hypothetical fund has $80,000 at the start of a year, earns 4%, and receives a $10,000 contribution at year-end. What are its year-end assets? (Answer: $93,200)
A fund starts with $200,000 and receives a $15,000 end-of-year contribution. Calculate year-end assets at 5% and at 2%, and give the difference. (Answer: $225,000 at 5%; $219,000 at 2%; difference $6,000)
An account starts with $50,000 and receives $5,000 at each year-end. Compare returns of 8% then $-4$% with $-4$% then 8%. What are the two ending balances and their difference? (Answer: $61,640 and $62,240, respectively; the second path is $600 higher)
A fund starts with $1,000,000, receives $100,000 at the start of the year, and pays $180,000 at year-end. Compare a 3% return with a 1% return. If the fixed year-end reserve is $940,000, what are the ending assets and surplus or deficit in each scenario? (Answer: at 3%, assets $953,000 and surplus $13,000; at 1%, assets $931,000 and deficit $9,000)
A fund starts with $400,000. At the start of year 1 it receives $20,000, earns 5%, and pays $30,000 at year-end. In year 2 it has no other cash flows and earns 4%. Under a stress, only year-1 return falls to 2%. Calculate year-2 ending assets in each scenario and the difference. (Answer: baseline $427,440; stressed $414,336; difference $13,104)

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