Why Investment Experience Changes Asset Shares
An asset share is an accumulated measure of the funds attributable to a policy or a defined group of policies, based on specified cash flows and assumptions. When investment experience differs from the return assumption used in a projection, the asset share changes. The effect is not limited to the interest earned in the year of the difference: later returns also apply to the amount gained or lost.
This tutorial focuses on isolating that investment effect. To make comparisons meaningful, hold the cash-flow pattern and all other assumptions fixed, then change only the investment returns. The resulting difference shows how sensitive the accumulated amount—and therefore the funds available to meet a benefit—is to investment experience. Any product or policy scenario below is hypothetical.
An asset share is calculated for a specified policy or cohort and cash-flow basis. This projection does not, by itself, imply a guaranteed policy benefit or a prescribed investment return. It is a way to track accumulated funds under the assumptions chosen.
Comparing a Base Projection With Investment Experience
Use one projection for the base returns and another for the alternative returns. Keep the initial asset share and each year’s cash flow identical in both projections. Write \(A_t\) for the base asset share, \(A'_t\) for the asset share under alternative returns \(j'_t\), and \(D_t=A'_t-A_t\) for their difference.
Subtracting the first recurrence from the second gives a useful year-by-year decomposition:
The first term carries forward the existing difference using the alternative return for year \(t\). The second term is the effect in year \(t\) of applying a different return to the base funds exposed to investment during that year. This decomposition makes clear why it is generally wrong to add up annual return differences without accounting for the amounts invested and the subsequent accumulation.
Worked Projections
Worked Example: Two Years With Different Return Experience
A hypothetical policy starts with an asset share of $0. It receives a net cash flow of $600 at the beginning of each of two years. The base projection assumes a 4% effective annual net return in each year. Investment experience is instead 5% in year 1 and 3% in year 2. Calculate both ending asset shares and the difference.
Method. Apply the annual recurrence with each beginning-of-year cash flow added before that year’s return. This avoids treating a beginning-of-year amount as though it were received at year-end.
Base projection, year 1. The funds invested are \(0+600=600\). Thus \(A_1=600(1.04)=624\).
Base projection, year 2. Add the second cash flow to the year 1 ending asset share: \(624+600=1{,}224\). Then \(A_2=1{,}224(1.04)=1{,}272.96\).
Alternative-return projection, year 1. The same $600 is invested, so \(A'_1=600(1.05)=630\).
Alternative-return projection, year 2. The funds before interest are \(630+600=1{,}230\). Therefore \(A'_2=1{,}230(1.03)=1{,}266.90\).
Comparison. The ending difference is \(A'_2-A_2=1{,}266.90-1{,}272.96=-6.06\). The alternative experience leaves $6.06 less at the end of year 2, despite the higher return in year 1. The lower second-year return applies to a larger accumulated amount, so the sequence of returns matters.
Check. The recurrence gives \(1{,}230(1.03)=1{,}266.90\), and subtracting the base amount gives \(-6.06\), consistent with the comparison.
Worked Example: A Return Gain Followed by a Return Loss
A hypothetical policy has an initial asset share of $10,000. At the beginning of year 1 it receives a net cash flow of $1,000. At the beginning of year 2 it pays a net outflow of $2,000, so \(C_2=-2{,}000\). The base return is 4% in both years. Alternative investment experience is 6% in year 1 and 2% in year 2. Find each ending asset share and trace the source of their difference.
Base projection, year 1. Funds before the return are \(10{,}000+1{,}000=11{,}000\), giving \(A_1=11{,}000(1.04)=11{,}440\).
Base projection, year 2. After the beginning-of-year outflow, \(11{,}440-2{,}000=9{,}440\) is invested. Thus \(A_2=9{,}440(1.04)=9{,}817.60\).
Alternative projection, year 1. The funds are again $11,000, so \(A'_1=11{,}000(1.06)=11{,}660\). The year 1 difference is \(D_1=11{,}660-11{,}440=220\).
Alternative projection, year 2. After the same $2,000 outflow, the invested amount is \(11{,}660-2{,}000=9{,}660\). Therefore \(A'_2=9{,}660(1.02)=9{,}853.20\).
The ending difference is \(9{,}853.20-9{,}817.60=35.60\). It can also be checked using the decomposition. The $220 difference carried from year 1 grows to \(220(1.02)=224.40\). The year 2 return difference contributes \(9{,}440(0.02-0.04)=-188.80\). Adding these terms gives \(224.40-188.80=35.60\).
Check. The alternative ending asset share exceeds the base amount by $35.60. The year 2 return shortfall reduces, but does not eliminate, the earlier gain.
Worked Example: A Small Return Increase Accumulated Over Three Years
A hypothetical policy starts with an asset share of $0 and receives $1,000 at the beginning of each of three years. The base return is 4% per year. A sensitivity projection uses 5% per year. Calculate the ending asset shares and show how the difference develops.
Base projection. Year 1 gives \(A_1=(0+1{,}000)(1.04)=1{,}040\). In year 2, \(A_2=(1{,}040+1{,}000)(1.04)=2{,}040(1.04)=2{,}121.60\). In year 3, \(A_3=(2{,}121.60+1{,}000)(1.04)=3{,}121.60(1.04)=3{,}246.464\).
Alternative projection. Year 1 gives \(A'_1=1{,}000(1.05)=1{,}050\). Year 2 gives \(A'_2=(1{,}050+1{,}000)(1.05)=2{,}050(1.05)=2{,}152.50\). Year 3 gives \(A'_3=(2{,}152.50+1{,}000)(1.05)=3{,}152.50(1.05)=3{,}310.125\).
The unrounded ending difference is \(3{,}310.125-3{,}246.464=63.661\), or $63.66 to the nearest cent. The difference decomposition provides a check. In year 1 it is \(1{,}000(0.01)=10\). In year 2 it is \(10(1.05)+2{,}040(0.01)=10.50+20.40=30.90\). In year 3 it is \(30.90(1.05)+3{,}121.60(0.01)=32.445+31.216=63.661\). The year 3 amount includes both the accumulated earlier differences and the year 3 return effect.
Check. The direct projections and the year-by-year decomposition both give an ending difference of $63.661 before rounding. A persistent return increase has a growing effect because it applies to accumulated funds and earlier return gains.
Why the Sensitivity Formula Holds
The decomposition follows directly from the asset-share recurrence; it does not require an assumption that the return difference is small. Start from the two projections for year \(t\), then add and subtract the base-year invested amount multiplied by the alternative return:
The split has a practical interpretation. A return difference may create a gain or loss on the funds in the current year; that difference then becomes part of the asset share exposed to future returns. If a future cash outflow occurs, it reduces the amount exposed to that future year’s return. Consequently, both the timing of cash flows and the order of the return assumptions matter.
If the cash flows and returns are the same in both projections through year \(t-1\), then \(D_{t-1}=0\). The first year with a different return therefore has \(D_t=(A_{t-1}+C_t)(j'_t-j_t)\). This special case is useful for isolating one year’s immediate return effect. In later years, the difference already accumulated must also be carried forward.
Interpreting the Result and Avoiding Traps
An ending asset-share difference measures a change in accumulated funds on the stated basis. It can help assess how much is available to support a hypothetical benefit at a specified date, but it does not determine the benefit itself. The benefit terms and any applicable guarantees must be considered separately. Likewise, this comparison changes investment returns only; changing cash flows or other experience at the same time would no longer isolate investment sensitivity.
A common error is to apply the annual return to the previous ending asset share but omit the beginning-of-year cash flow. In Worked Example 1, that incorrect year 1 calculation would be \(0(1.05)=0\), ignoring the $600 invested at the beginning of the year. Another common error is to add a return difference directly to the final asset share without allowing it to compound. The decomposition shows why the existing difference is multiplied by \(1+j'_t\) in each later year.
The recurrence is tied to the timing convention. If the $600 in Worked Example 1 were instead received at the end of each year, it would not earn that year’s return. One must then change the recurrence to reflect the stated timing rather than use the beginning-of-year formula mechanically. Always identify the date of each cash flow and the period over which it earns investment income before calculating.
Check Your Understanding
For each problem, use the stated beginning-of-year cash-flow convention and compare the requested asset shares.
- A hypothetical policy starts at $0 and receives $800 at the beginning of each of two years. Returns are 3% in year 1 and 5% in year 2. What is the ending asset share? (Answer: $1,705.20)
- A policy has an initial asset share of $5,000 and a beginning-of-year net outflow of $400 in year 1. If the effective annual net return is 4%, find the year-end asset share. (Answer: $4,784.00)
- A hypothetical policy starts at $2,000, receives $500 at the beginning of year 1, and pays $300 at the beginning of year 2. Compare the ending asset shares under a 3% base return in both years and alternative returns of 4% in both years. (Answer: Base $2,343.25; alternative $2,392.00; difference $48.75)
- A policy starts at $0 and receives $1,000 at the beginning of each of three years. The base return is 4% each year; alternative returns are 4%, 5%, and 6%. Find the ending difference, alternative minus base. (Answer: $84.06)
- A policy starts at $10,000 with no later cash flows. The base return is 4% in each of two years; alternative returns are 5% and 3%. Find the year 2 ending asset shares and their difference. (Answer: Base $10,816.00; alternative $10,815.00; difference \(-$1.00\))