Accumulating Unit Prices Under Deterministic Returns

Why Unit Prices Accumulate by Compounding

In an equity-linked policy, a unit price translates the value of an investment option into a value per unit. When the underlying investment earns a return, the price of each unit changes; the number of units held is a separate quantity. This tutorial focuses on how a unit price accumulates through a sequence of deterministic returns, before considering policy cash flows or deductions.

The key calculation is multiplicative. A return is earned on the value already accumulated, so each period’s return applies to the current unit price, not just to the initial price. This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $U_t$ be the unit price at the end of period $t$, and let $r_t$ be the effective investment return for period $t$, from time $t-1$ to time $t$. The return is measured on a consistent basis with the quoted unit prices. For this tutorial, returns are deterministic, there are no external cash flows or unit-level deductions during a period, and any investment income included in the stated return is reinvested. The gross return factor for period $t$ is $1+r_t$.

A return of $r_t=0.04$, for example, means that one unit’s value increases by 4% over that period. A negative return is also permitted, provided the resulting unit price remains nonnegative. If returns are stated net of investment expenses, the recursion uses those net returns; if they are gross, the resulting price is before those expenses. The basis must be clear and consistent.

The Unit-Price Recursion

By definition, the period return is the change in unit price divided by the price at the start of the period. Rearranging that definition gives the recursion.

r_t=\frac{U_t-U_{t-1}}{U_{t-1}} \quad\Longrightarrow\quad U_t=U_{t-1}(1+r_t).

Applying the recursion repeatedly gives the price after $n$ periods. Each step multiplies the current price by that period’s gross factor.

U_n=U_0\prod_{t=1}^{n}(1+r_t).

The cumulative return over the full $n$-period interval is the total change relative to the initial unit price. It is therefore the product of the gross factors, less one.

R_{0,n}=\frac{U_n-U_0}{U_0} =\prod_{t=1}^{n}(1+r_t)-1.

Why the product formula holds: For one period, the definition of return gives $U_1=U_0(1+r_1)$. If $U_k=U_0\prod_{t=1}^{k}(1+r_t)$, then the next-period recursion gives $U_{k+1}=U_k(1+r_{k+1})=U_0\prod_{t=1}^{k+1}(1+r_t)$. Thus, by induction, the product formula holds for every positive integer $n$ under the stated assumptions.

This result depends on matching the return periods to the unit-price dates. A monthly return must be applied once for each month; it cannot be treated as an annual return without converting the period basis. Also, if a quoted price excludes distributions while a return includes reinvested distributions, the price series and return definition are not on the same basis. Use returns consistent with the prices being accumulated.

Worked Example: Accumulating Through a Gain and a Loss

A hypothetical investment option has a unit price of $100.00 at the start of a three-period sequence. Its deterministic period returns are 5%, −10%, and 8%. Calculate each closing unit price and the cumulative return. The returns are effective returns for the stated periods, with investment income reinvested.

Method. Apply $U_t=U_{t-1}(1+r_t)$ in chronological order. The common mistake to avoid is adding the returns, or multiplying their arithmetic average by the number of periods, instead of multiplying gross factors.

At the end of period 1:

U_1=100.00(1+0.05)=100.00(1.05)=\$105.00.

At the end of period 2, the 10% loss applies to $105.00, not to the original $100.00:

U_2=105.00(1-0.10)=105.00(0.90)=\$94.50.

At the end of period 3:

U_3=94.50(1+0.08)=94.50(1.08)=\$102.06.

The cumulative gross factor is $1.05(0.90)(1.08)=1.0206$. Hence the cumulative return is $1.0206-1=0.0206$, or 2.06%, and the final unit price is $100.00(1.0206)=\$102.06$. This checks the period-by-period calculation.

The arithmetic average of the three returns is $(5\%-10\%+8\%)/3=1\%$. Multiplying that average by three gives 3%, not the actual cumulative return of 2.06%. More generally, multiplying an arithmetic average by the number of periods does not produce the accumulated return; the product of gross factors gives the actual result. In special cases the two calculations can coincide, but the average-based calculation is not generally valid.

Working Backward From Observed Prices

The same recursion can be rearranged to infer a period return from consecutive unit prices. This is useful when a price history is given but the returns are not stated. The calculation requires a positive starting price for the period.

r_t=\frac{U_t}{U_{t-1}}-1.

When one return in a sequence is unknown, first accumulate all known gross factors, then divide the required total gross factor by their product. Do not subtract known percentage returns from the cumulative return: percentage returns compound.

Worked Example: Solving for a Missing Period Return

A hypothetical unit price is $120.00 at time 0. The next two period returns are 10% and −5%. The price after a third period is observed to be $135.432. Find the third-period return and verify the result.

Method. The observed terminal price determines the total gross factor. Divide it by the initial price and by the two known gross factors to isolate the unknown factor.

The known first two periods give:

U_1=120.00(1.10)=\$132.00, \qquad U_2=132.00(0.95)=\$125.40.

The unknown third-period gross factor must take the price from $125.40 to $135.432:

1+r_3=\frac{135.432}{125.40}=1.08, \qquad r_3=0.08=8\%.

Verification: $125.40(1.08)=\$135.432$, the observed terminal price. The common mistake is to treat the cumulative change from $120.00 as a simple sum of returns. The correct method uses the known accumulated price after two periods as the base for the unknown return.

Period Length, Quoted Scale, and Interpretation

Returns can be monthly, quarterly, annual, or defined over another interval. The recursion itself does not change, but the return must be applied the correct number of times. For four quarterly returns covering a year, the effective annual return is the product of the four quarterly gross factors less one. It is not generally four times the arithmetic mean of those quarterly returns.

A quoted unit price may also be rebased. For example, a fund might change the price scale by dividing every quoted price by a fixed number. If every price is multiplied by the same positive constant $c$, then the return is unchanged:

\frac{cU_t}{cU_{t-1}}-1 =\frac{U_t}{U_{t-1}}-1.

Thus, a price quoted in cents rather than dollars can still produce the same percentage return, provided both prices use the same scale. The absolute quoted amount changes, but the proportional accumulation does not.

Worked Example: Accumulating Quarterly Returns Over One Year

A hypothetical option has a unit price of $50.00 at the start of a year. Its four successive quarterly returns are 3%, −2%, 1.5%, and 2.5%. Calculate the quarter-end prices, the year-end price to the nearest cent, and the effective return for the year. Assume each return covers exactly one quarter and includes reinvested investment income.

Method. Apply each quarterly gross factor once, in order. Keep unrounded values during the calculation and round only the final dollar price to cents.

After quarter 1:

U_1=50.00(1.03)=\$51.50.

After quarter 2:

U_2=51.50(0.98)=\$50.47.

After quarter 3:

U_3=50.47(1.015)=\$51.22705.

After quarter 4:

U_4=51.22705(1.025)=\$52.50772625.

Rounded to the nearest cent, the year-end price is $52.51. The effective annual return is calculated from the unrounded price:

\frac{52.50772625}{50.00}-1 =1.050154525-1 =0.050154525 =5.0154525\%.

The annual effective return is approximately 5.015% before rounding. The quarterly arithmetic average is $(3\%-2\%+1.5\%+2.5\%)/4=1.25\%$, and four times that average is 5%. That is not the actual annual accumulation here; the product of quarterly gross factors gives 5.0154525%. Using the unrounded intermediate prices also ensures that rounding at quarter ends does not introduce avoidable discrepancies.

Common Exam Traps

Several errors can produce plausible-looking but incorrect answers. Check the return definition, the period length, and the base to which each return is applied.

Adding returns as if they were dollar changes. A 10% gain followed by a 10% loss does not return the price to its starting value: the gross factor is $1.10(0.90)=0.99$, a cumulative loss of 1%.
Applying every return to the initial price. Each period’s return applies to the unit price at the beginning of that period, so the recursion is sequential.
Confusing an arithmetic average with accumulation. The arithmetic average describes the average listed periodic rate. It does not, in general, determine the cumulative return; multiply gross factors instead.
Mixing time units. A quarterly return is not an annual return. Apply it once per quarter in a quarterly projection, or convert it consistently before using an annual period.
Rounding intermediate prices too early. Carry sufficient precision through the sequence, then round the requested final amount.
Mixing price and return bases. Confirm that the stated return and the unit prices both include or exclude investment income and expenses consistently.

Key takeaway: Under deterministic periodic returns and consistent price definitions, each period updates the unit price by $U_t=U_{t-1}(1+r_t)$. Over several periods, accumulate by multiplying gross factors; the cumulative return is their product less one.

Check Your Understanding

For each problem, use the stated period returns and price basis consistently. Round dollar amounts to the nearest cent only when requested.

A hypothetical unit price starts at $40.00 and earns returns of 6% and then −4%. Find the final price and cumulative return. (Answer: $40.70; cumulative return 1.76%)
A unit price starts at $75.00. Its successive returns are −8%, 12%, and 5%. Find the price after each period and the final price to the nearest cent. (Answer: $69.00, $77.28, $81.14)
A unit price starts at $250.00, earns 8% in the first period and −5% in the second, and ends the third period at $271.89. Find the third-period return. (Answer: 6%)
Two hypothetical options both start at $100.00. Option A earns 30% and then −10%; Option B earns 10% in each of two periods. Their arithmetic average periodic returns are equal. Find each terminal price and explain whether the accumulated values are equal. (Answer: Option A, $117.00; Option B, $121.00; the accumulated values are not equal)
A quoted unit price is $30.00 at the start of a year. Quarterly returns are 1%, −2%, 3%, and 4%. Find the end-of-year price to the nearest cent and the cumulative return. (Answer: $31.81; cumulative return 6.027376%)

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