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Exam ALTAM · Equity-Linked Benefits · Tutorial 660 of 999

Unit Accounts and Unit Prices

Learn to value unit accounts at a specified date and apply a stated benefit formula without confusing unit holdings, unit prices, and dollars.

Advanced 10 min read

What You'll Learn

  • Define a unit, a unit price, and a policy fund value
  • Calculate the value of a single unit account at a specified valuation time
  • Add values across multiple unit funds using the correct holdings and prices
  • Apply a stated contract formula to an investment-linked benefit
  • Identify errors caused by averaging prices or mixing valuation dates

Introduction

An equity-linked policy may express the policyholder’s investment account as a number of units in one or more funds. The units are not a dollar amount: their dollar value depends on the applicable unit price. To calculate an investment-linked benefit, first value each holding at the relevant time, then apply the benefit formula specified by the hypothetical contract.

This distinction matters whenever unit prices differ across funds or dates. Multiplying the right number of units by a price from the wrong fund, or combining holdings and prices from different valuation times, can produce a plausible-looking but incorrect benefit.

Definition: A unit is a standardized interest in a unitized investment fund. The unit price is the dollar value assigned to one unit at a specified valuation time, on the pricing basis stated for the policy. If a policy holds \(N\) units at unit price \(p\), its value in that fund is \(Np\). The fund value of a policy with holdings in several funds is the sum of the values of its holdings, all measured at the same valuation time and in the same currency.

From Units to Dollars

For one fund, the calculation is direct: multiply the number of units held by the applicable price per unit. For several funds, do that calculation separately for each fund before adding the results. If fund \(j\) has \(N_j\) units and a unit price \(p_j\), the total fund value \(F\) is

$$ F=\sum_{j=1}^{m}N_jp_j. $$

Here \(m\) is the number of funds in which the policy has holdings. The number of units and the unit price must refer to the same fund. They must also be measured at a consistent valuation time: a holding recorded at one date cannot generally be valued using a price from another date unless the contract specifically requires it.

Valuation convention: Use the unit price and number of units specified for the benefit’s valuation time. If the contract defines an investment-linked benefit \(B\) as a function \(g\) of fund value, calculate \(F\) first and then apply that stated function: \(B=g(F)\). Do not assume a benefit formula that is not given.

The unit price is a price per unit; it is not the policyholder’s rate of investment return, the amount of a premium, or the fund value itself. A policy can have a high unit price and a small fund value if it holds few units, or a lower unit price and a larger fund value if it holds many units. The monetary value is determined by the product of the two.

Worked Example: One Fund and a Benefit Multiple

A hypothetical policy is recorded as holding 1,840 units in its investment fund. At the benefit valuation time, the applicable unit price is $12.50. Under the stated contract for this example, the benefit is 104% of the fund value. Find the fund value and the benefit. The holding and price are deterministic valuation inputs; no interest-rate, mortality, or probability distribution assumption is needed or used.

Method. Value the units at the specified price, then apply the contract’s benefit multiple. This avoids treating the number of units as if it were already a dollar amount.

Fund value. Each of the 1,840 units is worth $12.50, so

$$ F=1{,}840\times \$12.50=\$23{,}000.00. $$

Benefit. The contract pays 104% of this value, so

$$ B=1.04\times \$23{,}000.00=\$23{,}920.00. $$

The result is a fund value of $23,000.00 and a benefit of $23,920.00. As a check, the additional 4% is \(0.04\times \$23{,}000=\$920\), and \(\$23{,}000+\$920=\$23{,}920\). A common mistake would be to report $1,840 as the benefit; that is a unit count, not a dollar amount.

Valuing Holdings in More Than One Fund

A policy may hold units in more than one unitized fund. The total value is not generally the total number of units multiplied by one selected unit price. Each fund has its own holding and price, so each product must be calculated separately. The resulting dollar values can then be added.

Worked Example: Two Funds and a Specified Benefit Formula

A hypothetical policy has 725 units in Fund A and 1,260 units in Fund B. At the benefit valuation time, the applicable prices are $18.40 per unit for Fund A and $7.25 per unit for Fund B. The contract in this example defines the benefit as 98% of total fund value plus a fixed $250. Calculate the benefit. These are deterministic snapshot values; no interest, mortality, or probability distribution assumption is needed.

Method. Calculate each fund’s dollar value using its own units and price, add the values, and then apply the stated formula. This avoids mixing the price of one fund with the units of another.

For Fund A,

$$ F_A=725\times \$18.40=\$13{,}340.00. $$

For Fund B,

$$ F_B=1{,}260\times \$7.25=\$9{,}135.00. $$

Therefore, total fund value is

$$ F=F_A+F_B=\$13{,}340.00+\$9{,}135.00=\$22{,}475.00. $$

Apply the 98% factor and then add the fixed amount:

$$ B=0.98\times \$22{,}475.00+\$250.00 =\$22{,}025.50+\$250.00 =\$22{,}275.50. $$

The fund value is $22,475.00 and the benefit is $22,275.50. Check the percentage calculation by finding the 2% reduction: \(0.02\times \$22{,}475=\$449.50\); thus 98% is \(\$22{,}475-\$449.50=\$22{,}025.50\). Adding $250 gives the stated benefit. A common wrong calculation is to add the unit counts and multiply by one fund’s price. That treats units in different funds as interchangeable at one price, which they are not.

Why the Valuation Formula Holds

The calculation follows from the meaning of a unit price. At the valuation time, one unit in fund \(j\) has a value of \(p_j\) dollars. If the policy holds \(N_j\) units, the value of those units is the sum of \(N_j\) individual unit values. Since each is priced at \(p_j\), the sum is \(p_j\) added \(N_j\) times, or \(N_jp_j\). Holdings in different funds are distinct components, so their dollar values add.

$$ \underbrace{p_j+p_j+\cdots+p_j}_{N_j\text{ units}}=N_jp_j, \qquad F=\sum_{j=1}^{m}N_jp_j. $$

This reasoning assumes that the stated unit price applies to every unit in that holding at the valuation time, and that the contract’s pricing convention has been followed. It also assumes that the holdings and prices have been stated in compatible units and currency. If the contract specifies a different pricing basis, a conversion, or a separate adjustment, that instruction must be applied as stated rather than silently built into the unit-price calculation.

Comparing Values Across Valuation Times

The number of units and the unit price play different roles when an account is observed at two dates. If the holding is unchanged, a change in fund value reflects a change in the applicable price. If holdings differ between dates, both the unit count and the price affect the comparison. In either case, calculate the fund value at each date using the units and prices that belong to that date.

Worked Example: Comparing Two Account Values

A hypothetical policy holds units in two funds. At the first valuation time, it has 500 units in Fund R priced at $10.00 and 800 units in Fund S priced at $15.00. At the second valuation time, the holdings remain 500 and 800 units, while the prices are $12.00 and $13.50, respectively. Under the stated contract in this example, the benefit at the second valuation time is 102% of fund value. Find both fund values, the percentage change in value, and the benefit. The scenario is a deterministic comparison of snapshots, not a stochastic investment model; no interest or mortality assumption is used.

Method. Value each fund separately at each date. Compare the resulting totals rather than averaging the two prices. This avoids giving equal weight to prices when the policy holds unequal numbers of units.

At the first valuation time,

$$ F_0=(500\times \$10.00)+(800\times \$15.00) =\$5{,}000.00+\$12{,}000.00 =\$17{,}000.00. $$

At the second valuation time,

$$ F_1=(500\times \$12.00)+(800\times \$13.50) =\$6{,}000.00+\$10{,}800.00 =\$16{,}800.00. $$

The dollar change is

$$ F_1-F_0=\$16{,}800.00-\$17{,}000.00=-\$200.00. $$

The percentage change, measured relative to the first value, is

$$ \frac{F_1-F_0}{F_0}\times 100\% =\frac{-\$200.00}{\$17{,}000.00}\times 100\% \approx -1.18\%. $$

The benefit at the second valuation time is

$$ B=1.02\times \$16{,}800.00=\$17{,}136.00. $$

Thus, the fund value decreases by $200.00, or 1.18% when rounded to two decimal places, and the stated benefit is $17,136.00. The result is reasonable even though Fund R’s price rose: the decrease in Fund S’s value exceeds the increase in Fund R’s value. A tempting but incorrect approach is to average the two prices at each date and multiply by the total units. That would weight each fund equally, despite the policy holding 500 units of one and 800 of the other.

Common Valuation Traps

The arithmetic is usually short; the main challenge is matching each input to the right fund, time, and benefit definition. Before calculating, identify the valuation date, the relevant unit count and price for each holding, and the exact contract formula to apply. Keep intermediate fund values separate until every holding has been valued.

  • Confusing units with dollars. A holding of 900 units is not a $900 account. Multiply by the applicable price per unit.
  • Using one price for several funds. Each holding must be paired with the price for its own fund.
  • Averaging prices without regard to holdings. Fund value uses a holding-weighted sum of unit prices, not an unweighted average.
  • Mixing valuation times. Use holdings and prices from the same specified valuation time unless the contract directs otherwise.
  • Changing the stated benefit formula. Apply the multiplier, fixed amount, or other formula exactly as given; do not substitute a different benefit definition.
Key takeaway: At a specified valuation time, calculate each fund holding’s value as units multiplied by that fund’s unit price. Add the dollar values to obtain total fund value, then apply only the investment-linked benefit formula stated in the contract.

Check Your Understanding

For each hypothetical scenario, calculate the requested values using the specified holdings, prices, and benefit formula.

  1. A policy holds 860 units at $11.75 per unit. Its stated benefit is 103% of fund value. Find the fund value and benefit. (Answer: Fund value $10,105.00; benefit $10,408.15)
  2. A policy holds 400 units in Fund A at $8.60 and 950 units in Fund B at $12.40. The stated benefit is 97% of total fund value plus $300. Find the total fund value and benefit. (Answer: Fund value $15,220.00; benefit $15,063.40)
  3. A policy holds 450 units in Fund C and 900 units in Fund D at both valuation times. Prices change from $20.00 to $18.00 for Fund C and from $5.00 to $6.00 for Fund D. Find the fund value at each time and the percentage change. (Answer: First value $13,500.00; second value $13,500.00; change 0.00%)
  4. A policy holds 250 units in one fund priced at $16.00 and 750 units in another fund priced at $8.00. Its stated benefit is 105% of fund value. Find the fund value and benefit. (Answer: Fund value $10,000.00; benefit $10,500.00)
  5. A policy has 600 units in Fund E at $9.50 and 400 units in Fund F at $14.00. The stated benefit is 99% of total fund value plus $125. Find each fund value, the total fund value, and the benefit. (Answer: Fund E $5,700.00; Fund F $5,600.00; total $11,300.00; benefit $11,312.00)

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