Credited Interest on Policy Accounts

Why the Rate Basis Matters

A universal life policy account earns interest at a rate set under the policy’s crediting provisions. The insurer may declare a rate for a period, while the contract may also specify a guaranteed minimum. To determine the interest credited, a calculation must respect both the contract’s rule and the way each rate is quoted. In particular, a nominal annual rate convertible quarterly is not the same as an effective annual rate with the same stated percentage.

This tutorial considers annual account projections in which a balance earns interest over a year and no premiums, charges, withdrawals, or other transactions occur during that year. Those cash flows affect an account projection when present, but the focus here is how to convert and apply credited rates. In each example, the contract is stated to credit the greater of the declared effective annual rate and the guaranteed effective annual minimum. Actual policy provisions can define a different rule, so the contract’s wording always controls.

The rate-conversion and comparison techniques here are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The central discipline is to compare rates on a consistent basis before applying them to an account balance.

Nominal and Effective Annual Rates

Definition: An effective annual rate $i$ is the proportion by which a balance increases over one year when interest is applied once at year-end. A nominal annual rate $i^{(m)}$, convertible $m$ times per year, is a quoted annual rate under which the rate applied at each conversion is $i^{(m)}/m$. Its effective annual equivalent is $\left(1+i^{(m)}/m\right)^m-1$.

For example, a nominal annual rate of $5\%$ convertible quarterly applies $5\%/4=1.25\%$ at each quarterly conversion. Over a full year, those four increases compound. The effective annual rate is therefore $(1.0125)^4-1$, not simply $5\%$. A rate already stated as effective annually should not be converted again.

To keep the account projection clear, let $B_t$ denote the policy account balance at the start of year $t+1$. If the credited rate for that year is an effective annual rate $j_t$, and there are no account transactions during the year, then the balance at year-end is $B_{t+1}=B_t(1+j_t)$. The credited rate $j_t$ must be the rate specified by the policy’s rule, expressed on an effective annual basis.

Compare Rates Only After Conversion

Suppose the policy guarantees a minimum effective annual rate $g$, and the declared rate is quoted as nominal $i^{(m)}$ convertible $m$ times per year. Under a contract that credits the greater of the two annual rates, first calculate the declared rate’s effective annual equivalent. Then compare that equivalent with $g$. The greater rate is applied to the opening balance for the annual period.

j=\max\left\{\left(1+\frac{i^{(m)}}{m}\right)^m-1,\ g\right\}, \qquad B_{t+1}=B_t(1+j).

This rule is specific to the stated contract provision. A guarantee might instead be defined for each conversion period, or the policy might describe a different method for setting its declared rate. Do not assume that every contract uses a “greater of” rule; establish the contractual rule before calculating.

Why the Conversion Formula Holds

The conversion formula follows directly from applying the nominal rate at each of the $m$ conversion times. Start with a balance $B$. At the first conversion it becomes $B(1+i^{(m)}/m)$; after the second, it becomes $B(1+i^{(m)}/m)^2$. Repeating this multiplication for all $m$ conversions gives the year-end balance below.

B_{\text{year-end}}=B\left(1+\frac{i^{(m)}}{m}\right)^m.

The effective annual rate is defined as the single annual rate that produces the same year-end balance. Thus $B(1+i_{\text{eff}})=B(1+i^{(m)}/m)^m$. For $B>0$, dividing by $B$ and subtracting 1 gives $i_{\text{eff}}=(1+i^{(m)}/m)^m-1$. This derivation assumes the stated nominal rate applies uniformly at each conversion during the year.

Applying the Rate to an Account

Worked Example: Quarterly Declared Rate Versus an Annual Guarantee

A hypothetical policy account has a balance of $10,000 at the start of a year. The policy credits the greater of a declared rate of $5.0\%$ nominal per year convertible quarterly and a guaranteed rate of $5.0\%$ effective annually. Find the effective annual declared rate, identify the rate credited, and calculate the year-end balance.

Method. Convert the declared nominal rate to an effective annual rate, compare it with the guaranteed effective rate, and apply the selected rate for one year.

The rate per quarter is $0.05/4=0.0125$, or $1.25\%$. The declared rate’s effective annual equivalent is:

$$ (1+0.0125)^4-1 =1.0509453369140625-1 =0.0509453369140625. $$

As a percentage, this is $5.0945336914\%$, or $5.0945\%$ to four decimal places. It exceeds the $5.0\%$ effective annual guarantee, so the declared rate determines the credit. The year-end balance is:

B_{\text{year-end}} =\$10{,}000(1.0509453369140625) =\$10{,}509.453369\ldots =\$10{,}509.45.

Check. The interest credited is $\$10{,}509.45-\$10{,}000=\$509.45$ to the nearest cent. This is consistent with an effective rate of about $5.0945\%$ on $10,000. A common mistake is to treat the nominal $5.0\%$ as the effective annual rate; the quarterly compounding makes the effective rate higher. Money is rounded to the nearest cent only at the end.

Worked Example: The Guarantee Exceeds the Converted Declared Rate

A hypothetical policy account begins the year with $25,000. Under its stated rule, it earns the greater of a declared rate of $4.8\%$ nominal per year convertible semiannually and a guaranteed rate of $4.9\%$ effective annually. Determine the credited rate and year-end balance.

Method. Convert the declared rate before comparing it with the guarantee. The nominal rate per half-year is $0.048/2=0.024$, or $2.4\%$. Its effective annual equivalent is:

(1+0.024)^2-1 =1.048576-1 =0.048576 =4.8576\%.

The guarantee of $4.9\%$ is greater than $4.8576\%$, so $j=0.049$. The year-end account balance is:

B_{\text{year-end}} =\$25{,}000(1.049) =\$25{,}000+\$1{,}225 =\$26{,}225.00.

If the declared rate had been used instead, the balance would have been $\$25{,}000(1.048576)=\$26{,}214.40$, which is lower than the balance under the stated guarantee. This provides a direct check that applying the floor increases the credited amount. The calculation assumes the guarantee is an effective annual minimum, as specified in the problem.

More Than One Annual Crediting Period

When the declared rate or the guarantee changes from year to year, select a credited effective annual rate separately for each year. Apply each year’s rate to the balance accumulated at the end of the preceding year. This compounds the account’s interest; it does not reset the account to its original balance. The calculation below assumes no premiums, deductions, or other transactions during either year.

Worked Example: Two Years with Different Declared Rates

A hypothetical policy account starts with $18,000. The policy credits the greater of the declared effective annual rate and the guaranteed effective annual minimum each year. In year 1, the declared rate is $4.0\%$ nominal per year convertible quarterly and the guarantee is $4.0\%$ effective annually. In year 2, the declared rate is $3.8\%$ nominal per year convertible semiannually and the guarantee is $4.0\%$ effective annually. Find the account balance at the end of year 2.

Method. Convert each year’s declared rate, select that year’s greater rate, and apply the rates successively.

For year 1, the quarterly rate is $0.04/4=0.01$, so the effective annual declared rate is:

(1.01)^4-1=1.04060401-1=0.04060401=4.060401\%.

This exceeds the $4.0\%$ guarantee. The balance after year 1 is:

B_1=\$18{,}000(1.04060401)=\$18{,}730.87218.

For year 2, the semiannual rate is $0.038/2=0.019$, so the effective annual declared rate is:

(1.019)^2-1=1.038361-1=0.038361=3.8361\%.

This is below the $4.0\%$ guarantee, so the year 2 credited rate is $4.0\%$. Therefore:

B_2 =\$18{,}730.87218(1.04) =\$19{,}480.1070672 =\$19{,}480.11.

Check. The selected rates are $4.060401\%$ in year 1 and $4.0\%$ in year 2. The combined accumulation factor is $1.04060401(1.04)=1.0822281704$; multiplying this by $18,000 gives the same $19,480.11 after rounding. Applying the year 2 rate to the original $18,000 would omit interest already credited in year 1.

Common Exam Traps

The most frequent error is comparing quoted percentages before identifying their bases. A nominal rate of $5.0\%$ convertible quarterly and a $5.0\%$ effective annual guarantee are not equal rates: the former is equivalent to $5.0945\%$ effective annually. The correct comparison is between $5.0945\%$ and $5.0\%$, not between the two printed figures.

Here is the wrong calculation for the first worked example:

\text{Wrong: } \$10{,}000(1.05)=\$10{,}500.00.

This treats $5.0\%$ nominal convertible quarterly as though it were $5.0\%$ effective annually and ignores the four quarterly conversions. The correction is to calculate $(1+0.05/4)^4-1=0.0509453369140625$, compare that effective rate with the annual guarantee, and apply it to the opening balance. The resulting balance is $10,509.45.

Other checks help prevent related errors:

Do not divide an effective annual rate by the number of conversions unless the policy or problem explicitly defines a periodic rate that way.
Do not compare a nominal rate directly with an effective annual guarantee.
Apply the guarantee according to its stated basis and the contract’s rule; do not presume that all products use the greater-of rule.
In successive annual projections, apply each year’s selected rate to the balance carried forward from the preceding year.
Round the final money amount to cents, rather than rounding intermediate rates so aggressively that the balance changes.

Key takeaway: Identify how each rate is quoted, convert nominal rates to effective annual rates when needed, apply the policy’s stated guarantee rule, and then multiply the account balance by one plus the selected annual rate.

Check Your Understanding

For each problem, assume the policy credits the greater of the declared effective annual rate and the stated guaranteed effective annual minimum. Unless stated otherwise, there are no account transactions during the crediting period.

A hypothetical account starts with $8,000. The declared rate is $3.6\%$ nominal convertible quarterly, and the guarantee is $3.7\%$ effective annually. Convert the declared rate and find the year-end balance. (Answer: declared effective rate $3.6489\%$; credited rate $3.7\%$; $8,296.00)
A $10,000 account has a declared rate of $4.0\%$ nominal convertible quarterly and a $4.05\%$ effective annual guarantee. Find the effective declared rate and the year-end balance. (Answer: declared effective rate $4.0604\%$; credited rate $4.0604\%$; $10,406.04)
A hypothetical account of $20,000 has a declared rate of $6.0\%$ nominal convertible semiannually and a guaranteed rate of $6.1\%$ effective annually. Determine the credited rate and year-end balance. (Answer: declared effective rate $6.09\%$; credited rate $6.1\%$; $21,220.00)
A $10,000 account has a $4.0\%$ effective annual guarantee. In year 1 the declared rate is $4.0\%$ nominal convertible quarterly; in year 2 it is $3.0\%$ effective annually. Find the balance after two years. (Answer: year 1 balance $10,406.04; year 2 credited rate $4.0\%$; final balance $10,822.28)
A hypothetical contract compares a declared rate of $5.0\%$ nominal convertible semiannually with a $5.1\%$ effective annual guarantee. For a $12,000 opening balance, calculate the declared effective rate, identify the credited rate, and find the year-end balance. (Answer: declared effective rate $5.0625\%$; credited rate $5.1\%$; $12,612.00)

What You'll Learn