Monthly Crediting and Charge Timing

Why Monthly Timing Matters

Universal life account projections often involve a credited interest rate and cash flows that recur monthly. A calculation can use the correct annual rate and the correct premium and charge amounts yet still produce the wrong account value if it applies interest to the wrong balance. The essential question is: which amounts are in the account during the period that earns interest?

This tutorial treats the credited rate and the timing convention as explicit model inputs. Premiums are amounts allocated to the account after any premium load. Monthly deductions are specified dollar amounts; the calculations do not determine mortality charge rates or other charge formulas. Unless an example says otherwise, assume the policy remains in force, the account has sufficient funds for every deduction, and no minimum-account floor or other transaction applies.

Definition: Let $A_{m-1}$ be the account value at the start of month $m$, before that month’s transactions. Let $P_m$ be the net premium allocated to the account during month $m$, and let $Q_m$ be the monthly deduction. Let $j$ be the effective credited rate for one month. Under an end-of-month convention, the opening account earns interest before the premium and deduction are posted: $A_m=A_{m-1}(1+j)+P_m-Q_m$. Under a beginning-of-month convention, the premium and deduction are posted before the account earns the month’s interest: $A_m=(A_{m-1}+P_m-Q_m)(1+j)$.

Finding the Monthly Credited Rate

The monthly rate must match the way the annual credited rate is stated. If the credited rate is an annual effective rate $i_c$, the equivalent effective monthly rate $j$ satisfies $(1+j)^{12}=1+i_c$. Therefore,

j=(1+i_c)^{1/12}-1.

If instead the credited rate is a nominal annual rate $i_c^{(12)}$, convertible monthly, then the monthly rate is $j=i_c^{(12)}/12$. These are different rate conventions. For example, an annual effective rate of 12% does not produce a monthly rate of exactly 1%; the equivalent monthly rate is $(1.12)^{1/12}-1$. A nominal rate of 12% convertible monthly does produce a monthly rate of exactly 1%.

In either case, apply one monthly rate to the balance specified by the timing convention. Do not apply an annual rate directly to a one-month interval, and do not divide an annual effective rate by 12 unless the problem explicitly defines it as a nominal rate convertible monthly.

Worked Example: One Month Under Two Timing Conventions

A hypothetical policy begins the month with an account value of $10,000. The credited rate is 6% effective annually. At the month’s transaction date, $200 of net premium is allocated and a $50 monthly deduction is taken. Find the month-end account value under (a) end-of-month transactions and (b) beginning-of-month transactions. Assume no other cash flows.

Method. First convert the annual effective rate to an effective monthly rate. Then apply the relevant recurrence. The common mistake to avoid is dividing the annual effective rate by 12.

The monthly rate is $j=(1.06)^{1/12}-1=0.0048675506$, or approximately 0.48675506% per month. The net transaction amount is $P_1-Q_1=200-50=150$.

(a) End-of-month transactions. The opening account earns interest, and then the net $150 is added:

A_1=10{,}000(1.0048675506)+200-50 =10{,}048.6755+150 =10{,}198.6755.

Rounded to the nearest cent, the account value is $10,198.68.

(b) Beginning-of-month transactions. The account first receives the premium and deduction, leaving $10{,}000+200-50=10{,}150$, and that amount earns the month’s interest:

A_1=(10{,}000+200-50)(1.0048675506) =10{,}150(1.0048675506) =10{,}199.4056.

Rounded to the nearest cent, the account value is $10,199.41. The beginning-of-month convention gives a value higher by about $0.73 because the net $150 is in the account for the month and earns $150j\approx \$0.73$. This check confirms the direction and size of the difference.

Projecting More Than One Month

Apply the recurrence one month at a time. The closing value for one month becomes the opening value for the next; do not restart the projection from the original account value. If a premium or deduction changes from month to month, use its amount for that particular month.

Worked Example: Two Months with Different Net Transactions

A hypothetical account starts at $5,000. The credited rate is a nominal annual rate of 12% convertible monthly. In month 1, $300 of net premium is allocated and a $40 deduction is taken. In month 2, $100 of net premium is allocated and a $40 deduction is taken. Calculate the month-2 account value under both conventions.

Method. Convert the nominal monthly rate, calculate each month in sequence, and carry forward the account value. The common mistake to avoid is using month 1’s starting value again in month 2.

The monthly rate is $j=0.12/12=0.01$. The net transaction amounts are $300-40=260$ in month 1 and $100-40=60$ in month 2.

End-of-month transactions. In month 1, $A_1=5{,}000(1.01)+260=5{,}050+260=5{,}310$. In month 2, $A_2=5{,}310(1.01)+60=5{,}363.10+60=5{,}423.10$.

Beginning-of-month transactions. In month 1, $A_1=(5{,}000+260)(1.01)=5{,}260(1.01)=5{,}312.60$. In month 2, $A_2=(5{,}312.60+60)(1.01)=5{,}372.60(1.01)=5{,}426.326$. Rounded to the nearest cent, this is $5,426.33.

The beginning-of-month result exceeds the end-of-month result by $5{,}426.326-5{,}423.10=\$3.226$, or $3.23 to the nearest cent. The extra amount is interest on each month’s net transaction, with the month-1 difference itself also earning interest in month 2.

Transaction Dates Can Differ

A problem may specify that a premium and a deduction occur at different points in the month. In that case, do not force both transactions into a single beginning- or end-of-month net amount. Apply interest only to the amounts that are in the account during the interest-crediting period.

For example, if a premium is posted at the beginning of the month and a deduction is taken at the end, the premium earns interest for the month but the deduction does not reduce the balance that earns that month’s interest. The recurrence is $(A_{m-1}+P_m)(1+j)-Q_m$. This is not the same as either convention in which both transactions occur together.

Worked Example: Premium at the Beginning, Deduction at the End

A hypothetical account opens the month at $8,000. The monthly credited rate is 0.5%. A net premium of $250 is posted at the beginning of the month, and a $70 deduction is posted at the end. Find the closing value. Compare it with the value if both transactions were posted at the beginning.

Method. Apply interest to the opening balance plus the beginning premium, then subtract the end-of-month deduction. The common mistake to avoid is allowing the end-of-month deduction to reduce the balance before it earns interest.

With the stated timing, the amount earning interest is $8{,}000+250=8{,}250$. Interest is $8{,}250(0.005)=\$41.25$. Therefore,

A_1=(8{,}000+250)(1.005)-70 =8{,}291.25-70 =8{,}221.25.

If both transactions were posted at the beginning, the balance earning interest would instead be $8{,}000+250-70=8{,}180$, giving $8{,}180(1.005)=\$8{,}220.90$. The stated split timing produces a closing value higher by $8{,}221.25-8{,}220.90=\$0.35$, exactly the interest $70(0.005)$ that the deduction avoids reducing during the month. Both calculations reconcile to the stated transaction dates.

Why the Timing Difference Has This Form

The relationship between the two standard conventions follows directly from their recurrences. Let $N_m=P_m-Q_m$ be the net transaction amount in month $m$. For the same opening value, the beginning-of-month value for that month is $(A_{m-1}+N_m)(1+j)$, while the end-of-month value is $A_{m-1}(1+j)+N_m$. Subtracting gives:

(A_{m-1}+N_m)(1+j)-\bigl(A_{m-1}(1+j)+N_m\bigr)=jN_m.

Thus, for a single month, moving a positive net amount to the beginning increases the closing value by its monthly interest, $jN_m$. If the net amount is negative, moving it to the beginning reduces the closing value by the interest on the amount withdrawn. This result assumes the same interest rate, the same beginning account value, and no other transactions or account-value constraints.

Over several months, the differences accumulate because an earlier timing difference remains in the account and can itself earn later interest. If both conventions use the same net transactions and monthly rate, the month-$n$ difference is:

A_n^{\text{begin}}-A_n^{\text{end}} =\sum_{k=1}^{n} jN_k(1+j)^{n-k}.

To see why, each month $k$ contributes an immediate difference $jN_k$. That amount remains in the account for the remaining $n-k$ months, so it is accumulated by the factor $(1+j)^{n-k}$. Adding the contributions from all months gives the expression. It applies when the two projections differ only in the timing convention and use identical transaction amounts and rates.

Common Timing and Rate Errors

Using the annual rate for one month. A one-month projection requires a monthly rate. Convert an annual effective rate with the twelfth-root formula; divide by 12 only for a nominal annual rate convertible monthly.
Crediting interest to a transaction that occurs at month-end. Under the end-of-month convention, the month’s interest applies to the opening account, not to the premium just posted at the end.
Subtracting an end-of-month deduction too early. If a deduction is explicitly taken at month-end, it does not reduce the balance earning interest during that month.
Combining transactions with different dates. A beginning premium and an end deduction must be applied separately, as in the third worked example.
Mixing recurrence conventions between months. Use the stated convention consistently unless the transaction dates change. Keep each closing account value as the next month’s opening value.
Ignoring model limits. These recurrences assume sufficient funds and no floor or additional account-value rule. If a problem specifies a minimum balance, insufficient-funds treatment, or another transaction, that rule must also be modeled.

Key takeaway: A monthly account projection is determined by the monthly credited rate, the net amounts allocated or deducted, and the dates on which those amounts enter or leave the account. Write the transaction timeline before choosing the recurrence.

Check Your Understanding

For each problem, use the stated timing and round any dollar result to the nearest cent.

A hypothetical account opens at $12,000. The monthly credited rate is 0.4%, a $180 net premium is allocated, and a $30 deduction is taken at month-end. What is the closing account value under the end-of-month convention? (Answer: $12,198.00)
A hypothetical policy has a nominal annual credited rate of 10% convertible monthly. It opens the month at $6,000; a $120 net premium and a $24 deduction both occur at the beginning of the month. Find the closing account value. (Answer: $6,146.80)
A hypothetical account starts at $4,000 and earns 0.5% per month. Under the end-of-month convention, the net transaction is $100 in month 1 and $-$50$ in month 2. Find the month-2 account value. (Answer: $4,090.60)
A hypothetical account starts at $9,000 and earns 0.6% per month. A $200 net premium is posted at the beginning of the month and a $45 deduction at the end. Find the closing account value. (Answer: $9,210.20)
A hypothetical account starts at $7,000, and the credited rate is 4% effective annually. Under the end-of-month convention, a $100 net premium is posted after the month’s interest. Find the closing account value for one month. (Answer: $7,122.92)

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