Level Annuities Crossing an Interest-Rate Change

Why a Rate Change Requires a Split

A level annuity has equal payments, but equal payments do not necessarily have equal present values or accumulate by the same factor. If the effective annual interest rate changes during the payment term, the factor applied to a payment depends on which rate applies in each period between that payment and the chosen focal date. The reliable method is to divide the payments at the rate-change boundary and value each segment using only the rates that apply to it.

This tutorial considers annual payments and annual effective rates. The same reasoning applies at other regular payment intervals when each interval’s effective rate is specified. It is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Suppose payments of $R$ occur at times $1,2,\ldots,n$, and the effective annual rate is $i_1$ for the first $m$ periods and $i_2$ for the remaining $n-m$ periods. The rate for period $t-1$ to $t$ is the rate applying to that interval. Thus, a payment at time $m$ is included in the first segment, while payments at times $m+1,\ldots,n$ are in the second segment. This convention assumes payments occur at the stated period ends and the new rate applies from time $m$ onward.

For a level annuity-immediate of $k$ payments, $a_{\overline{k}|}(i)$ denotes the present value at the beginning of its first payment interval, when each payment is made at the end of a period and the effective rate is $i$ in every period. For $i\ne 0$, $a_{\overline{k}|}(i)=(1-v^k)/i$, where $v=1/(1+i)$; when $i=0$, its value is $k$.

Present Value at the Start of the Annuity

For payments at times $1$ through $n$, split the stream after the payment at time $m$. The first $m$ payments form an annuity-immediate valued at time 0 using $i_1$. To value the remaining payments, first value them at time $m$ using $i_2$, then discount that amount through the first $m$ periods using $i_1$.

PV_0 = R\,a_{\overline{m}|}(i_1) + v_1^m R\,a_{\overline{n-m}|}(i_2), \qquad v_1=\frac{1}{1+i_1}.

This expression does not apply unchanged if the rate changes at a different point in the timeline, if the payment dates are irregular, or if payments are not level. In those cases, identify each payment’s actual discount factors from the rate schedule rather than forcing the stream into this formula.

Worked Example: Present Value When the Rate Rises After Two Payments

Payments of $700 are made at the ends of years 1 through 5. The annual effective rate is 0% for years 1 and 2, then 25% for years 3 through 5. Find the present value at time 0. State the method and avoid applying the later rate to payments made before the change.

Method. Split after the payment at time 2. The first two payments have no discounting because the rate in both periods is 0%. Value the last three payments at time 2 using 25%, then bring that value back two periods using the first rate.

PV_0 = 700a_{\overline{2}|}(0) + (1.00)^{-2}\,700a_{\overline{3}|}(0.25).

The first segment is $700(2)=$1,400$. For the second segment, $v_2=1/1.25=0.8$, so $a_{\overline{3}|}(0.25)=(1-0.8^3)/0.25=(1-0.512)/0.25=1.952$. Its value at time 2 is $700(1.952)=$1,366.40$. Discounting that value through the first two years at 0% leaves it unchanged.

$$ PV_0=$1,400+$1,366.40=$2,766.40. $$

As a check, the payments at times 1 and 2 each have present value $700. The payments at times 3, 4, and 5 have present values $700/1.25=$560$, $700/1.25^2=$448$, and $700/1.25^3=$358.40$. Their total is $1,400+560+448+358.40=$2,766.40$. The common mistake is to discount all five payments at 25%; that wrongly applies the new rate to the first two periods.

Why the Split Formula Holds

The formula follows from valuing every payment at the same date. At time $m$, the later payments are an annuity-immediate of $n-m$ payments, each discounted using $i_2$. Their value at time $m$ is $R\,a_{\overline{n-m}|}(i_2)$. To express that value at time 0, discount it over the first $m$ periods, all of which use $i_1$. The first segment is already an annuity-immediate valued at time 0.

Choose the focal date.
For present value, use time 0. For an accumulated value, use the specified future date.

Assign each payment to its interval.
Use the rate that applies between the payment dates. Under the convention here, payment $m$ belongs to the first segment.

Value the segments consistently.
Calculate each segment at a date where its applicable constant-rate annuity formula is valid.

Move values to the focal date.
Discount or accumulate each segment through every intervening period, using the rate specified for each period.

Equivalently, a payment at time $t$ has present value equal to its amount multiplied by the product of the discount factors for periods 1 through $t$. The split formula groups those payment-by-payment products into two sums. It therefore does not introduce a new valuation principle; it organizes the same period-by-period discounting into manageable annuity segments.

Accumulated Values and Unknown Payments

The same segmentation works when the target is an accumulated value. Each payment must be carried forward to the focal date through the periods remaining after it is made. The growth factor is the product of the factors $1+i$ for those periods. In particular, a payment made at the focal date has no further accumulation.

Worked Example: Accumulating Deposits Across a Rate Change

Equal deposits $R$ are made at times 0, 1, 2, and 3. The annual effective rate is 10% from time 0 to time 1, then 20% from time 1 to time 3. The accumulated value at time 3 is $4,179.20. Find $R$. Use the deposit dates as stated; the deposit at time 0 earns interest in all three periods.

Method. Accumulate each deposit separately to time 3, then factor out $R$. A deposit at time 1 earns 20% for two periods; deposits at times 2 and 3 earn 20% for one and zero periods, respectively.

4,179.20 = R\left[1.10(1.20)^2+(1.20)^2+1.20+1\right].

The time-0 deposit grows to $R(1.10)(1.20)^2=1.584R$. The time-1 deposit grows to $R(1.20)^2=1.44R$. The time-2 deposit grows to $1.20R$, and the time-3 deposit remains $R$. Therefore the total accumulation factor is $1.584+1.44+1.20+1=5.224$.

R=\frac{4,179.20}{5.224}=$800.

Check: $5.224($800)=$4,179.20$. The common mistake is to treat the time-0 deposit as earning only the two 20% periods, omitting the 10% period from time 0 to time 1. Another is to assign the time-1 deposit an extra 10% period: that deposit is made at the boundary and earns only the subsequent two periods at 20%.

Worked Example: Accumulating Payments to a Later Date

Four payments of $R$ are made at times 1, 2, 3, and 4. The annual effective rate is 10% for the first two periods and 0% for the next two. The accumulated value at time 4 is $4,920. Find $R$. The rate change occurs at time 2, after the second payment.

Method. Accumulate every payment to time 4. The time-1 payment earns 10% from time 1 to 2, but nothing in the two later periods. The time-2 payment also earns nothing after it is made. The last two payments have no accumulation after their payment dates.

$$ 4,920=R(1.10+1+1+1)=4.10R. $$

Thus $R=4,920/4.10=$1,200$. The individual accumulated amounts are $1,320, $1,200, $1,200, and $1,200, which sum to $4,920. As a check, $4.10($1,200)=$4,920$. The common mistake is to apply 10% to all four payments through time 4, even though the rate is 0% in periods 3 and 4.

Boundary Checks and Exam Traps

A rate-change problem is often lost through a timeline error rather than difficult arithmetic. Before using an annuity factor, mark the payment dates and the intervals between them. A rate stated “for the first $m$ years” applies to the first $m$ intervals; the payment at time $m$ is made at the end of that segment. It is not automatically carried at the new rate merely because it occurs on the change date.

Another trap is using one rate for an entire annuity because the payments are level. Level describes the payment amounts, not the discount factors. When discounting across a rate change, the factor for a payment after the boundary includes both the early-period discount factors and the later-period discount factors. When accumulating, it includes the post-payment rates up to the focal date, in chronological order.

If one segment has a zero rate, its annuity factor is simply the number of payments in that segment. Do not use $(1-v^n)/i$ by substituting $i=0$, since that expression would divide by zero; use the limiting value $n$, or add the equal payments directly. Keep unrounded factors during the calculation and round the final monetary answer to cents unless the problem specifies otherwise.

Key takeaway: At a rate change, split the level stream at the boundary, value each part under the rates that actually apply, and move every segment to one common focal date. Check the boundary payment and the number of periods in each rate regime before calculating.

Check Your Understanding

For each problem, identify the applicable rate periods before calculating. Round money to the nearest cent where needed.

Payments of $300 occur at times 1 through 4. The annual effective rate is 5% for the first period and 0% for the next three. Find the present value at time 0. (Answer: $1,142.86)
Payments of $400 occur at times 1 through 5. The annual effective rate is 10% for the first three periods and 20% for the next two. Find the present value at time 0. (Answer: $1,453.88)
Deposits of $R$ occur at times 0, 1, 2, and 3. The annual effective rate is 5% for the first period and 10% for the next two. The accumulated value at time 3 is $4,580.50. Find $R$. (Answer: $1,000)
Payments of $1,000 occur at times 1, 2, 3, and 4. The annual effective rate is 0% for the first period and 10% for the next three. Find the accumulated value at time 4. (Answer: $4,641.00)
A level annuity-immediate has six payments of $R$ at times 1 through 6. The rate is 8% for the first three periods and 0% for the next three. Write an expression for its present value at time 0 using annuity factors; state which payment is in the first segment. (Answer: $R\,a_{\overline{3}|}(0.08)+v^3R\,a_{\overline{3}|}(0)$, where $v=1/1.08$; the payment at time 3 is in the first segment.)

What You'll Learn