Fractional Periods at Deferred Annuity Boundaries

Why the Fractional Boundary Matters

A deferred annuity is straightforward when its deferral ends on a standard compounding date: accumulate to the annuity’s starting point, then value the payments there. The calculation needs more care when that boundary falls between compounding dates. The accumulation to the boundary may include a fractional conversion period, and the first payment may occur either at that boundary or later. Confusing either point changes the value.

The essential method is to mark the dates first. Then calculate the accumulation factor to the boundary under the stated interest convention, and value the payments from the boundary using their actual timing. The annuity factor describes the payment stream; it does not decide when the first payment occurs.

Definition: Suppose interest is quoted at nominal annual rate $i^{(m)}$, convertible $m$ times per year. Let $j=i^{(m)}/m$ be the rate per conversion period. Under the standard simple-interest convention for a fractional conversion period, an interval of $N+\alpha$ conversion periods, where $N$ is a nonnegative integer and $0\leq\alpha<1$, has accumulation factor $$ (1+j)^N(1+\alpha j). $$ This uses compound interest for the $N$ complete periods and simple interest for the remaining fraction $\alpha$ of a period.

This convention must match the rate information in the problem. If an effective annual rate $i$ is specified and the model assumes compound accumulation over arbitrary real times, the accumulation factor over $t$ years is $(1+i)^t$. Do not substitute that rule for the nominal-rate fractional-period convention unless the problem specifies it. A contract may state a different fractional-period rule; in that case, use the stated rule.

Build the Timeline Before Using an Annuity Factor

For a deferred annuity, distinguish the boundary date from the first payment date. A payment at the boundary is an annuity-due payment relative to that date. A first payment one payment period later is an annuity-immediate payment. If the first payment occurs some other length of time after the boundary, represent that interval explicitly rather than forcing it into either standard factor.

For example, if payments are made every half-year and the first payment is one half-year after the boundary, the value at the boundary is an ordinary annuity value using the half-year rate. If the first payment is made at the boundary instead, the value there is an annuity-due value. The accumulation from time zero to a fractional boundary is a separate calculation.

Mark the boundary and every payment date.
State whether the first payment is at the boundary, one payment period later, or at another specified interval.

Count complete conversion periods.
Write the time to the boundary as $N+\alpha$ conversion periods, with $0\leq\alpha<1$.

Accumulate to the boundary.
Under the nominal-rate fractional-period convention, use $(1+j)^N(1+\alpha j)$.

Value the payments at the boundary.
Use an annuity factor only when its payment frequency and first-payment timing match the timeline.

Discount the boundary value to time zero.
Divide by the accumulation factor to the boundary, then check the result against the individual cash flows.

Why the Fractional Accumulation Formula Holds

Under the nominal-rate model, one complete conversion period multiplies a fund by $1+j$. After $N$ complete periods, the fund has therefore been multiplied by $(1+j)^N$. For the remaining fraction $\alpha$ of a conversion period, simple interest at rate $j$ per full period adds interest $\alpha j$ per unit invested. The fractional-period multiplier is consequently $1+\alpha j$. Multiplying the complete-period and fractional-period multipliers gives the stated factor.

\text{Accumulation over }N+\alpha\text{ periods} =(1+j)^N(1+\alpha j).

The fractional factor is not $(1+j)^\alpha$ under this convention. That exponent would represent compound interpolation within the conversion period, a different model. The distinction can affect a deferred annuity’s present value even when the fractional interval is short.

Worked Example: Payments Begin at a Fractional Boundary

A fund earns nominal interest at $8\%$ convertible semiannually. It must provide three payments of $ \$1{,}000$, made at times $2.25$, $2.75$, and $3.25$ years. Find the present value at time zero. Use simple interest for a fractional conversion period and round the final answer to the nearest cent.

Method and timeline. The boundary is $t=2.25$, and the first payment is made exactly at that boundary. The subsequent payments are each one half-year apart. The half-year rate is $j=0.08/2=0.04$. The boundary is $4.5$ semiannual periods from time zero: four complete periods and one-half of a period.

Accumulation to the boundary.

$$ A(2.25)=(1.04)^4(1+0.5(0.04)) =1.16985856(1.02) =1.1932557312. $$

Value of the payments at the boundary. There is a payment immediately at the boundary and then payments at one and two half-year intervals. Thus the value at the boundary is an annuity-due value at the half-year rate:

V_{2.25} =1000\left(1+\frac{1}{1.04}+\frac{1}{(1.04)^2}\right) =1000(1+0.96153846+0.92455621) =2886.09467.

Present value and check. Discount the boundary value by the accumulation factor:

V_0=\frac{2886.09467}{1.1932557312} =2418.67.

As a cash-flow check, the three payments are at times $2.25$, $2.75$, and $3.25$, so each later payment is discounted by one or two full half-year periods relative to the first. The calculated value uses exactly those spacings and the stated fractional accumulation to time $2.25$. The common mistake avoided here is to treat the first payment as occurring one half-year after the boundary and use an annuity-immediate factor.

Worked Example: The First Payment Follows the Boundary

A deferred annuity pays $ \$500$ every six months, with four payments at times $1.75$, $2.25$, $2.75$, and $3.25$ years. The nominal annual interest rate is $10\%$ convertible semiannually. The deferral boundary is $t=1.25$. Find the present value, using simple interest within a fractional conversion period, to the nearest cent.

Method and timeline. The first payment is one full semiannual period after the boundary, so the annuity is immediate relative to $t=1.25$. The half-year rate is $j=0.10/2=0.05$. The boundary lies $2.5$ conversion periods after time zero.

Accumulate to the boundary. There are two complete half-year periods and one-half of a period:

$$ A(1.25)=(1.05)^2(1+0.5(0.05)) =1.1025(1.025) =1.1300625. $$

Value at the boundary. The first payment is one period later, so use an annuity-immediate factor for four payments at rate $0.05$:

V_{1.25} =500a_{\overline{4}|0.05} =500\left(\frac{1-(1.05)^{-4}}{0.05}\right) =500(3.5459505) =1772.97525.

Present value and check.

V_0=\frac{1772.97525}{1.1300625} =1568.92.

The timeline check is that the first payment occurs at $t=1.75$, not at $t=1.25$. The factor $a_{\overline{4}|0.05}$ discounts four payments from one through four half-year periods after the boundary. Using $\ddot{a}_{\overline{4}|0.05}$ would incorrectly move the first payment to the boundary.

Worked Example: A Fraction of a Conversion Period to the First Payment

A hypothetical benefit fund earns nominal interest at $12\%$ convertible quarterly. Its boundary date is $t=1.375$ years, and it pays $ \$750$ at $t=1.5$, $1.75$, $2.0$, and $2.25$ years. Find the present value, using simple fractional-period interest, to the nearest cent.

Method and timeline. The quarterly rate is $j=0.12/4=0.03$. The boundary is $5.5$ quarters from time zero. The first payment is half a quarter after the boundary, while each gap between later payments is one full quarter. The fractional first-payment interval must be included separately.

Accumulation to the boundary.

$$ A(1.375)=(1.03)^5(1+0.5(0.03)) =1.1592740743(1.015) =1.1766631854. $$

Value of the payments at the boundary. A half-quarter of simple interest accumulates by $1+0.5(0.03)=1.015$. Therefore the first payment is discounted by $1/1.015$ to the boundary, and the remaining payments are each one additional full quarter apart:

V_{1.375} =\frac{750}{1.015}a_{\overline{4}|0.03} =\frac{750}{1.015}(3.7170984) =2746.62.

Present value and check.

V_0=\frac{2746.62}{1.1766631854} =2334.25.

The first cash flow is half a quarter after the boundary, not one full quarter after it. The factor $1.015$ handles that initial half-period; the annuity factor then accounts for the quarter-spaced payments relative to the first-payment position. The common mistake avoided is to use a standard annuity-immediate factor at the boundary without adjusting for the half-quarter first interval.

Common Traps and a Final Check

Fractional-boundary questions usually test more than arithmetic. A correct calculation has to use a consistent interest model and a consistent timeline. Before finalizing a value, check each of the following:

Do not round the fractional period into a full period. For $N+\alpha$ periods, retain both $N$ and $\alpha$.
Do not mix fractional-period conventions. Under the nominal-rate convention stated here, use $1+\alpha j$, not $(1+j)^\alpha$.
Do not infer the first payment date from the word “deferred.” Read the stated payment dates or wording that defines them.
Do not use an annuity factor with the wrong payment frequency. The rate per payment interval must correspond to the actual gap between payments.
Do not count the boundary payment twice. A payment at the boundary is already included in an annuity-due value there.

A useful independent check is to discount each cash flow to time zero using its actual date and the stated accumulation convention. The sum should agree with the method that first values the payments at the boundary and then discounts that value to time zero. If it does not, recheck the boundary accumulation and the location of the first payment.

Key takeaway: A fractional deferral boundary determines the accumulation factor to the annuity’s starting point; the first payment date determines the annuity value at that point. Solve these as separate steps, using the same interest model throughout.

Check Your Understanding

For each problem, draw the timeline before selecting an annuity factor. Round dollar answers to the nearest cent.

A fund earns nominal interest at $6\%$ convertible semiannually. A two-payment annuity pays $ \$400$ at the boundary $t=1.25$ and again one half-year later. Find its present value under the simple fractional-period convention. (Answer: $ \$732.11$)
Nominal interest is $8\%$ convertible quarterly. A deferred annuity boundary is $t=0.625$, and four quarterly payments of $ \$250$ begin one quarter after the boundary. Find the present value under the simple fractional-period convention. (Answer: $ \$905.91$)
An effective annual rate of $9\%$ applies with compound accumulation over arbitrary real times. Three annual payments of $ \$1{,}000$ begin at the boundary $t=1.4$ and continue at one-year intervals. Find the present value. (Answer: approximately $ \$2{,}445.52$)
Nominal interest is $12\%$ convertible semiannually. Three payments of $ \$600$ occur at $t=2.75$, $3.25$, and $3.75$ years. Find the present value under the simple fractional-period convention. (Answer: $ \$1{,}233.36$)

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