Direct-Sum Checks for Deferred Annuity Values

Why Check a Deferred Annuity by Summing Its Payments?

A deferred annuity factor compresses several discounted payments into one expression. That is efficient, but it can hide a timing error: using the right factor with the wrong deferral count, or discounting from the wrong first-payment date. A direct-sum check makes the timeline visible. It also provides a reliable way to confirm a compact calculation before using it in a larger valuation.

The setting here is deterministic: payments are known, occur at fixed dates, and are discounted at a constant effective rate per payment period. There are no expenses, uncertain payments, or changes in the rate. The goal is not to replace annuity factors, but to verify that the factor matches the dated cash flows.

Definition: Suppose a level annuity-immediate pays $R$ at the end of each period for $n$ consecutive periods. If the first payment is at time $m+1$, where $m$ complete payment periods are deferred, its present value at time 0 is the sum of the payments discounted from their actual dates. With effective interest rate $i$ per period and $v=1/(1+i)$, this is $R(v^{m+1}+\cdots+v^{m+n})$. A deferred annuity-due with first payment at time $m$ instead has present value $R(v^m+\cdots+v^{m+n-1})$.

Direct Sums and the Compact Factor

For an annuity-immediate, the payment dates are $m+1,m+2,\ldots,m+n$. Each payment $R$ at time $k$ contributes $Rv^k$ to the present value at time 0. Thus the direct sum is $R\sum_{j=1}^{n}v^{m+j}$. Factoring out the discount to time $m$ gives $Rv^m\sum_{j=1}^{n}v^j$.

The ordinary annuity-immediate factor is $a_{\overline{n}|}=v+v^2+\cdots+v^n$. Consequently, the direct sum can be written $Rv^m a_{\overline{n}|}$. The annuity factor values the payments at the end of the deferral period; $v^m$ then discounts that value from time $m$ to time 0. This identity is a useful check only when the payment dates really are $m+1$ through $m+n$.

\begin{aligned} PV_0 &=R\sum_{j=1}^{n}v^{m+j}\\ &=Rv^m\sum_{j=1}^{n}v^j\\ &=Rv^m a_{\overline{n}|} =Rv^m\frac{1-v^n}{i}, \qquad i\ne 0. \end{aligned}

The last equality follows because $v+v^2+\cdots+v^n$ is a finite geometric series with first term $v$ and common ratio $v$. Its sum is $v(1-v^n)/(1-v)$. Since $v=1/(1+i)$, $1-v=i/(1+i)=iv$; substituting gives $(1-v^n)/i$. If $i=0$, each payment has discount factor 1, so the present value is $nR$; the quotient formula is not used.

For a deferred annuity-due, the first payment occurs at time $m$, not $m+1$. Its direct sum is $R(v^m+\cdots+v^{m+n-1})=Rv^m\ddot{a}_{\overline{n}|}$, where $\ddot{a}_{\overline{n}|}=1+v+\cdots+v^{n-1}$. This distinction matters at the boundary: an immediate annuity deferred for $m$ periods has no payment at time $m$, whereas a due annuity with the same $m$ has its first payment then.

A Reliable Direct-Sum Check

Use the payment dates, rather than the factor name alone, to organize a check. The following steps apply to a level annuity with a fixed rate per payment period.

Mark the focal date and payment dates.
Count the actual periods from the focal date to each payment. For a present value at time 0, a payment at time $k$ is discounted by $v^k$.

Write the finite sum.
Include one term for each payment, with its own date and amount. This exposes a missing payment or an off-by-one exponent.

Calculate the compact value independently.
Use $v^m a_{\overline{n}|}$ for an immediate annuity whose first payment is at $m+1$, or $v^m\ddot{a}_{\overline{n}|}$ for a due annuity whose first payment is at $m$.

Compare before rounding.
The direct sum and compact calculation should agree apart from numerical rounding. If they do not, check the payment count, the first payment date, and the rate per period.

Worked Example: Checking a Deferred Annuity-Immediate

A fund will pay \$1,000 at the end of each of three consecutive years, with the first payment at the end of year 3. The effective annual interest rate is 25%. Find the present value at time 0 by a direct sum and verify it with a compact factor.

Method: list the dated payments first. There are two complete years with no payment, so $m=2$. The three payments occur at times 3, 4, and 5. The annual discount factor is $v=1/1.25=0.8$.

Direct sum. Discount each \$1,000 payment from its own date:

\begin{aligned} PV_0 &=1000(v^3+v^4+v^5)\\ &=1000(0.8^3+0.8^4+0.8^5)\\ &=1000(0.512+0.4096+0.32768)\\ &=1000(1.24928)=\$1,249.28. \end{aligned}

Compact check. Here $a_{\overline{3}|}=(1-v^3)/i=(1-0.512)/0.25=1.952$. Therefore $1000v^2a_{\overline{3}|}=1000(0.64)(1.952)=1000(1.24928)=\$1,249.28$. The two methods agree. The common mistake to avoid is using $v^3a_{\overline{3}|}$: that would discount the whole factor one period too far, because the annuity factor already starts with a payment one period after its valuation date.

Worked Example: Checking a Deferred Annuity-Due at the Boundary

A hypothetical grant pays \$500 at times 2, 3, and 4 years. The effective annual interest rate is 20%. Find the present value at time 0 by direct summation and verify it using an annuity-due factor.

Method: identify whether there is a payment at the deferral date. The first payment is at time 2, so there is a payment at the end of the two-year deferral period. This is a deferred annuity-due with $m=2$ and $n=3$. The discount factor is $v=1/1.20=5/6$.

Direct sum. The payments are at times 2, 3, and 4:

\begin{aligned} PV_0 &=500(v^2+v^3+v^4)\\ &=500\left(\frac{25}{36}+\frac{125}{216}+\frac{625}{1296}\right)\\ &=500\left(\frac{900+750+625}{1296}\right)\\ &=500\left(\frac{2275}{1296}\right) =\$877.7006\ldots \end{aligned}

Compact check. The annuity-due factor is $\ddot{a}_{\overline{3}|}=1+v+v^2=1+5/6+25/36=91/36$. Then $v^2\ddot{a}_{\overline{3}|}=(25/36)(91/36)=2275/1296$, matching the direct sum. Rounded to the nearest cent, the present value is \$877.70. A common mistake is to use $v^2a_{\overline{3}|}$, which describes payments at times 3, 4, and 5—not the stated dates.

Using a Nominal Rate: Work in Payment Periods

A direct-sum check also helps when the quoted rate is nominal and convertible more than once per year. First convert the nominal rate to the effective rate for one payment period. If payments are quarterly and the nominal annual rate is $i^{(4)}$, the quarterly rate is $i^{(4)}/4$, and the quarterly discount factor is $v=1/(1+i^{(4)}/4)$. Count both the deferral and payment dates in quarters. Do not use a quarterly payment count with an annual discount factor.

Worked Example: Verifying Quarterly Payments at a Nominal Rate

A hypothetical agreement pays \$1,000 at the end of each quarter for three quarters. The first payment is at the end of the third quarter. The nominal annual interest rate is 40% convertible quarterly. Find the present value at time 0 and verify it in two ways.

Method: convert, then count periods. The effective quarterly rate is $0.40/4=0.10$, so $v=1/1.10=10/11$. The two-quarter deferral means the three payments occur at quarters 3, 4, and 5.

Direct sum.

\begin{aligned} PV_0 &=1000(v^3+v^4+v^5)\\ &=1000\left(\frac{1000}{1331} +\frac{10000}{14641} +\frac{100000}{161051}\right)\\ &=1000\left(\frac{121000+110000+100000}{161051}\right)\\ &=1000\left(\frac{331000}{161051}\right) =\$2,055.248\ldots \end{aligned}

Compact check. The factor is $a_{\overline{3}|}=(1-v^3)/0.10=2.486851\ldots$. Thus $1000v^2a_{\overline{3}|}=1000(100/121)(3310/1331)=1000(331000/161051)$, the same result. Rounded to the nearest cent, the present value is \$2,055.25. The trap is to discount by calendar years using 40% as though it were an effective annual rate. The payments are quarterly, so both the conversion and the date count must use quarters.

Common Errors and When the Check Helps

Direct summation is especially useful when a problem describes dates in words rather than naming an annuity type. It forces each timing decision to be explicit. It is also a good diagnostic when a value seems unexpectedly large or small.

Shifting every payment by one period. For an immediate annuity deferred $m$ periods, the exponents are $m+1$ through $m+n$. For a due annuity with the first payment at time $m$, they are $m$ through $m+n-1$.
Counting the deferral as a payment-free period without checking the boundary. A payment at time $m$ belongs to a due annuity in this setup; it is not the first payment of an immediate annuity deferred $m$ periods.
Mixing time units. If payments are monthly or quarterly, use a discount factor and deferral count in those same periods.
Comparing rounded intermediate results. Keep sufficient precision through the calculation and round the final money amount to cents.

The direct sum assumes a level amount paid at each stated date and a constant rate per period. If payment amounts vary, discount each amount separately rather than multiplying a level annuity factor by one payment. If rates differ by period, use the appropriate discount accumulation to each payment date rather than one constant $v^k$. Under the constant-rate, level-payment assumptions used here, equality of the direct sum and compact factor is a strong check that the timing and arithmetic are consistent.

Key takeaway: A deferred annuity value is a sum of payments discounted from their actual dates. For an annuity-immediate with $m$ full periods before its first payment, that sum is $Rv^m a_{\overline{n}|}$; for a deferred annuity-due whose first payment is at time $m$, it is $Rv^m\ddot{a}_{\overline{n}|}$. Use the direct sum to verify the dates before trusting the factor.

Check Your Understanding

For each question, identify the payment periods and compare the direct sum with the corresponding compact factor.

An annuity-immediate pays \$400 at times 3 and 4. The effective annual interest rate is 10%. Find its present value at time 0. (Answer: \$573.73)
A deferred annuity-due pays \$250 at times 3, 4, and 5. The effective annual interest rate is 25%. Find its present value at time 0. (Answer: \$312.32)
A nominal annual rate of 12% is convertible monthly. Six monthly payments of \$90 are made at the ends of months 3 through 8. Find the present value at time 0, to the nearest cent. (Answer: \$511.32)
An annuity-immediate pays \$300 at times 5, 6, 7, and 8. The effective annual interest rate is 20%. Find its present value at time 0, to the nearest cent. (Answer: \$374.54)
An annuity-immediate pays \$1,000 at times 3, 4, and 5. The effective annual interest rate is 10%. Find its value at time 2, and verify by discounting the payments to that focal date. (Answer: \$2,486.85)

What You'll Learn