Present Values for Regular Level Withdrawals

Why Withdrawal Timing Determines the Required Fund

A fund set aside to make regular withdrawals must be large enough, at the valuation date, to cover the present value of every planned payment. The amount depends not only on the withdrawal size, number of withdrawals, and interest rate, but also on when the first withdrawal occurs. A withdrawal made immediately uses part of the fund before it earns interest; a withdrawal at the end of the first period does not.

This tutorial considers a finite sequence of equal withdrawals at regular intervals. The valuation date is the date at which the required fund is measured. Unless stated otherwise, assume withdrawals occur annually, the effective annual interest rate is constant, and there are no fees, taxes, deposits, or other cash flows.

Definition: An annuity-immediate is a sequence of equal payments made at the ends of successive periods. An annuity-due is a sequence of equal payments made at the beginnings of successive periods. For a fund valued at time 0, an $n$-payment annuity-immediate has payments at times $1,2,\ldots,n$; an $n$-payment annuity-due has payments at times $0,1,\ldots,n-1$.

Let $R$ be the amount of each withdrawal, $n$ the number of withdrawals, and $i$ the effective interest rate per payment period. The discount factor per period is $v=1/(1+i)$. The formulas below require a constant rate applying over each period between the valuation date and a payment. If the payment interval differs from a year, $i$ must be the effective rate for that interval.

Key result: The present value at time 0 of $n$ level withdrawals of amount $R$ is $R a_{\overline{n}|}$ if the withdrawals are at the ends of periods, and $R\ddot{a}_{\overline{n}|}$ if they are at the beginnings of periods. The factors use the effective rate per payment period.

Why the Immediate and Due Factors Work

The present value of a payment is its amount multiplied by the appropriate discount factor. Thus, for an annuity-immediate, the first withdrawal is discounted one period, the second two periods, and the last $n$ periods. For an annuity-due, the first withdrawal is made at the valuation date and is not discounted.

a_{\overline{n}|}=v+v^2+\cdots+v^n =\frac{1-v^n}{i}, \qquad \ddot{a}_{\overline{n}|}=1+v+\cdots+v^{n-1} =\frac{1-v^n}{d}, \quad d=\frac{i}{1+i}.

To justify the immediate formula, multiply the finite sum $v+v^2+\cdots+v^n$ by $1-v$. All intermediate terms cancel, leaving $v-v^{n+1}=v(1-v^n)$. Since $1-v=i v$, dividing by $1-v$ gives $(1-v^n)/i$. For the due factor, the corresponding sum begins with 1 and ends with $v^{n-1}$. Multiplying by $1-v$ gives $1-v^n$, so the sum is $(1-v^n)/(1-v)$. Because $1-v=i/(1+i)=d$, this is $(1-v^n)/d$.

The due factor also satisfies $\ddot{a}_{\overline{n}|}=(1+i)a_{\overline{n}|}$: every immediate payment is shifted one period earlier in the due stream. This identity is a useful check, but it does not decide which factor applies. The actual dates of the withdrawals do.

Calculate the Required Fund from the Payment Dates

A reliable method is to list the withdrawal dates, match them to the factor, and then multiply by the withdrawal amount. If the prompt describes a withdrawal at the valuation date, include it at time 0. Do not discount that payment by one period.

Set the valuation date and period.
Identify the date at which the fund is required and the length of one payment period.

Write the withdrawal dates.
Payments at times $1,\ldots,n$ are immediate; payments at times $0,\ldots,n-1$ are due.

Apply the matching factor.
Use $R a_{\overline{n}|}$ or $R\ddot{a}_{\overline{n}|}$, with the rate per payment period.

Check the cash flows.
Confirm that the factor includes exactly $n$ payments and that its first payment is discounted by the correct number of periods.

Worked Example: Withdrawals Starting Immediately

A hypothetical account must pay five annual withdrawals of $800. The first withdrawal is made immediately, at the valuation date, and the remaining withdrawals are made at the ends of the next four years. The effective annual interest rate is 4%. Find the fund required at the valuation date, rounded to cents.

Method and timing. The payment dates are $0,1,2,3,4$. There are five withdrawals, and the first is at time 0, so this is an annuity-due. Use $R\ddot{a}_{\overline{5}|}$, with $R=800$ and $i=0.04$.

v=\frac{1}{1.04}, \qquad v^5=\frac{1}{1.04^5} =\frac{1}{1.2166529024} \approx 0.8219271068.

The effective discount rate is $d=i/(1+i)=0.04/1.04=0.03846153846\ldots$. Therefore,

\ddot{a}_{\overline{5}|} =\frac{1-v^5}{d} =\frac{1-1/1.04^5}{0.04/1.04} \approx 4.629895223.

The required fund is

800\ddot{a}_{\overline{5}|} =800(4.629895223\ldots) =3703.916178\ldots \approx \$3{,}703.92.

Check. The five payments have present values $800$, $800/1.04$, $800/1.04^2$, $800/1.04^3$, and $800/1.04^4$. Their sum is the same annuity-due value, approximately $3,703.9162, which rounds to $3,703.92. The method avoids discounting the immediate withdrawal.

Worked Example: Withdrawals at Each Year-End

A hypothetical reserve is to fund six withdrawals of $400, each made at the end of a year. The effective annual interest rate is 3%. Find the reserve needed now, rounded to cents.

Method and timing. The payment dates are $1,2,3,4,5,6$, so the stream is an annuity-immediate. Use $400a_{\overline{6}|}$.

1.03^6=1.194052296529, \qquad v^6=\frac{1}{1.03^6}\approx 0.8374842567.

Then

a_{\overline{6}|} =\frac{1-v^6}{0.03} =\frac{1-1/1.03^6}{0.03} \approx 5.417191445.

The present value is

400a_{\overline{6}|} =400(5.417191445\ldots) =2166.876578\ldots \approx \$2{,}166.88.

Check. The first payment is discounted once and the sixth payment six times, so the present value is $400(v+v^2+v^3+v^4+v^5+v^6)$. This is precisely $400a_{\overline{6}|}$. The method avoids treating an end-of-year payment as though it were available at the valuation date.

Worked Example: Finding the Level Withdrawal from a Fund

A hypothetical fund of $7,000 is available now to make three equal annual withdrawals, each at the end of a year. The effective annual interest rate is 10%, and the account is to be fully exhausted by the third withdrawal. Find the withdrawal amount, rounded to cents.

Method and timing. The payments occur at times 1, 2, and 3, so use the annuity-immediate factor. Set the present value of the three withdrawals equal to the fund and solve for $R$.

a_{\overline{3}|} =\frac{1-v^3}{i} =\frac{1-1/1.1^3}{0.10} =\frac{1-1000/1331}{0.10} =\frac{3310}{1331} \approx 2.486851991.

The value equation is $7000=R a_{\overline{3}|}$, so

R=\frac{7000}{a_{\overline{3}|}} =\frac{7000(1331)}{3310} \approx 2814.803625 \approx \$2{,}814.80.

Check. The exact value equation is $R(v+v^2+v^3)=7000$. Substituting the unrounded value of $R$ gives a present value of $7,000. The method avoids dividing by the due factor, which would assume a first withdrawal at time 0 instead of time 1.

Common Timing and Calculation Traps

The most frequent error is selecting a factor from the word “withdrawal” rather than from the payment dates. Withdrawals can form either an immediate or a due annuity. A payment at the valuation date is part of the due stream, while a payment one full period later is the first payment of an immediate stream.

Wrong: For five withdrawals beginning immediately, using $800a_{\overline{5}|}$ discounts the first withdrawal one year and omits the time-0 payment pattern. Correction: use $800\ddot{a}_{\overline{5}|}$, because the dates are $0$ through $4$.
Wrong: For six year-end withdrawals, using $\ddot{a}_{\overline{6}|}$ counts a payment now. Correction: use $a_{\overline{6}|}$, because the dates are $1$ through $6$.
Wrong: Using an annual rate with monthly or quarterly payments without conversion. Correction: use the effective rate for one payment interval. The formulas discount period by period.
Wrong: Rounding $v^n$ or the annuity factor early, then carrying that rounded value through a large payment calculation. Correction: retain the exact expression or sufficient digits until the final monetary result.

A present-value calculation matters because it measures the amount that, at the stated interest assumption, is equivalent at the valuation date to the scheduled withdrawals. The result is conditional on the model: if the interest rate changes, if withdrawals are not level, or if there are additional cash flows, this simple factor calculation may no longer describe the fund. For the level finite streams considered here, the key discipline is to align the factor with the actual payment dates and count.

Key takeaway: First locate the valuation date, then list the withdrawal dates. Use $a_{\overline{n}|}$ for payments at period ends and $\ddot{a}_{\overline{n}|}$ for payments at period beginnings. Multiply by the level withdrawal, or divide the fund by the factor when solving for the withdrawal.

Check Your Understanding

For each problem, identify the payment timing before calculating. Round monetary answers to cents.

A fund must make four annual withdrawals of $900 at the ends of years 1 through 4. The effective annual interest rate is 6%. Find the fund required now. (Answer: $3,118.60)
Three withdrawals of $1,500 are made immediately and at the beginnings of the next two years. The effective annual interest rate is 8%. Find the present value at the first withdrawal date. (Answer: $4,174.90)
A fund of $10,000 supports four equal withdrawals at the ends of successive years, with the first one year from now. The effective annual interest rate is 5%. Find the maximum level withdrawal that exhausts the fund at the fourth payment. (Answer: $2,820.12)
Five annual withdrawals of $500 are made immediately and then at the ends of the next four years. At an effective annual interest rate of 4%, find the due-annuity present value and the difference between it and the present value of five $500 withdrawals at the ends of years 1 through 5. (Answer: Due value $2,314.95; immediate value $2,225.91; difference $89.04)

What You'll Learn