Combining Immediate and Due Streams

Why Separate Streams Need Separate Timing

A financial schedule may contain more than one regular payment stream. For example, an account could receive deposits at the end of each year and a separate contribution at the beginning of each year. Even if both streams have level payments and the same payment frequency, they do not have the same value: each payment earns interest for a different length of time.

The reliable method is to identify the timing of each stream, value each one with the matching annuity factor, and express all values at the same focal date. The stream values can then be added. This tutorial assumes certain payments and a constant effective annual interest rate; no mortality assumption or probability distribution is involved.

Definition: An annuity-immediate with $n$ equal payments has payments at the ends of periods $1,2,\ldots,n$. An annuity-due with $n$ equal payments has payments at the beginnings of periods $0,1,\ldots,n-1$. The annuity-due payments are each one period earlier than the corresponding annuity-immediate payments.

Let $i$ be the effective interest rate per payment period and let $v=1/(1+i)$. A level annuity-immediate with payment amount $R$ has present value $R a_{\overline{n}|}$, where $a_{\overline{n}|}$ is the present-value factor for $n$ payments at times $1$ through $n$. A level annuity-due has present value $R\ddot{a}_{\overline{n}|}$, where $\ddot{a}_{\overline{n}|}$ applies to payments at times $0$ through $n-1$.

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} =(1+i)a_{\overline{n}|}.

The formulas use the interest rate per payment period. If payments are annual and the effective annual rate is $i$, the formulas above apply directly. If the payment interval differs from a year, first identify the effective rate for that interval; using an annual rate as if it were a monthly or quarterly rate would give incorrect values.

Combining Values at a Common Date

Each stream can be valued at the present date or at another focal date. Present-value factors are convenient when the focal date is time $0$. Accumulated-value factors are convenient when the focal date is after the final payment. For dates between those points, value the payments directly at the chosen date, or move a value from another date by the appropriate power of $1+i$.

Key rule: Value every stream at the same focal date before adding. If a stream’s value is known at time $t$, its value $k$ periods later is multiplied by $(1+i)^k$; its value $k$ periods earlier is multiplied by $v^k$. Use this rule only when the interest rate and period length are consistent across the move.

For instance, if there are $n$ end-of-period payments, their accumulated value at time $n$ is $R s_{\overline{n}|}$. The corresponding $n$ beginning-of-period payments accumulate to $R(1+i)s_{\overline{n}|}$ at time $n$, since each due payment is one period earlier and earns one extra period of interest compared with its immediate counterpart.

Why the Due Factor Is Larger

The relationship between the two present-value factors follows from matching the payment amounts and comparing their dates. The immediate payments occur at times $1$ through $n$; the due payments occur at times $0$ through $n-1$. Every due payment is shifted one period earlier, so its present value is multiplied by one fewer factor of $v$.

\ddot{a}_{\overline{n}|} =1+v+\cdots+v^{n-1} =(1+i)(v+v^2+\cdots+v^n) =(1+i)a_{\overline{n}|},

The middle equality holds because $(1+i)v=1$, and multiplying each term of $a_{\overline{n}|}$ by $1+i$ shifts its discount power down by one. Thus, for a positive interest rate and positive payments, an annuity-due has a greater present value than an otherwise identical annuity-immediate. The same timing argument explains why its accumulated value at a fixed date after the final payment is greater.

Worked Examples

Worked Example: Combining Two Streams at Present

An account earns an effective annual interest rate of $5\%$. It will receive four annual payments of $$400$ at the ends of years $1$ through $4$, and four annual payments of $$250$ at the beginnings of years $0$ through $3$. Find the combined present value at time $0$. The payments are certain; no mortality assumption or probability distribution applies.

Method. Value each stream at time $0$, using an annuity-immediate factor for the end-of-year payments and an annuity-due factor for the beginning-of-year payments.

For the $$400$ annuity-immediate, $n=4$ and $i=0.05$. The discount factor is $v=1/1.05$, so:

PV_{\mathrm{immediate}} =400a_{\overline{4}|} =400\left(\frac{1-(1.05)^{-4}}{0.05}\right) \approx \$1{,}418.3802.

For the $$250$ annuity-due, use the due factor for the same four-payment term:

PV_{\mathrm{due}} =250\ddot{a}_{\overline{4}|} =250(1.05)a_{\overline{4}|} \approx \$930.8120.

Both values are already at time $0$, so add them:

PV_{\mathrm{combined}} =400a_{\overline{4}|}+250\ddot{a}_{\overline{4}|} \approx 1{,}418.3802+930.8120 =\$2{,}349.1922.

Rounded to the nearest cent, the combined present value is $$2,349.19$. As a check, the payment dates are $0,1,2,3,4$: the cash flows are $$250$ at time $0$, $$650$ at times $1,2,3$, and $$400$ at time $4$. Discounting those amounts to time $0$ gives the same total. The key mistake avoided is using the immediate factor for both streams.

Worked Example: Combining Streams at a Future Date

An account earns an effective annual interest rate of $6\%$. A saver deposits $$500$ at the end of each of years $1,2,3$, and separately deposits $$200$ at the beginning of each of years $0,1,2$. Find the account value at time $4$. All deposits are certain; no mortality assumption or probability distribution applies.

Method. Accumulate each stream to time $4$. The end-of-year deposits have earned for $3,2,1$ years, respectively. The beginning-of-year deposits have earned for $4,3,2$ years, respectively.

For the $$500$ immediate stream, first accumulate its three payments to time $3$ with $s_{\overline{3}|}$, then move that value one more year to time $4$:

s_{\overline{3}|} =\frac{(1.06)^3-1}{0.06} =3.1836, \qquad V_4^{\mathrm{immediate}} =500(1.06)s_{\overline{3}|} =500(3.374616) =\$1{,}687.308.

For the $$200$ due stream, its payments occur at times $0,1,2$. At time $4$, they have accumulated for $4,3,2$ years. Equivalently, the due stream accumulates to time $3$ as $(1.06)s_{\overline{3}|}$ per dollar of payment, then earns one more year:

V_4^{\mathrm{due}} =200(1.06)^2s_{\overline{3}|} =200(3.57709296) =\$715.418592.

Add the two values, since both are at time $4$:

V_4 =1{,}687.308+715.418592 =\$2{,}402.726592.

Rounded to the nearest cent, the account value is $$2,402.73$. Checking the payment dates confirms the accumulation periods: the immediate deposits grow by $(1.06)^3,(1.06)^2,(1.06)$; the due deposits grow by $(1.06)^4,(1.06)^3,(1.06)^2$. The common error avoided is stopping the due stream’s accumulation at time $3$ instead of carrying it to the requested date $4$.

Worked Example: Overlapping Streams at Present

A fund earns an effective annual interest rate of $4\%$. It pays $$1,000$ at the beginnings of years $0,1,2$, and also pays $$700$ at the ends of years $1,2,3$. Find the total present value at time $0$. The payments are certain; no mortality assumption or probability distribution applies.

Method. Apply the annuity-due factor to the three payments at times $0,1,2$, and the annuity-immediate factor to the three payments at times $1,2,3$. Then add the two present values.

a_{\overline{3}|} =\frac{1-(1.04)^{-3}}{0.04} \approx 2.775091, \qquad \ddot{a}_{\overline{3}|} =1.04a_{\overline{3}|} \approx 2.886094.

Therefore the present value of the due stream is $1{,}000\ddot{a}_{\overline{3}|}$, and the present value of the immediate stream is $700a_{\overline{3}|}$:

PV =1{,}000\ddot{a}_{\overline{3}|} +700a_{\overline{3}|} \approx 2{,}886.0947+1{,}942.5637 =\$4{,}828.6584.

Rounded to the nearest cent, the present value is $$4,828.66$. A boundary check combines the cash flows at each date: $$1,000$ at time $0$, $$1,700$ at times $1$ and $2$, and $$700$ at time $3$. Discounting gives $1{,}000+1{,}700v+1{,}700v^2+700v^3$, which agrees with the factor calculation. The key mistake avoided is treating overlapping payments as if the two streams had the same timing.

Common Timing Trap

A frequent error is to notice that two streams have the same payment amount and number of payments, then use one factor for both. That ignores the payment boundaries. For example, at a positive interest rate, discounting an annuity-due as though it were immediate changes a payment at time $0$ into one at time $1$, and shifts every other payment one period later.

Exam trap: If a $$100$ payment is due now, its present value is $$100$, not $$100v$. The incorrect calculation $100a_{\overline{n}|}$ for a due stream discounts every payment one period too far. Use $\ddot{a}_{\overline{n}|}$, or write the actual payment dates and discount each cash flow directly.

Before adding values, label the focal date on the timeline. A present value and an accumulated value cannot be added directly if they refer to different dates. Move one of them first, and keep the number of interest periods equal to the number of periods between the two dates.

Mark the payment dates.
For each stream, identify whether payments occur at period beginnings or endings.

Select a focal date.
Use the date requested, or another date that makes the calculations convenient.

Value each stream separately.
Use its matching annuity factor, and move its value to the focal date if needed.

Add and check.
Add values at the common date, then compare with a direct timeline calculation when practical.

Check Your Understanding

For each question, identify the payment timing before selecting a factor. Unless stated otherwise, payments are certain and no mortality assumption or probability distribution applies.

An account earns $5\%$ effective annual interest. Find the present value of $$200$ at the ends of years $1,2,3$ together with $$100$ at the beginnings of years $0,1,2$. (Answer: $$830.59$)
At $4\%$ effective annual interest, find the value at time $3$ of $$300$ deposited at the ends of years $1,2,3$, and $$200$ deposited at the beginnings of years $0,1,2$. (Answer: $$1,585.77$)
At $6\%$ effective annual interest, find the present value of $$500$ paid at the ends of years $1$ through $4$, and $$150$ paid at the beginnings of years $0$ through $3$. (Answer: $$2,283.50$)
An account earns $8\%$ effective annual interest. Find the value at time $4$ of $$400$ paid at the ends of years $2,3,4$, together with $$250$ paid at the beginnings of years $1,2,3$. (Answer: $$2,175.09$)
At $3\%$ effective annual interest, a fund pays $$1,000$ at the ends of years $1,2$, and $$400$ at the beginnings of years $0,1,2$. Find the combined present value, accounting for the fact that payments from the two streams occur on some of the same dates. (Answer: $$3,078.86$)

What You'll Learn