Varying Payments under Changing Interest Rates

Why Changing Rates Require Payment-by-Payment Valuation

For a fixed effective rate, standard annuity factors can compress a sequence of payments into one calculation. When the effective rate differs by period, however, a payment at the end of period 3 is discounted through periods 1, 2, and 3, using the rate applicable in each one. The payment pattern may be arithmetic or geometric, but the interest-rate path determines its value date by date.

This tutorial considers payments at the ends of consecutive periods, with one effective rate specified for each period. The period could be a year, a month, or another consistent unit. Payment amounts and rates must be expressed on that same timeline. The method is to discount or accumulate each payment using the factors for all periods between its payment date and the valuation date.

Definition: Let $i_k$ be the effective interest rate earned during period $k$, for $k=1,\ldots,n$, and let $C_k$ be a payment at the end of period $k$. Define the cumulative accumulation factor to time $k$ by $A_k=\prod_{j=1}^{k}(1+i_j)$, with $A_0=1$. The present value at time 0 of the payment stream is $PV_0=\sum_{k=1}^{n} C_k/A_k$. This definition assumes the stated rate applies for the whole of its period and that each payment occurs at the stated period end.

The factor $A_k$ is a product, not a single period’s factor. In particular, the period-3 payment is discounted by $A_3=(1+i_1)(1+i_2)(1+i_3)$. Rates may differ in every period. The usual condition $i_k>-1$ ensures each accumulation factor is positive. If payments occur at dates other than the ends of these consecutive periods, the period-by-period formula must be adapted to the actual dates; that is a different timing setup.

Valuation Formula and Its Justification

Each payment can be moved to time 0 on its own. A payment $C_k$ at time $k$ is divided by the factor earned from time 0 through time $k$. Adding those time-0 values gives the total present value.

PV_0=\frac{C_1}{1+i_1} +\frac{C_2}{(1+i_1)(1+i_2)} +\cdots+ \frac{C_n}{\prod_{j=1}^{n}(1+i_j)} =\sum_{k=1}^{n}\frac{C_k}{A_k}.

The same payments can be valued at time $n$. A payment at time $k$ accumulates only through periods $k+1$ to $n$; it earns no interest before it is paid. Therefore,

AV_n=\sum_{k=1}^{n}C_k\prod_{j=k+1}^{n}(1+i_j),

where a product over no periods equals 1. This convention makes the final payment $C_n$ appear unchanged in the accumulated value.

Justification: Under the effective-rate model, one dollar at time 0 grows to $A_k$ dollars at time $k$. Thus $C_k/A_k$ at time 0 is equivalent to $C_k$ at time $k$. Adding these equivalent time-0 amounts gives $PV_0$. To move that value to time $n$, multiply by $A_n$; term by term, $A_n(C_k/A_k)=C_k\prod_{j=k+1}^{n}(1+i_j)$. Hence $AV_n=A_nPV_0$, which agrees with accumulating each payment separately.

Arithmetic and geometric patterns specify the amounts $C_k$, not the discount factors. For example, an arithmetic-increasing sequence with first payment $C_1$ and fixed increase $g$ has $C_k=C_1+(k-1)g$. A geometric-increasing sequence with first payment $C_1$ and growth rate $r$ has $C_k=C_1(1+r)^{k-1}$. Under changing interest rates, substitute these amounts into the payment-by-payment valuation formula. A single constant-rate annuity factor generally cannot replace the resulting sum.

Worked Examples

Worked Example: An Arithmetic Gradient with Three Different Rates

A fund will pay $100 at the end of year 1, $120 at the end of year 2, and $140 at the end of year 3. The effective annual rates for years 1, 2, and 3 are 4%, 5%, and 6%, respectively. Find the present value at time 0. Use the payment-by-payment method and check by accumulating to year 3.

Method. The payments form an arithmetic sequence with first payment $100 and increase $20 each year. First calculate the cumulative factors:

A_1=1.04,\qquad A_2=1.04(1.05)=1.092,\qquad A_3=1.092(1.06)=1.15752.

Discount each payment by its cumulative factor:

PV_0 =\frac{100}{1.04}+\frac{120}{1.092}+\frac{140}{1.15752} =96.153846+109.890110+120.948234 =326.992190.

The unrounded present value is approximately $326.992190, so the value to the nearest cent is $326.99. For the check, accumulate the year-1 payment through years 2 and 3, the year-2 payment through year 3, and leave the year-3 payment unchanged:

$$ AV_3=100(1.05)(1.06)+120(1.06)+140 =111.30+127.20+140 =378.50. $$

The present value multiplied by the full factor to year 3 is $326.992190\ldots(1.15752)=378.50$, confirming the result. The common mistake avoided here is discounting a later payment only by its own year’s rate.

Worked Example: Present and Accumulated Values of an Arithmetic Sequence

A hypothetical grant pays $50, $60, and $70 at the ends of years 1, 2, and 3. The effective annual rates are 4%, 5%, and 10%, respectively. Find the present value at time 0 and the accumulated value at time 3.

Method. The first two cumulative factors are $A_1=1.04$ and $A_2=1.04(1.05)=1.092$. The third is $A_3=1.092(1.10)=1.2012$. Thus,

PV_0 =\frac{50}{1.04}+\frac{60}{1.092}+\frac{70}{1.2012} =48.076923+54.945055+58.275058 =161.297036.

The present value is approximately $161.297036, or $161.30 to the nearest cent. For the accumulated value, move each payment forward only through the periods after it is paid:

$$ AV_3=50(1.05)(1.10)+60(1.10)+70 =57.75+66.00+70 =193.75. $$

As a check, $161.297036\ldots(1.2012)=193.75$. The third payment’s discount denominator is the product of all three annual factors, not merely $1.10$; using only the final-year rate would give it the wrong time-0 value.

Worked Example: A Geometric Payment Pattern under Changing Rates

A hypothetical research fund pays $80 at the end of year 1. Each later annual payment is 10% greater than the preceding payment, so the next payments are $88, $96.80, and $106.48. The effective annual rates for years 1 through 4 are 2%, 4%, 5%, and 3%, respectively. Find the present value at time 0 and verify it by accumulating to year 4.

Method. The geometric payment growth determines the payment amounts, while the four rates determine the cumulative discount factors. Compute them in order:

A_1=1.02,\quad A_2=1.02(1.04)=1.0608,\quad A_3=1.0608(1.05)=1.11384,\quad A_4=1.11384(1.03)=1.1472552.

Discount the four payments separately:

\begin{aligned} PV_0 &=\frac{80}{1.02}+\frac{88}{1.0608} +\frac{96.8}{1.11384}+\frac{106.48}{1.1472552}\\ &=78.431373+82.956259+86.906558+92.812828\\ &=341.107018. \end{aligned}

The present value is approximately $341.107018, or $341.11 to the nearest cent. For an independent check at year 4, the four payments accumulate to:

\begin{aligned} AV_4 &=80(1.04)(1.05)(1.03)+88(1.05)(1.03)+96.8(1.03)+106.48\\ &=89.9808+95.172+99.704+106.48\\ &=391.3368. \end{aligned}

Multiplying the unrounded present value by $A_4$ gives $341.107018\ldots(1.1472552)=391.3368$, as required. Notice that the second cumulative factor is $1.0608$, not $1.061$; retaining the correct factor is important when carrying the later calculations.

Common Mistakes and Practical Checks

A changing rate does not mean that each payment is valued using only the rate in its payment period. Every earlier period’s rate also affects the value of money carried from time 0. For the first example, the incorrect shortcut would be:

\frac{100}{1.04}+\frac{120}{1.05}+\frac{140}{1.06} =96.153846+114.285714+132.075472 =342.515032.

That calculation is wrong because it ignores the earlier periods when discounting the second and third payments. The correct denominators are $1.092$ and $1.15752$, respectively, giving $PV_0\approx326.992190$, not $342.515032$.

Other checks help prevent timing and arithmetic errors:

Label the period rates. A stated sequence of rates must match the periods in chronological order. Do not apply the year-1 rate to every payment unless the problem says the rate is constant.
Build cumulative factors sequentially. Calculate $A_k=A_{k-1}(1+i_k)$. This makes it easier to spot an omitted or duplicated period.
Keep payment growth separate. The arithmetic increment or geometric growth rate determines $C_k$; it does not replace the interest rate for that period.
Check at a common date. Present value and accumulated value should satisfy $AV_n=A_nPV_0$, apart from rounding. Carry unrounded values through the calculation and round money only at the end.

This approach is useful whenever a stream of scheduled payments spans periods with different effective rates. It avoids forcing a single rate onto a changing rate path, while retaining the familiar arithmetic or geometric payment descriptions. It is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the payment dates and use the interest rate for every period between time 0 and the relevant payment or valuation date. Round dollar answers to the nearest cent where needed.

A fund pays $40, $50, and $60 at the ends of periods 1, 2, and 3. The effective rates are 0%, 25%, and 20%, respectively. Find the present value at time 0. (Answer: $120.00)
A contract pays $105, $110.25, and $115.7625 at the ends of three successive years. The effective annual rate is 5% in each year. Find the present value. (Answer: $300.00)
A project pays $25, $40, and $60 at the ends of years 1, 2, and 3. The effective annual rates are 2%, 3%, and 4%. Find the accumulated value at year 3. (Answer: $128.38)
A trust pays $75, $90, and $105 at the ends of periods 1, 2, and 3. The rates are 4%, 6%, and 2%, respectively. Find the accumulated value at the end of period 3. (Answer: $277.89)
A payment stream consists of $100, $110, $120, and $130 at the ends of periods 1 through 4. The effective rate is 0% in period 1 and 10% in each of periods 2, 3, and 4. Find the present value at time 0. (Answer: $396.85)

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