How Interest Rates Change Annuity Values

Why the Interest Rate Changes an Annuity’s Value

An annuity’s value depends not only on its payment amounts and timing but also on the interest rate used to value those payments. Holding the payments, payment dates, and valuation date fixed, increasing a positive effective rate has a predictable effect: it lowers present value and raises accumulated value for nonnegative payments. The reason is visible payment by payment—later cash flows are discounted more heavily to an earlier date and earn more interest when carried to a later date.

The rate comparison is meaningful only when the rate is expressed per payment period and both valuations use the same cash-flow schedule and focal date. The discussion below concerns fixed, nonnegative payments. Signed cash flows require more care, as shown later.

Definition: Let $i>0$ be the effective interest rate per payment period, and let $v=1/(1+i)$. For payments $C_k$ at times $k=1,\ldots,n$, the present value at time 0 is $\sum_{k=1}^{n} C_kv^k$. The accumulated value at time $n$ is $\sum_{k=1}^{n}C_k(1+i)^{n-k}$. These expressions value the same fixed payments at different focal dates.

The formulas come directly from valuing each payment separately. A payment at time $k$ is moved back $k$ periods to time 0 by multiplying by $v^k$. To move it forward to time $n$, it earns interest for $n-k$ periods, so it is multiplied by $(1+i)^{n-k}$. Adding the values of all payments gives the value of the stream.

For an annuity-immediate with level payment $C$, the payments occur at times $1,\ldots,n$. Thus its present value is $C\sum_{k=1}^n v^k$, and its accumulated value at time $n$ is $C\sum_{k=1}^n(1+i)^{n-k}$. For an annuity-due, payments occur at times $0,\ldots,n-1$; each payment is still valued according to its own time relative to the focal date. The rate comparisons below do not depend on the payments being level or on their being immediate or due.

Why Present Value Decreases as the Rate Increases

Suppose two positive effective rates satisfy $i_2>i_1>0$. Then $1+i_2>1+i_1$, so for every payment time $k>0$,

\frac{1}{(1+i_2)^k}<\frac{1}{(1+i_1)^k}.

If $C_k\geq 0$, multiplying this inequality by $C_k$ shows that the present value contribution from that payment cannot increase. Adding the inequalities across all payment dates proves that the total present value cannot increase either. It decreases strictly if at least one positive payment occurs after time 0.

A payment made at time 0 is different: it is already at the present-value date, so its value is unchanged by the interest rate. A stream consisting only of a time-0 payment therefore has the same present value at every rate.

Theorem: For fixed nonnegative payments at fixed dates, present value at a fixed date is nonincreasing as the positive effective rate increases. It decreases strictly if at least one positive payment is made after that date. Payments made at the valuation date are unaffected.

Worked Example: Comparing Present Values at Two Rates

A fund will receive $100$ at the end of year 1, $200$ at the end of year 2, and $300$ at the end of year 3. Find the present value at an effective annual rate of $10\%$ and at an effective annual rate of $20\%$. Compare the results.

Method. Discount each payment to time 0 using its own payment time. The rates are annual and the payments are annual, so no rate conversion is needed.

At $i=0.10$, $v=1/1.10$. The three contributions are

\frac{100}{1.10}=90.9091,\qquad \frac{200}{(1.10)^2}=\frac{200}{1.21}=165.2893,\qquad \frac{300}{(1.10)^3}=\frac{300}{1.331}=225.3944.

Adding them gives $90.9091+165.2893+225.3944=481.5928$, so the present value is $\$481.59$, rounded to the nearest cent.

At $i=0.20$, the contributions are

\frac{100}{1.20}=83.3333,\qquad \frac{200}{(1.20)^2}=\frac{200}{1.44}=138.8889,\qquad \frac{300}{(1.20)^3}=\frac{300}{1.728}=173.6111.

Their sum is $83.3333+138.8889+173.6111=395.8333$, giving a present value of $\$395.83$ to the nearest cent. The higher rate produces the lower present value, as each payment is discounted more heavily. Check: each of the three contributions at $20\%$ is below its corresponding contribution at $10\%$.

Common mistake avoided: The payments are not multiplied by an accumulation factor when valuing them at time 0. Accumulating a payment moves it forward; discounting moves it back.

Why Accumulated Value Increases as the Rate Increases

Now compare the same fixed payments at a fixed focal date $n$. A payment made at time $k<n$ accumulates for $n-k$ periods. If $i_2>i_1>0$, then

(1+i_2)^{n-k}>(1+i_1)^{n-k}\qquad\text{when }k<n.

Multiplication by a nonnegative payment preserves that inequality. Adding the contributions shows that accumulated value cannot decrease as the rate increases. It increases strictly if at least one positive payment is made before the focal date. A payment made at the focal date has no time to earn interest and is unchanged.

Theorem: For fixed nonnegative payments at fixed dates, accumulated value at a fixed date is nondecreasing as the positive effective rate increases. It increases strictly if at least one positive payment is made before that date. A payment at the focal date contributes its amount at every rate.

Worked Example: Comparing Accumulated Values of Deposits

A saver deposits $200$ at time 0, $200$ at time 1, and $200$ at time 2. Find the accumulated value at time 3 at effective rates of $4\%$ and $8\%$ per period.

Method. Carry each deposit forward to time 3. The deposits earn interest for 3, 2, and 1 periods, respectively.

At $i=0.04$, the three accumulated contributions are

$$ 200(1.04)^3=200(1.124864)=224.9728, $$

$$ 200(1.04)^2=200(1.0816)=216.32,\qquad 200(1.04)=208. $$

The total is $224.9728+216.32+208=649.2928$, or $\$649.29$ to the nearest cent.

At $i=0.08$, the contributions are

$$ 200(1.08)^3=200(1.259712)=251.9424, $$

$$ 200(1.08)^2=200(1.1664)=233.28,\qquad 200(1.08)=216. $$

The total is $251.9424+233.28+216=701.2224$, or $\$701.22$ to the nearest cent. The accumulated value rises by $\$51.93$ when the rate rises from $4\%$ to $8\%$. Check: every deposit is made before time 3, so every contribution increases with the rate.

Common mistake avoided: Do not give every deposit the same number of periods to earn interest. The time-0 deposit earns for three periods, while the time-2 deposit earns for only one.

Payment Timing and the Focal Date

The direction of the change is determined by how far each payment is from the focal date. For present value, payments after time 0 are discounted back; for accumulated value, payments before the focal date are carried forward. A payment at the focal date is neither discounted nor accumulated. This remains true whether a payment schedule is described as an annuity-immediate or an annuity-due.

The following example includes payments at the starting date, between the starting and focal dates, and at the focal date. The endpoint payments make clear which contributions are affected by the rate.

Worked Example: A Payment at the Focal Date

Payments of $200$, $300$, and $500$ occur at times 0, 2, and 4, respectively. Find their accumulated value at time 4 using effective rates of $5\%$ and $10\%$ per period.

Method. Accumulate each payment only for the periods remaining until time 4. The time-0 payment earns for four periods, the time-2 payment earns for two, and the time-4 payment earns for zero.

At $i=0.05$, the contributions are

200(1.05)^4=200(1.21550625)=243.10125,\qquad 300(1.05)^2=300(1.1025)=330.75,\qquad 500(1.05)^0=500.

The total is $243.10125+330.75+500=1073.85125$, which is $\$1,073.85$ to the nearest cent.

At $i=0.10$, the contributions are

200(1.10)^4=200(1.4641)=292.82,\qquad 300(1.10)^2=300(1.21)=363,\qquad 500(1.10)^0=500.

The total is $292.82+363+500=1155.82$, or $\$1,155.82$ to the nearest cent. The first two contributions rise, while the payment made at time 4 remains $\$500$ at either rate. Check: the difference between totals, $\$81.97$, is exactly the sum of the increases in the first two contributions.

Common mistake avoided: The last payment does not earn four periods of interest. It is paid at the focal date, so its accumulation factor is $(1+i)^0=1$.

Assumptions, Limits, and an Exam Trap

The monotonicity results require nonnegative payments and a fixed set of payment dates. If a cash-flow stream contains both positive and negative amounts, multiplying individual contributions by rate-dependent factors does not guarantee that the total moves in one direction. A negative contribution can move opposite to a positive one, and the overall result depends on their sizes and timing.

The rate must also match the period used to count the exponents. If payments are annual, an annual effective rate belongs in the annual payment sums. Using a rate for a different period without converting it first changes the model rather than simply comparing rates.

A frequent error is to reason that a higher rate must make every annuity value larger because money earns more interest. That statement is true for accumulated value at a later date, but not for present value at an earlier date. For example, two payments of $100$ at times 1 and 2 have present value

\frac{100}{1.10}+\frac{100}{(1.10)^2} =90.9091+82.6446=173.5537

at $10\%$, and

\frac{100}{1.20}+\frac{100}{(1.20)^2} =83.3333+69.4444=152.7778

at $20\%$. The second present value is lower, not higher. The correct question is always: Are the payments being moved to an earlier date or a later date?

The comparison is aligned with the SOA Exam FM (Financial Mathematics) syllabus: identify the focal date, use the appropriate discounting or accumulation for each payment, and then compare the resulting values. The payment-by-payment method is also a useful check on a calculation made with an annuity factor.

Check Your Understanding

For each problem, keep the payment amounts and dates fixed while comparing rates. Round dollar values to the nearest cent when needed.

A fund receives $500$ at the end of each of the next two years. Find its present value at effective annual rates of $5\%$ and $10\%$. (Answer: $\$929.71$ at $5\%$; $\$867.77$ at $10\%$.)
Deposits of $200$ are made at times 0, 1, and 2. Find their accumulated value at time 3 at rates of $4\%$ and $8\%$ per period. (Answer: $\$649.29$ at $4\%$; $\$701.22$ at $8\%$.)
A stream pays $100$ at time 2 and $300$ at time 3. Find its present value at time 0 at rates of $25\%$ and $50\%$ per period. (Answer: $\$217.60$ at $25\%$; $\$133.33$ at $50\%$.)
Payments of $500$ at time 1 and $200$ at time 3 are accumulated to time 4. Find the values at $5\%$ and $10\%$ per period. (Answer: $\$788.81$ at $5\%$; $\$885.50$ at $10\%$.)
A signed stream pays $-300$ at time 1 and $100$ at time 2. Its present value is $-300/(1+i)+100/(1+i)^2$. Evaluate it at $10\%$ and $20\%$, and explain why the nonnegative-payment rule does not apply. (Answer: $-\$190.08$ at $10\%$; $-\$180.56$ at $20\%$; the present value increases because the stream includes a negative payment.)

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