Timeline Checks For Forward-Rate Equations

Why a Timeline Check Matters

A forward-rate equation can look plausible while comparing values measured at different dates. The safest check is to mark the date of every cash flow, label the interval to which each rate applies, and then move both strategies to the same focal date. Only values at a common date can be compared directly.

This tutorial focuses on that audit. It uses spot accumulation factors and forward rates to check no-arbitrage equations; the central task is not merely to substitute rates, but to verify that every factor applies over the right time interval. The methods are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Let $D(0,t)$ be the time-0 price of one unit payable at time $t$, and let $A(0,t)=1/D(0,t)$ be the corresponding accumulation factor from time 0 to time $t$. For $0\leq s<t$, the forward accumulation factor from $s$ to $t$ is $A(s,t)=D(0,s)/D(0,t)$. If $f_{s,t}$ is the annual effective forward rate for that interval, then $A(s,t)=(1+f_{s,t})^{t-s}$. The factor $t-s$ is the interval length in years.

The forward rate is attached to its stated start and end dates. In particular, a forward rate from time 1 to time 3 applies for two years; it is not automatically a rate from time 0 to time 3. When maturities are fractional, the interval length is still measured in years, and the stated effective-rate convention must be used consistently.

The Timeline Identity

Suppose an investment is to deliver one unit at time $t$. It can be priced directly using $D(0,t)$. Alternatively, it can be priced by first buying a unit payable at time $s$, then arranging the forward investment from $s$ to $t$. No-arbitrage requires these strategies to give the same value.

$$ D(0,s)D(s,t)=D(0,t) $$

Since $D(s,t)=1/A(s,t)$, this implies $A(s,t)=D(0,s)/D(0,t)$. Equivalently, the forward accumulation factor must complete the accumulation already earned by time $s$ to the total accumulation required by time $t$:

$$ A(0,s)A(s,t)=A(0,t) $$

Under annual effective rates at integer maturities, this becomes the familiar equation

(1+i_s)^s(1+f_{s,t})^{t-s}=(1+i_t)^t

Here $i_s$ and $i_t$ are the effective annual spot rates to times $s$ and $t$, respectively, and $f_{s,t}$ is the annual effective forward rate for the interval from $s$ to $t$. The equation assumes the rates are quoted on this effective annual basis and that the cash flows being compared have the same timing and certainty. If the problem specifies a different compounding convention, use that convention’s accumulation factors instead.

Mark every date.
Write the time-0 date, the forward interval’s start and end, and the date at which each amount is received or paid.

Choose a focal date.
Use a date at which both strategies can be valued, often the final cash-flow date.

Apply each factor only over its interval.
For example, a forward rate from time 1 to time 3 accumulates for two years, not three.

Compare the resulting values.
If the strategies are equivalent under the stated no-arbitrage assumptions, their values at the focal date must agree.

Worked Timeline Checks

Worked Example: Checking a One-Year Forward Rate

The one-year spot rate is 5% effective, and the two-year spot rate is 4% effective. Find the annual effective forward rate from time 1 to time 2. Then check the result by comparing the value at time 2 of two investments, each starting with $10,000 at time 0.

Method. Use time 2 as the focal date. The direct two-year investment accumulates at the two-year spot rate. The alternative first accumulates to time 1 at the one-year spot rate, then uses the one-year forward rate from time 1 to time 2.

The direct accumulation factor is

$$ A(0,2)=(1.04)^2=1.0816 $$

The first-year accumulation factor is

$$ A(0,1)=1.05 $$

Therefore the required forward accumulation factor is

A(1,2)=\frac{A(0,2)}{A(0,1)} =\frac{1.0816}{1.05} =1.0300952381\ldots

Because the interval is one year, the forward rate is

f_{1,2}=1.0300952381\ldots-1 =0.0300952381\ldots

Thus $f_{1,2}=3.0095\%$ to four decimal places. Now check the values at time 2. The direct investment is

\$10{,}000(1.04)^2=\$10{,}816

For the rolled strategy, the value at time 1 is $\$10{,}000(1.05)=\$10{,}500$. Its value at time 2 is

\$10{,}500(1.0300952381\ldots)=\$10{,}816

The two strategies agree at the focal date. The calculation avoids the common mistake of using the one-year spot rate as if it were the forward rate for the second year.

Worked Example: Keeping a Nonadjacent Forward Rate in Its Own Interval

The one-year spot rate is 3% effective. The annual effective forward rate from time 1 to time 3 is 4%. A deposit of $6,000 is made at time 0. Find its value at time 3 if it is invested at the one-year spot rate and then rolled over at the stated forward rate. Check the result using the total accumulation factor.

Method. Label the timeline $0\to1\to3$. The first interval is one year and uses the 3% spot rate. The forward interval is two years and uses the 4% forward rate.

At time 1, the deposit has grown to

\$6{,}000(1.03)=\$6{,}180

From time 1 to time 3, the forward accumulation factor is

(1.04)^{3-1}=(1.04)^2=1.0816

Therefore the value at time 3 is

\$6{,}180(1.0816)=\$6{,}684.288

Rounded to the nearest cent, the value is $\$6{,}684.29$. The total accumulation factor from time 0 to time 3 under this strategy is

$$ (1.03)(1.04)^2 =1.03(1.0816) =1.114048 $$

Multiplying this total factor by the initial deposit gives $\$6{,}000(1.114048)=\$6{,}684.288$, confirming the result. A tempting but incorrect calculation is $\$6{,}000(1.04)^3=\$6{,}749.184$. That applies the forward rate from time 0, even though it is only specified from time 1 to time 3. The timeline identifies the error: the first year is not part of the forward interval.

Worked Example: Checking a Forward Rate Between Fractional Dates

Suppose the time-0 discount factors are $D(0,0.5)=0.98$ and $D(0,1.5)=0.90$. Find the annual effective forward rate from time 0.5 to time 1.5. Check the result by comparing two investments that begin with $9,800 at time 0 and have equal values at time 1.5.

Method. Use the discount-factor ratio to find the forward accumulation factor. The interval from 0.5 to 1.5 is exactly one year, even though both dates are fractional maturities.

The forward accumulation factor is

A(0.5,1.5) =\frac{D(0,0.5)}{D(0,1.5)} =\frac{0.98}{0.90} =\frac{49}{45} =1.088888\ldots

For a one-year interval, the annual effective forward rate is the factor minus 1:

f_{0.5,1.5}=\frac{49}{45}-1 =\frac{4}{45} =0.088888\ldots

Thus the forward rate is $8.8889\%$ to four decimal places. To check, the time-0 investment of $\$9{,}800$ grows to $\$9{,}800/0.98=\$10{,}000$ at time 0.5, since $D(0,0.5)=0.98$. Rolling that amount forward gives

\$10{,}000\left(\frac{49}{45}\right) =\$10{,}888.888\ldots

Alternatively, using the time-1.5 discount factor, the direct value at time 1.5 is

\frac{\$9{,}800}{0.90} =\$10{,}888.888\ldots

Both strategies therefore have a value of $\$10{,}888.89$ at time 1.5, rounded to the nearest cent. The check relies on the given discount factors and the one-year interval; it does not treat either fractional maturity as a full number of years.

Common Timeline Errors

A forward-rate equation should be read as a statement about dated cash flows, not as a pattern of exponents to memorize. A useful audit is to ask what each factor does to a dollar and over which dates it does so.

Starting the forward rate too early. A rate for the interval from time $s$ to time $t$ cannot be applied to the interval from time 0 to time $t$ unless $s=0$.
Using the wrong interval length. The exponent on an annual effective forward factor is $t-s$, not $t$, when the forward interval begins after time 0.
Comparing amounts at different dates. A value at time $s$ cannot be directly compared with a value at time $t$. Accumulate or discount one amount to the other date first.
Reversing the discount-factor ratio. The forward accumulation factor is $D(0,s)/D(0,t)$. Reversing the ratio would produce a discount factor, not an accumulation factor.
Ignoring the rate convention. The power $t-s$ applies to an annual effective rate. If a rate is quoted with a different compounding convention, convert or use its matching accumulation factor.

Key takeaway: Label the cash-flow dates, identify the forward interval, and carry both strategies to one focal date. The forward accumulation factor must fill the gap between the accumulation already achieved at the interval’s start and the total accumulation required at its end.

Check Your Understanding

For each problem, mark the dates and compare values at the stated focal date.

The one-year spot rate is 4% effective and the two-year spot rate is 6% effective. Find the annual effective forward rate from time 1 to time 2, then find the time-2 value of a $2,500 deposit made at time 0. (Answer: $f_{1,2}=8.0385\%$ to four decimal places; time-2 value $=\$2{,}809.00$.)
The discount factors are $D(0,1)=0.96$ and $D(0,3)=0.84$. Find the annual effective forward rate from time 1 to time 3. (Answer: $A(1,3)=8/7$, so $f_{1,3}=\sqrt{8/7}-1\approx6.9045\%$ to four decimal places.)
A forward rate of 2.5% effective applies from time 2 to time 4. What is the value at time 4 of $7,000 available at time 2? State the interval length used. (Answer: The interval is two years; value $=\$7{,}000(1.025)^2=\$7{,}354.38$, to the nearest cent.)
The discount factors are $D(0,0.25)=0.99$ and $D(0,1.25)=0.95$. Find the annual effective forward rate from time 0.25 to time 1.25 and check the time-1.25 value of $9,900 invested at time 0 by both routes. (Answer: $A(0.25,1.25)=0.99/0.95=1.0421053$, so $f_{0.25,1.25}\approx4.2105\%$; both routes give $\$9{,}900/0.95=\$10{,}421.05$, to the nearest cent.)

What You'll Learn