Why Forward-Rate Problems Require More Than a Formula
A forward rate is implied by today’s spot prices for payments at different future dates. A reliable solution does not stop after calculating that rate: it checks the interval length, uses a consistent rate convention, and compares the strategies by their cash flows at each date. These steps are particularly important when the forward interval is not one year or when a quoted rate is nominal rather than effective.
Assume throughout that the relevant zero-coupon investments are default-free, can be bought or sold at the stated prices, and have no transaction costs or taxes. Let \(D(t)\) be the time-0 price of a zero-coupon payment of one unit at time \(t\). Thus a payment of \(C\) at time \(t\) has present value \(C D(t)\). Times are measured in years.
The formula uses the length of the forward interval, \(t_2-t_1\), as the exponent. It does not use \(t_2\) unless the interval begins at time 0. If effective annual spot rates \(i_{t}\) are given instead of discount factors, then \(D(t)=(1+i_t)^{-t}\), and so $$ (1+f_{t_1,t_2})^{t_2-t_1} =\frac{(1+i_{t_2})^{t_2}}{(1+i_{t_1})^{t_1}}. $$
Why the Discount-Factor Ratio Gives the Forward Accumulation
A forward investment with unit principal has dated cash flows \(-1\) at \(t_1\) and \(1+f_{t_1,t_2}\) raised to the interval length at \(t_2\). Its time-0 value, using the zero-coupon prices, is $$ -D(t_1)+D(t_2)(1+f_{t_1,t_2})^{t_2-t_1}. $$
In a no-arbitrage market, a forward contract entered with no initial payment must have value zero at inception. Setting the expression equal to zero and rearranging gives the discount-factor formula. Equivalently, one can replicate the forward cash flows with zero-coupon investments at the two dates. If the formula fails, the forward cash flows and the spot portfolio do not have the same value, creating a potential arbitrage.
Record the start and end of the forward interval, then find its length \(t_2-t_1\).
Divide the earlier discount factor by the later one, or use the corresponding spot-rate expression.
Take the power \(1/(t_2-t_1)\) for an effective annual rate. Convert separately if the problem asks for a nominal rate.
Compare the forward strategy with the spot strategy at each date, not merely by comparing rate labels.
Worked Applications
Worked Example: Infer a Forward Rate and Match the Investment
The discount factors are \(D(1)=0.90\) and \(D(3)=0.81\). Find the effective annual forward rate from year 1 to year 3. Then verify the implied accumulation by comparing two ways to invest \$8,100 from time 0 to year 3.
Method. Use the ratio of discount factors for the two endpoints. The forward interval is from year 1 to year 3, so its length is \(3-1=2\) years.
The accumulation factor over those two years is $$ \frac{D(1)}{D(3)}=\frac{0.90}{0.81}=\frac{10}{9}=1.111111\ldots $$
Therefore the annual effective forward rate is $$ f_{1,3}=\left(\frac{10}{9}\right)^{1/2}-1 =1.0540926\ldots-1 =0.0540926\ldots. $$ As a percentage, this is \(5.409\%\) to three decimal places.
Cash-flow check. A one-year spot investment accumulates by \(1/D(1)=1/0.90=10/9\). Thus \$8,100 grows to $$ 8100\left(\frac{10}{9}\right)=\$9,000 $$ at year 1. Investing that \$9,000 at the implied two-year forward factor gives $$ 9000\left(\frac{10}{9}\right)=\$10,000 $$ at year 3.
The direct three-year spot investment gives the same result: $$ 8100\left(\frac{1}{D(3)}\right) =8100\left(\frac{1}{0.81}\right) =\$10,000. $$ Both strategies start with \$8,100 and produce \$10,000 at year 3; the rolled strategy passes through \$9,000 at year 1. This verifies the cash-flow match. The common mistake avoided here is treating the interval as three years rather than two.
Worked Example: Convert the Implied Forward Rate and Compare a Quote
Suppose \(D(1)=9/10\) and \(D(3)=40/49\). Find the annual effective forward rate from year 1 to year 3 and its equivalent nominal annual rate convertible quarterly. A market quote is \(4.8\%\) nominal convertible quarterly. Compare the two rates and compare the two-year accumulation on \$10,000.
Method. First infer the effective annual rate from the discount-factor ratio. Then convert it to a nominal quarterly rate; compare rates only after putting them on the same convention.
The forward interval is two years, and its total accumulation factor is $$ \frac{D(1)}{D(3)} =\frac{9/10}{40/49} =\frac{441}{400} =(1.05)^2. $$ Thus the implied effective annual forward rate is \(5\%\).
Let \(j^{(4)}\) be the equivalent nominal annual rate convertible quarterly. Its quarterly rate must give the same annual accumulation: $$ \left(1+\frac{j^{(4)}}{4}\right)^4=1.05. $$ Solving, $$ j^{(4)}=4\left((1.05)^{1/4}-1\right) =4(1.0122722\ldots-1) =0.0490889\ldots. $$ The equivalent nominal rate is \(4.909\%\) to three decimal places. The market quote of \(4.8\%\) nominal convertible quarterly is lower.
Over the two-year interval, the market quote accumulates at $$ \left(1+\frac{0.048}{4}\right)^8=(1.012)^8=1.1001302\ldots, $$ whereas the implied forward accumulates at \((1.05)^2=1.1025\). On \$10,000, these give approximately \$11,001.30 and \$11,025.00, respectively, a difference of \$23.70 to the nearest cent. This is a comparison of accumulations over the same two-year period. It is not, by itself, a complete arbitrage construction: that also requires specifying the contract cash flows and any initial payment.
The main trap is comparing \(4.8\%\) nominal directly with \(5\%\) effective. The quarterly quote first needs to be converted to its annual accumulation, \((1.012)^4\), or to an equivalent annual effective rate. This example treats the quarterly compounding periods as aligned with the two-year contract interval.
Worked Example: Identify and Construct an Arbitrage
Let \(D(1)=0.95\) and \(D(2)=0.88\). A market offers a zero-initial-payment forward investment from year 1 to year 2 at an effective annual rate of \(9\%\): the investor pays \$100 at year 1 and receives \$109 at year 2. Determine whether the quote is consistent with no-arbitrage. If not, construct a zero-risk portfolio and calculate its time-0 gain.
Method. Calculate the implied one-year forward rate, then compare the forward cash flows with zero-coupon positions that offset them date by date.
The implied accumulation factor and effective annual rate are $$ \frac{D(1)}{D(2)}=\frac{0.95}{0.88}=1.07954545\ldots, \qquad f_{1,2}=1.07954545\ldots-1=7.954545\ldots\%. $$ The market rate of \(9\%\) is higher than the implied rate, so the zero-initial-payment forward is inconsistent with the stated zero-coupon prices.
At time 0, buy a one-year zero-coupon bond with face amount \$100. Its price is \(100(0.95)=\$95\). Also short a two-year zero-coupon bond with face amount \$109. The short sale produces proceeds of \(109(0.88)=\$95.92\). The net time-0 cash flow is therefore $$ -\$95+\$95.92=+\$0.92. $$
At year 1, the one-year bond pays \$100, exactly the amount required by the forward investment. At year 2, the forward pays \$109, exactly the amount owed on the short two-year bond. The year-1 and year-2 net cash flows are both zero, while the portfolio receives \$0.92 at time 0. This is an arbitrage under the assumptions stated above. The common mistake avoided is to label a rate difference as an arbitrage without constructing positions whose cash flows actually cancel at every future date.
Rate Conventions and Exam Traps
A quoted nominal annual rate \(j^{(m)}\), convertible \(m\) times per year, has accumulation factor \((1+j^{(m)}/m)^{m\Delta}\) over an interval of \(\Delta\) years when the compounding periods align with the interval. Its equivalent effective annual rate is \((1+j^{(m)}/m)^m-1\). Conversely, an effective annual forward rate \(f\) corresponds to $$ j^{(m)}=m\left((1+f)^{1/m}-1\right). $$ If a contract interval does not align with the stated conversion periods, follow the timing convention specified in the problem rather than silently treating a nominal quote as an effective rate.
The following wrong calculation illustrates the most frequent interval error. If \(D(1)=0.90\) and \(D(3)=0.81\), the factor \(0.90/0.81=10/9\) applies from year 1 to year 3. It covers two years, not three. The incorrect calculation \((10/9)^{1/3}-1\) uses the wrong exponent; the correct calculation is \((10/9)^{1/2}-1=5.409\%\).
For an unequal interval, the same rule applies without modification: use \(t_2-t_1\) in the exponent. Also check the direction of the discount-factor ratio. The earlier discount factor is in the numerator because a unit payable sooner generally costs more today than a unit payable later. If discount factors are provided directly, there is no need to first derive spot rates unless the question requests them.
Check Your Understanding
For each problem, state the interval length before calculating. Answers are rounded only at the final step.
- Given \(D(1)=0.96\) and \(D(3)=0.84\), find the effective annual forward rate from year 1 to year 3. (Answer: \(6.905\%\))
- Given \(D(2)=0.92\) and \(D(5)=0.80\), find the effective annual forward rate from year 2 to year 5. (Answer: \(4.769\%\))
- An effective annual forward rate is \(6\%\). Find its equivalent nominal annual rate convertible semiannually. (Answer: \(5.913\%\))
- Given \(D(1)=0.97\) and \(D(2)=0.91\), find the implied effective annual rate from year 1 to year 2, then state whether a zero-initial-payment market forward at \(6.5\%\) is above or below the implied rate. (Answer: implied rate \(6.593\%\); the market quote is below.)
- The effective annual spot rates are \(3\%\) for two years and \(4\%\) for four years. Find the effective annual forward rate from year 2 to year 4. (Answer: \(\left((1.04)^4/(1.03)^2\right)^{1/2}-1 \approx 5.010\%\))