What a Break-Even Forward Rate Means
A forward rate can be the unknown rate that makes two investment strategies produce the same value at a specified future date. The break-even rate depends on the cash flows being compared. If one strategy incurs a fee, or receives a separate credit, its value must reflect that adjustment at the date the cash flow occurs. A fee at the start of a forward investment does not have the same effect as the same fee at the end.
The essential method is to choose a common comparison date, write each strategy’s value at that date, and equate the values. This tutorial uses an annual effective quoted forward rate \(f\). If a forward investment runs for \(h\) years, its accumulation factor is \((1+f)^h\). This convention applies whether \(h\) is a whole number or a fraction of a year. If a problem specifies a different quote convention, use that convention’s accumulation factor instead.
Assume that at the start of a forward interval, one strategy has \(B\) dollars available. It pays a fixed fee \(C\) immediately from that amount, invests the remaining \(B-C\) for \(h\) years at annual effective forward rate \(f\), and pays a fixed fee \(K\) at the end. Its terminal value is
If the other strategy’s value at the end of the interval is \(A\), break-even requires
Provided \(B>C\), the solution is
The equation is a useful template, not a substitute for a timeline. In particular, it only includes fees at the start and end of the forward interval. A cash flow at another date must first be accumulated or discounted to the comparison date, as appropriate.
Why the Break-Even Equation Works
An annual effective rate \(f\) multiplies an invested amount by \(1+f\) over one year. Under the annual effective compounding model, over \(h\) years the accumulation factor is \((1+f)^h\). If \(C\) is deducted before investment, the invested principal is \(B-C\); if \(K\) is paid at the end, it reduces the terminal amount dollar for dollar. Thus the adjusted strategy has value \((B-C)(1+f)^h-K\) at the comparison date.
Break-even means this value equals the alternative value \(A\). Adding \(K\) to both sides and dividing by \(B-C\) gives \((1+f)^h=(A+K)/(B-C)\). Taking the \(h\)th root and subtracting 1 gives the stated solution. The root is necessary because the unknown is an annual rate while the equation first determines the accumulation factor over the entire interval.
Write both strategies’ values at the same date, usually the end of the forward interval.
Deduct a start-of-interval fee before applying interest. Deduct an end-of-interval fee after accumulation; add an end credit after accumulation.
Find the required interval accumulation factor first, then convert it to the quoted annual effective rate using the interval length.
Substitute the rate into both strategies and confirm that their comparison-date values agree.
Worked Examples
Worked Example: A Fee Deducted When the Forward Investment Begins
At time 0, an investor has $20,000. Strategy A invests the full amount for 3 years at an effective annual rate of 4%. Strategy B invests for 1 year at an effective annual rate of 3.5%, pays a $200 fee at the end of that year, and invests the remainder for 2 more years at an annual effective forward rate \(f\). Find the break-even \(f\). Assume annual compounding, no other cash flows, and no taxes.
Method. Compare the strategies at time 3. First calculate Strategy A’s terminal value:
For Strategy B, the amount at time 1 before the fee is
After paying the fee, the amount invested over the 2-year forward interval is
Equate Strategy B’s time-3 value to Strategy A’s time-3 value:
The break-even annual effective quoted forward rate is 4.758% to three decimal places. The calculation avoids the mistake of applying interest to the $200 fee: the fee is paid at time 1, so only $20,500 is invested from time 1 to time 3.
Check. At the unrounded rate, the forward accumulation factor is approximately 1.0974282927. Therefore Strategy B ends with \(20{,}500(1.0974282927)=\$22{,}497.28\), equal to Strategy A.
Worked Example: A Fee Paid at the End of the Forward Interval
At time 1, two alternatives each begin with $15,000. Strategy A invests for 1 year at an effective annual rate of 5%. Strategy B invests for the same year at annual effective forward rate \(f\), then pays a $150 fee at the end of the year. Find the break-even quote. Assume annual compounding and no other cash flows.
Method. Both values are compared at time 2. Strategy A’s terminal value is
Strategy B accumulates its full $15,000 before paying the fee. Its terminal value is \(15{,}000(1+f)-150\). Set this equal to Strategy A’s value:
The break-even annual effective forward rate is 6.000%. This is higher than Strategy A’s 5% rate because Strategy B must earn enough to cover both the comparison value and its terminal fee.
Check. Strategy B’s terminal value at 6% is \(15{,}000(1.06)-150=\$15{,}900-\$150=\$15{,}750\), matching Strategy A. A common error is to deduct the fee before applying interest. That would model a fee paid at time 1, not the stated fee paid at time 2.
Worked Example: A Terminal Credit Over a Fractional-Year Interval
At time 1, Strategy A is expected to provide $10,500 at time 1.5. Strategy B invests $10,000 from time 1 to time 1.5 at annual effective forward rate \(f\), and receives a separate $100 credit at time 1.5. Find the break-even annual effective quote. Assume annual effective compounding over the half-year interval and no other cash flows.
Method. The forward investment lasts \(h=0.5\) years, so its accumulation factor is \((1+f)^{0.5}\), not \(1+f\). The credit is paid at the comparison date and is added after accumulation. Thus
Subtract the credit and divide by the invested amount:
Square both sides to convert the half-year accumulation factor to the annual effective quote:
The break-even annual effective forward rate is 8.160%. The main trap is treating a half-year interval as a full year and reporting 4% as the annual quote. The 4% is the required half-year accumulation factor’s increase; the annual effective rate that produces this factor over half a year is 8.16%.
Check. The forward investment grows to \(10{,}000(1.0816)^{0.5}=10{,}000(1.04)=\$10{,}400\). Adding the $100 credit gives $10,500.
Cash-Flow Timing and Quote Conventions
A break-even equation is only as reliable as its timing. A fee at the start of the interval reduces the principal that earns interest. A fee at the end reduces the accumulated value. A credit at the end increases it. For an adjustment at an intermediate date, first carry it to the comparison date using the rate that applies over the remaining period. Do not move a cash flow along the timeline without accounting for its accumulation or discounting.
The examples use annual effective quotes. Under this convention, the accumulation factor for an interval of \(h\) years is \((1+f)^h\). If a problem instead states a nominal annual rate \(j^{(m)}\) convertible \(m\) times per year, and the interval contains \(mh\) compounding periods, its accumulation factor is \((1+j^{(m)}/m)^{mh}\). The break-even equation must use the factor for the stated quote, not a factor chosen merely because the interval is expressed in years.
A break-even rate answers a comparison question under the stated cash flows and assumptions. If the fee changes, the timing changes, or the comparison date changes, the break-even quote may change as well. It need not equal a forward rate derived from other market rates when the strategies include fees or other adjustments. To avoid a sign or timing error, calculate the terminal value of each strategy separately before combining them into one equation.
Check Your Understanding
For each problem, assume annual effective compounding unless another convention is specified. Compare values at the stated end date.
- At the start of a 1-year forward interval, Strategy A is worth $12,000 at the end. Strategy B invests $11,000 over the interval and pays a $100 fee at the end. Find the break-even annual effective forward rate. (Answer: 10.000%)
- Strategy A’s end value after 2 years is $22,990. Strategy B has $20,000 at the start of the forward interval and pays a $1,000 fee immediately before investing for 2 years. Find the break-even annual effective rate. (Answer: 10.000%)
- A $10,000 investment runs for 0.5 years. It also receives a $100 credit at the end, and must equal a comparison value of $10,500. Find the break-even annual effective forward rate. (Answer: 8.160%)
- At the start of a 1-year forward interval, Strategy B invests $20,000 and pays a $1,000 fee at the end. Its value must equal Strategy A’s $22,800 terminal value. Find the break-even annual effective forward rate. (Answer: 19.000%)
- At the start of a 2-year forward interval, Strategy B has $20,000, pays a $1,000 fee immediately, and pays a further $190 fee at the end. Strategy A’s terminal value is $22,800. Find the break-even annual effective forward rate. (Answer: 10.000%)