From Discount Factors to a Forward Rate
A spot curve gives discount factors for cash flows at different future dates, all measured from the same valuation date. Those factors also determine the accumulation implied between any two of those dates. The key is to use the discount factor at the start of the interval divided by the factor at its end.
This relationship is useful when a problem supplies discount factors rather than spot-rate quotations. It also gives a direct check on the interval: the resulting accumulation must carry a value at the start date into the equivalent value at the end date. The rate obtained is effective for the interval specified; if the interval lasts more than one year, it is not automatically an annual rate.
Assume the discount factors are positive and are defined consistently from the same valuation date. The formula applies to an investment with no intervening cash flows, using effective compound accumulation over the specified interval. It does not require the forward period to start at time 0.
Why the Ratio Gives the Forward Accumulation
Suppose an amount \(C\) is invested at time \(s\) and accumulates to \(C F_{s,t}\) at time \(t\). The value at time 0 of the amount invested is \(C D(s)\). The value at time 0 of the accumulated amount is \(C F_{s,t}D(t)\). With discount factors and accumulation measured consistently, those two values must agree.
For \(C\ne 0\), divide both sides by \(C D(t)\). This gives the forward factor and, after subtracting 1, the effective rate for the interval:
The order of the ratio follows from the value equation: \(D(s)\) is the value of the amount at the start, and \(D(t)\) is the discount factor applied to the amount at the end. Since the forward factor accumulates an amount, it must be the start-date factor divided by the end-date factor. Reversing that ratio would instead produce a discount factor from \(t\) back to \(s\).
Worked Calculations
Worked Example: A One-Year Forward Rate From Adjacent Discount Factors
At a valuation date, a term structure has discount factors \(D(2)=0.90\) and \(D(3)=0.85\). Find the effective rate for the one-year interval from time 2 to time 3. Check the result by comparing the values at time 0 of two equivalent payments.
Method. Use the ratio of the discount factor at the start of the interval to the factor at its end:
The effective rate is the accumulation factor minus 1:
The interval lasts one year, so this is a one-year effective rate. Rounded to four decimal places as a percentage, it is 5.8824%.
For a check, suppose $1,700 is paid at time 2. Its accumulated amount at time 3 is \(1{,}700(18/17)=1{,}800\). The time-0 value of the first payment is \(1{,}700(0.90)=\$1{,}530\); the time-0 value of the second is \(1{,}800(0.85)=\$1{,}530\). The values agree.
Common mistake avoided: using \(0.85/0.90\) reverses the direction and gives a factor below 1. The forward accumulation from time 2 to time 3 must use \(D(2)/D(3)\), not its reciprocal.
Worked Example: A Forward Rate Over a Three-Year Interval
Suppose \(D(2)=0.810\) and \(D(5)=0.729\). Find the effective rate for the entire interval from time 2 to time 5. Then find the amount at time 5 equivalent to $4,500 at time 2, and check equivalence at time 0.
Method. The interval is three years, but the discount-factor ratio still gives the accumulation factor directly:
Thus the effective rate for the full three-year interval is
The answer is 11.1111% for the entire interval from time 2 to time 5. This is not an annual rate. Converting a multi-year interval rate to an annual rate is a separate calculation.
The equivalent amount at time 5 is \(4{,}500(10/9)=\$5{,}000\). Check their values at time 0: \(4{,}500(0.810)=\$3{,}645\), and \(5{,}000(0.729)=\$3{,}645\). The equality confirms that the calculated factor is consistent with both discount factors.
Common mistake avoided: treating \(11.1111\%\) as an annual rate because the dates are expressed in years. The subtraction from the factor gives the effective rate for the complete interval, regardless of its length.
Worked Example: Finding an Unknown Discount Factor
A discount curve has \(D(1)=0.960\). The effective rate for the interval from time 1 to time 3 is 25% for the full interval. Find \(D(3)\), then find the value at time 0 of $1,000 payable at time 3. Check the implied value of the payment at time 3 when valued at time 1.
Method. First convert the interval rate to its accumulation factor:
The ratio relationship is \(F_{1,3}=D(1)/D(3)\). Rearrange it to solve for the unknown end-date discount factor:
Therefore, the time-0 value of $1,000 payable at time 3 is \(1{,}000(0.768)=\$768\).
To check the interval valuation, the value at time 1 of $1,000 payable at time 3 is its time-0 value divided by \(D(1)\): \(\$768/0.960=\$800\). This also follows by discounting the payment over the interval using the factor \(1/1.25=0.8\): \(\$1{,}000(0.8)=\$800\).
Common mistake avoided: multiplying \(D(1)\) by 1.25 to find \(D(3)\). The forward accumulation factor is \(D(1)/D(3)\), so solving for the end-date factor requires division: \(D(3)=D(1)/F_{1,3}\).
Reading and Checking the Result
A careful calculation begins by identifying the two dates and the valuation date to which both discount factors refer. Then label the start and end explicitly before forming the ratio. This small step prevents a reversed quotient, especially when the dates are not adjacent.
Write down its start date \(s\) and end date \(t\), with \(s<t\).
Confirm that \(D(s)\) and \(D(t)\) are both measured from the same valuation date.
Evaluate \(D(s)/D(t)\) first; subtract 1 only after the factor is found.
Describe the result as the effective rate for that interval. Call it a one-year effective rate only if the interval is one year.
Multiply an amount at \(s\) by \(D(s)\), and its accumulated amount at \(t\) by \(D(t)\). The two values at time 0 should match.
Several errors can be detected before the final answer is reported. A forward accumulation factor must be positive when both discount factors are positive. The rate is that factor minus 1, not the factor itself. Also, discount factors must not be taken from different valuation dates: the ratio has the intended interpretation only when they share a common reference date.
For a one-year interval, the distinction between the interval rate and an annual rate causes no unit issue. For an interval of multiple years, however, the result from the ratio is the effective rate over the whole interval. Labeling it clearly is essential; do not silently change its time basis.
Check Your Understanding
For each problem, use discount factors measured from the same valuation date. Report effective rates to four decimal places as percentages when needed.
- Given \(D(1)=0.94\) and \(D(2)=0.90\), find the forward accumulation factor and effective rate from time 1 to time 2. (Answer: factor \(47/45=1.044444\ldots\); rate \(4.4444\%\).)
- Given \(D(2)=0.88\) and \(D(4)=0.80\), find the effective rate for the entire interval from time 2 to time 4. Is it an annual rate? (Answer: factor \(1.10\); interval rate \(10\%\), not an annual rate.)
- Given \(D(3)=0.75\) and \(D(6)=0.60\), $2,400 at time 3 accumulates to what amount at time 6? Check by comparing the time-0 values. (Answer: $3,000; both time-0 values equal $1,800.)
- Given \(D(1)=0.98\), the effective rate for the interval from time 1 to time 2 is 20%. Find \(D(2)\) and the time-0 value of $500 due at time 2. (Answer: \(D(2)=0.816666\ldots\); present value \(=\$408.33\) to the nearest cent.)
- Given \(D(2)=0.84\) and \(D(4)=0.70\), find the effective rate for the full interval from time 2 to time 4. Then verify that $700 at time 2 is equivalent to the appropriate amount at time 4. (Answer: factor \(1.20\); interval rate \(20\%\); equivalent amount at time 4 \(=\$840\), with both time-0 values equal to \(\$588\).)