Why a Bootstrap Output Can Look Implausible
A sequential bootstrap can produce a complete set of discount factors and still leave you with a result that deserves investigation. A factor may be negative, a spot rate may be negative, or spot rates may rise and fall across maturities. These observations do not all have the same meaning: a negative discount factor is a serious warning under ordinary positive discounting, while a negative or nonmonotonic spot rate is not automatically an error.
The aim of diagnosis is not to force the curve into a preferred shape. Instead, check the solved factors, convert them consistently to spot rates, and reprice the input bonds. Then examine the quote and cash-flow data if a result remains suspect. This method is aligned with the SOA Exam FM (Financial Mathematics) syllabus.
A discount factor greater than one corresponds to a negative spot rate under this convention. Spot rates are averages over the period from time zero to maturity, so their movement across maturities need not be monotonic. In particular, neither a steadily decreasing discount-factor sequence nor an upward-sloping spot-rate curve is a universal requirement. Always interpret the factors and rates under the stated compounding convention.
A Diagnostic Framework
For a bond with price \(P\), cash flow \(C_j\) at time \(t_j\), and discount factor \(D(t_j)\), the pricing equation is
When the final cash flow at a new maturity \(t_n\) is the only cash flow whose discount factor is unknown, rearranging this equation gives
This identity explains one useful warning sign: if the price is less than the present value of the earlier cash flows, the numerator is negative and the solved final factor is negative. The algebra may be correct, but the input data, cash-flow setup, or assumed pricing framework then needs investigation. A zero residual alone is not enough to establish that a result is sensible, because the final factor was chosen to make the pricing equation hold.
Confirm that the solved discount factor is positive and that the correct cash flows and times were included. Under ordinary deterministic discounting, a nonpositive factor is not a valid input to the spot-rate conversion above.
Calculate each spot rate from its own maturity: \(z_t=D(t)^{-1/t}-1\). Do not mistake a factor above one for an error; it implies a negative spot rate under annual effective compounding.
Apply the factors to each bond's dated cash flows and compare the resulting present value with its quoted price. Record the residual and its sign.
Review the quote, units, settlement and coupon dates, redemption amount, and equation. Do not alter the curve merely to make its rates monotonic.
Worked Example: A Negative Factor Hidden by an Exact Repricing
Worked Example: A Negative Factor Hidden by an Exact Repricing
A bootstrap has already established \(D(1)=0.98\) and \(D(2)=0.94\). A three-year bond with face value $100 pays an annual coupon of $10 and is reported at a price of $18. Find the implied \(D(3)\), test the result, and identify what it suggests.
Method. Write the bond's cash flows at times 1, 2, and 3. They are $10, $10, and $110, respectively. Solve the pricing equation for the last factor.
The present value of the first coupon is
The present value of the second coupon is
Together, the earlier cash flows have present value \(9.80+9.40=19.20\). Therefore,
Substitution confirms the reported price algebraically:
The residual is approximately zero, but \(D(3)<0\). The standard spot-rate conversion requires \(D(3)>0\), so this result cannot be converted into a real annual effective spot rate. More fundamentally, the earlier coupons alone have a present value of $19.20, already exceeding the reported price of $18.00.
Conclusion and trap avoided. An exact repricing is not proof that a bootstrap output is valid: the solved factor can force the equation to fit. Here, investigate a misread price, a missing or mistimed cash flow, or incompatible input data. Do not describe this negative factor as merely a negative interest rate; a negative spot rate corresponds to a positive factor greater than one, not a negative factor. For comparison, if the price were $118.20 instead, then \(D(3)=(118.20-19.20)/110=0.90\), a positive factor that can be converted to a spot rate.
Negative Rates and Factors Above One
A factor above one can be a coherent output when the associated spot rate is negative. This is a feature of the stated rates and prices, not by itself evidence of a bad bootstrap. The example below checks both the factor-to-rate conversion and the pricing of a coupon bond.
Worked Example: Valid Negative Spot Rates and Increasing Factors
One-year and two-year zero-coupon bonds each promise $100 at maturity. Their observed prices are $101.00 and $102.01, respectively. Under annual effective spot-rate compounding, find the discount factors and spot rates, then reprice a two-year bond that pays $4 at time 1 and $104 at time 2.
Step 1: Obtain the discount factors. A zero-coupon bond's price is its redemption payment times the discount factor. Thus,
Step 2: Convert each factor to its spot rate.
For the two-year rate,
The factors exceed one and increase from \(1.01\) to \(1.0201\), while both annual effective spot rates are negative. The two factors are consistent with the same annual effective rate because \(1.0201=(1.01)^2\).
Step 3: Reprice the coupon bond. Its cash flows are $4 at time 1 and $104 at time 2, so
The calculated price is $110.1304, or $110.13 to the nearest cent. The zero-coupon prices also reprice directly: \(100(1.01)=101.00\) and \(100(1.0201)=102.01\). The result is unusual only if one incorrectly assumes discount factors must be below one; no such assumption is being made here.
Nonmonotonic Spot Rates Need Not Be Errors
A spot rate is the single annual effective rate that equates a maturity's discount factor to one constant rate over the entire term. It is not the rate for only the final year. Consequently, one spot rate may be below the preceding rate even when all factors are positive. The next example uses zero-coupon prices to make the distinction transparent and then checks a coupon-bond quote against the same factors.
Worked Example: A Nonmonotonic Spot Curve That Reprices
Suppose zero-coupon bonds paying $100 at years 1, 2, and 3 have prices $98, $97, and $94. Under annual effective spot-rate compounding, calculate the factors and spot rates. Then check a three-year, 5% annual-coupon bond with face value $100 and quoted price $108.45.
Step 1: Calculate the discount factors. Divide each price by its $100 maturity payment:
Step 2: Calculate the annual effective spot rates.
For year 2,
For year 3,
The spot rates first fall, from about 2.0408% to 1.5346%, and then rise to about 2.0839%. That nonmonotonic pattern is not, by itself, an error.
Step 3: Reprice the coupon bond. The bond pays $5 at year 1, $5 at year 2, and $105 at year 3. Its value from the factors is
The calculated price exactly matches the quoted $108.45, so the residual is \(108.45-108.45=0.00\). The zero-coupon prices also match because \(100(0.98)=98\), \(100(0.97)=97\), and \(100(0.94)=94\). These checks support the consistency of the stated prices and factors; they do not establish that spot rates must have this shape in every market or model.
Reading Repricing Residuals Carefully
For each input bond, calculate the model price from the stated dated cash flows and factors, then subtract it from the quoted price. A residual close to zero indicates that the factors reproduce that quote, subject to the precision used. A material nonzero residual indicates that something does not match: possible causes include an incorrect factor, a coupon or redemption amount entered incorrectly, a cash flow assigned to the wrong date, or a quote that uses a different price convention from the one assumed.
There is an important limitation. In a sequential bootstrap, the newest factor is often solved directly from the bond being checked. Repricing that same bond with the solved factor will then give a zero residual by construction. The check is valuable for finding arithmetic or transcription mistakes, but it is not an independent test of whether the quote and model assumptions are reasonable. Where available, also verify the earlier factors against the bonds that established them, and examine whether the new quote implies an admissible factor.
A useful diagnostic is to compare the price with the present value of known earlier cash flows before dividing by the final cash flow. If the remaining price is small but positive, the implied factor may be positive yet extreme. If it is zero, the implied final factor is zero. If it is negative, the implied factor is negative. Each result calls for a different response; do not silently replace it with a preferred curve value.
Common Mistakes and Exam Traps
- Assuming every discount factor must be below one. A factor above one implies a negative spot rate under annual effective compounding, but does not alone prove an error.
- Assuming spot rates must increase or decrease with maturity. Spot rates summarize different time horizons. They may be nonmonotonic, so investigate inputs and repricing rather than imposing a shape.
- Calling a negative factor a negative interest rate. A negative spot rate requires a positive factor above one under the stated conversion. A negative factor cannot be converted to a real spot rate using \(D(t)^{-1/t}-1\).
- Treating a zero residual as a complete validation. A solved factor can make the bond equation hold even when that factor is negative or otherwise suspicious.
- Comparing prices without matching cash-flow timing. Verify coupon dates and redemption timing before attributing a discrepancy to the curve. A correctly priced cash flow at the wrong time is still the wrong equation.
- Rounding too early. Keep sufficient precision in factors while repricing, then report money to the cent if appropriate. Early rounding can create a small residual that is only a calculation artifact.
Check Your Understanding
For each problem, use the stated annual effective spot-rate convention and distinguish an unusual rate pattern from an inconsistency.
- A one-year zero-coupon bond pays $100 and costs $99.50. Find its discount factor and spot rate. (Answer: \(D(1)=0.995\); \(z_1\approx0.5025\%\).)
- Two zero-coupon bonds paying $100 at years 1 and 2 cost $102.00 and $104.04, respectively. Find both factors and spot rates. What does the factor pattern indicate? (Answer: \(D(1)=1.02\), \(D(2)=1.0404\); \(z_1=z_2\approx-1.9608\%\); factors above one are consistent with negative spot rates.)
- Given \(D(1)=0.97\) and \(D(2)=0.94\), find the model price of a two-year bond with face value $100 and annual coupon $6. Its cash flows are $6 at year 1 and $106 at year 2. (Answer: \(6(0.97)+106(0.94)=5.82+99.64=\$105.46\).)
- A three-year bond with face value $100 pays annual coupons of $4 and is quoted at $100. The known factors are \(D(1)=0.99\) and \(D(2)=0.96\). Solve for \(D(3)\), using the final cash flow of $104, and state whether the factor is positive. (Answer: \(D(3)=[100-4(0.99)-4(0.96)]/104=92.20/104\approx0.88654\), which is positive.)
- A three-year bond with face value $100 pays annual coupons of $8 and is quoted at $15. The known factors are \(D(1)=0.99\) and \(D(2)=0.98\). Solve for \(D(3)\) and explain the diagnostic. (Answer: \(D(3)=[15-8(0.99)-8(0.98)]/108=(15-15.76)/108\approx-0.007037\); the negative factor cannot be converted to a real spot rate and signals a quote, cash-flow, or setup inconsistency to investigate.)