Discount Factors Above One and Negative Spot Rates

Why a Discount Factor Can Exceed One

In a bootstrap, a discount factor is solved from observed prices and the bond’s future cash flows. It is tempting to assume that every such factor must be below one because discounting is often associated with reducing a future payment. That assumption is not valid when the corresponding spot rate is negative: under annual effective compounding, a future unit payment can have a present value greater than one.

The important distinction is between a mathematical implication and a data-quality conclusion. A discount factor above one implies a negative spot rate under the stated compounding convention. It does not, by itself, prove that the bootstrap is wrong—or prove that its inputs were handled correctly. The cash flows, their dates, the price convention, and the solution all need to be checked.

The interpretation in this tutorial uses annual effective spot rates and is aligned with the SOA Exam FM (Financial Mathematics) syllabus. It does not assume that a bond’s coupon rate is its spot rate or that the same rate applies at every maturity.

Definition: Let $D(t)$ be the price at time 0 of one unit of currency payable at time $t$ years. Under annual effective spot-rate compounding, the spot rate $i_t$ for maturity $t>0$ is defined by $$ D(t)=(1+i_t)^{-t}, \qquad i_t=D(t)^{-1/t}-1. $$ For $D(t)>0$, $D(t)>1$ if and only if $i_t<0$. A zero rate gives $D(t)=1$, and a positive rate gives $0<D(t)<1$.

Why the Sign Relationship Holds

The discount factor is the present value of a unit payment at maturity. If the annual effective spot rate is $i_t$, one unit invested for $t$ years accumulates to $(1+i_t)^t$. Therefore the amount needed today to produce one unit at time $t$ is its reciprocal. This gives the defining relationship between the spot rate and the discount factor.

D(t)=\frac{1}{(1+i_t)^t} \quad\Longrightarrow\quad i_t=D(t)^{-1/t}-1

For a positive discount factor, $D(t)>1$ means $D(t)^{-1/t}<1$, so $i_t<0$. Conversely, if $i_t<0$ and $1+i_t>0$, then $(1+i_t)^t<1$, making $D(t)>1$. This sign test relies on the annual effective spot-rate convention and a positive accumulation base. If a rate is quoted using another compounding convention, convert using that convention’s relationship rather than applying this formula without adjustment.

A discount factor is a price, not a claim that a payment literally grows in every economic setting. In the deterministic valuation model used here, it is the factor that prices a specified future cash flow at a specified date. A negative spot rate is the rate implied by that factor.

Reading a Bootstrapped Factor

For a bond with known cash flows $C_k$ at times $t_k$, the model prices the bond as the sum of each cash flow multiplied by its discount factor. When the factors for earlier dates are already known, this equation can be solved for the new factor. The solution can exceed one if the observed price, after accounting for earlier cash flows, requires it.

P=\sum_k C_kD(t_k)

Here $P$ must be the price appropriate to the valuation date and cash-flow schedule. The equation does not permit silently substituting a coupon rate for a spot rate, omitting a redemption payment, or mixing clean and dirty prices. Any of those errors can produce a factor—and an implied rate—that does not represent the intended bond.

Worked Example: A Zero-Coupon Bond Priced Above Its Redemption Value

A hypothetical two-year zero-coupon bond pays $100 at maturity and has a price of $102 today. Assuming annual effective spot-rate compounding, find $D(2)$ and the two-year spot rate. Check the price using the resulting factor.

Method. A zero-coupon bond has only one cash flow, so its price is the redemption payment multiplied by the discount factor for its maturity: $$ 102=100D(2). $$

Solving, $$ D(2)=\frac{102}{100}=1.02. $$ Because this factor exceeds one, the two-year annual effective spot rate is negative. Calculate it from the two-year factor, not by treating the factor as a one-year rate: $$ i_2=D(2)^{-1/2}-1 =\frac{1}{\sqrt{1.02}}-1 \approx -0.0098525. $$ Thus $i_2\approx -0.9852\%$, rounded to four decimal places in percentage terms.

Check. The factor reprices the promised payment to $100(1.02)=\$102$, the stated price. The negative sign is also consistent with $D(2)>1$. The common mistake this method avoids is reporting a positive spot rate merely because the factor is called a discount factor.

Coupon Bonds Can Also Produce Factors Above One

A coupon bond’s price reflects all its payments, not just redemption. In a bootstrap, known earlier factors first price the earlier coupons; the remaining price is then assigned to the cash flow at the new maturity. That remaining amount may imply a factor above one. The coupon rate alone does not determine whether the new spot rate is positive or negative.

Worked Example: Bootstrapping a Two-Year Factor Above One

A hypothetical two-year bond has face value $100, pays annual coupons of 5% of face value, and redeems at par. Its coupon-date price is $112.15. The one-year discount factor is $D(1)=1.01$. Find $D(2)$ and the two-year annual effective spot rate, then reprice the bond.

Method. The bond pays $5 at year 1 and $105 at year 2. Apply the cash-flow pricing equation, using the known one-year factor: $$ 112.15=5(1.01)+105D(2). $$

The present value of the first coupon is $$ 5(1.01)=5.05. $$ The amount of price remaining for the year-2 payment is $$ 112.15-5.05=107.10. $$ Therefore, $$ D(2)=\frac{107.10}{105}=1.02. $$ The two-year annual effective spot rate is $$ i_2=\frac{1}{\sqrt{1.02}}-1\approx -0.9852\%. $$ The one-year spot rate is $i_1=1/1.01-1\approx -0.9901\%$. These are different maturities; the two-year rate is calculated from $D(2)$, not by reusing the one-year rate.

Check. Reprice each payment: $$ 5(1.01)+105(1.02)=5.05+107.10=112.15. $$ The price is reproduced exactly. The common mistake this method avoids is assuming that a coupon bond’s discount factors must be below one, or using its 5% coupon rate as its spot rate.

Separate a Valid Negative Rate from an Input Error

A factor above one can be a valid consequence of the price and cash flows under the model, but the sign of the factor should prompt a deliberate input check. For a coupon bond, verify the amount and timing of every coupon, the redemption amount, the settlement date, and whether the quoted price is clean or dirty. Also verify that all prices and cash flows use the same currency units.

In particular, a clean price excludes accrued interest, while a dirty price includes it. A bond cash-flow equation at settlement must use the price on the same basis as the valuation being performed. If a clean quote is given and accrued interest is specified, add that interest to obtain the dirty price before using the equation. A mistaken price convention can change the solved factor and can even change the sign of the implied spot rate.

Worked Example: Checking a Factor When the Quote Is Clean

A hypothetical bond is valued halfway through an annual coupon period. Its next cash flow is $6 in 0.5 years, and its final cash flow is $106 in 1.5 years. The known factor is $D(0.5)=1.01$. The quoted clean price is $110.00, and accrued interest is $3.00. Find $D(1.5)$ and the annual effective spot rate for 1.5 years. Then show what using the clean price directly would do.

Method. First convert to the dirty price used in the cash-flow equation: $$ P_{\text{dirty}}=110.00+3.00=113.00. $$ The equation is $$ 113.00=6(1.01)+106D(1.5). $$

The known near-term cash flow has present value $$ 6(1.01)=6.06. $$ The residual price is $113.00-6.06=106.94$, so $$ D(1.5)=\frac{106.94}{106}\approx 1.0088679. $$ Convert this factor using its actual maturity: $$ i_{1.5}=D(1.5)^{-1/1.5}-1 \approx -0.005869, $$ or approximately $-0.5869\%$, rounded to four decimal places in percentage terms.

Price-convention check. If the clean price were incorrectly used as though it were dirty, the calculation would instead give $$ D(1.5)=\frac{110.00-6.06}{106} =\frac{103.94}{106} \approx 0.9805660. $$ That erroneous factor is below one and implies a positive annual effective spot rate of approximately $1.317\%$. The factor changes from above one to below one solely because accrued interest was omitted. The correct factor reprices the dirty price: $$ 6(1.01)+106(1.0088679)\approx 6.06+106.94=113.00. $$ The common mistake this method avoids is treating a clean quote as the full settlement-date price.

Common Mistakes and a Practical Check

The sign of a factor is informative, but it is not a substitute for checking the bootstrap equation. In particular, do not infer a negative rate from a bond’s price being above par alone. A coupon bond’s price depends on the amounts and timing of all its payments. The relevant sign conclusion comes from the solved discount factor for a particular maturity.

Using the wrong maturity in the conversion. For maturity $t$, use $i_t=D(t)^{-1/t}-1$. A factor at 1.5 years is not converted with an exponent of $-1$.
Reversing the sign interpretation. Under annual effective compounding, $D(t)>1$ corresponds to a negative spot rate, not a positive one.
Confusing coupon and spot rates. A coupon rate determines scheduled coupon amounts; it is not the spot rate implied by a particular discount factor.
Ignoring price conventions or timing. Use a price consistent with settlement and with the cash flows being discounted. For a clean quote with accrued interest, account for the interest before using a dirty-price equation.
Accepting a result without repricing. Substitute solved factors back into the original cash-flow equation. A mismatch indicates an arithmetic, cash-flow, timing, or price-input error.

A useful review sequence is: write the dated cash flows, write the price equation, isolate the unknown factor, convert that factor using its maturity and stated compounding convention, and reprice. A factor above one is not automatically an error; a factor that fails to reproduce the correctly specified price is.

Check Your Understanding

For each problem, use annual effective spot-rate compounding unless stated otherwise. Check both the factor and the rate sign.

A hypothetical three-year zero-coupon bond pays $100 and costs $103.0301. Find $D(3)$ and the annual effective three-year spot rate. (Answer: $D(3)=1.030301$, $i_3=-0.9901\%$.)
A hypothetical two-year bond pays annual coupons of $4 on face value $100 and redeems at par. Its coupon-date price is $109.06, and $D(1)=1.005$. Find $D(2)$ and $i_2$. (Answer: $D(2)=1.01$, $i_2\approx-0.4963\%$.)
A two-year zero-coupon bond pays $100 and has price $99. Find its annual effective spot rate. Is the rate negative? (Answer: $D(2)=0.99$, $i_2=0.99^{-1/2}-1\approx0.5038\%$, so it is positive.)
A hypothetical two-year bond pays an $8 coupon at year 1 and $108 at year 2. Its price is $120.16 and $D(1)=0.98$. Find $D(2)$, $i_2$, and verify the price. (Answer: $D(2)=1.04$, $i_2\approx-1.9419\%$; $8(0.98)+108(1.04)=120.16$.)
A hypothetical bond valued halfway through an annual coupon period pays $5 in 0.5 years and $105 in 1.5 years. The known factor is $D(0.5)=1.02$; the clean price is $107.50 and accrued interest is $2.50. Find the correct $D(1.5)$ and spot rate using the dirty price. (Answer: Dirty price $=\$110.00$, $D(1.5)=(110-5(1.02))/105\approx0.9990476$, and $i_{1.5}\approx0.0635\%$, positive.)

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