Positive, Zero, and Negative Spot Rates

Why the Sign of a Spot Rate Matters

A discount factor converts a payment at a specified future date into its value at the valuation date. With a positive spot rate, a future dollar is worth less than a dollar now. With a zero rate, the two amounts are equal under the stated model. With a negative rate, the future dollar can be worth more than a dollar now. These outcomes follow directly from the spot-rate convention; they are not errors in the calculation.

The sign of a spot rate also does not, by itself, tell you whether discount factors decrease as maturity increases. Each spot rate describes a different maturity. A longer-maturity spot rate can be low enough that the discount factor for that longer maturity is greater than the discount factor for a shorter one, even when both spot rates are positive.

This tutorial uses continuously compounded spot rates, consistent with the earlier material on continuous spot rates and discount factors. Rate and maturity units must match: here, $t$ is measured in years and the annual spot rate is applied over $t$ years. This topic is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: For a maturity $t>0$, let $\delta_t$ be the continuously compounded spot rate from the valuation date to time $t$. The discount factor for that maturity is $D(t)=e^{-\delta_t t}$. For a zero-coupon payment of $F$ at time $t$, with no intervening payments, its value at the valuation date is $P(0,t)=F D(t)=F e^{-\delta_t t}$. The rate $\delta_t$ is specific to maturity $t$; it need not be the same at other maturities.

What Positive, Zero, and Negative Rates Imply

For a positive maturity $t$, the sign of $\delta_t$ determines whether the discount factor is below, equal to, or above 1. Since $e^x$ is positive and increases with $x$, changing the sign of the exponent $-\delta_t t$ gives the following results.

\begin{aligned} \delta_t>0 &\quad\Longrightarrow\quad -\delta_t t<0 \quad\Longrightarrow\quad 0<D(t)<1,\\ \delta_t=0 &\quad\Longrightarrow\quad D(t)=1,\\ \delta_t<0 &\quad\Longrightarrow\quad -\delta_t t>0 \quad\Longrightarrow\quad D(t)>1. \end{aligned}

Because $P(0,t)=F D(t)$, the same comparisons apply to the price of a zero-coupon payment relative to its redemption value $F$. A positive rate gives a price below redemption, a zero rate gives a price equal to redemption, and a negative rate gives a price above redemption. These comparisons assume the same valuation date, maturity, payment currency, and zero-coupon cash flow. They do not compare investments with different payment dates or other cash flows.

Spot-rate sign	Discount factor	Zero-coupon price relative to redemption
Positive	Between 0 and 1	Below redemption
Zero	Equal to 1	Equal to redemption
Negative	Greater than 1	Above redemption

The relationship can be justified by imagining an amount invested from time 0 to time $t$ at a constant continuously compounded rate $\delta_t$. One unit grows to $e^{\delta_t t}$. Therefore, the amount required at time 0 to produce one unit at time $t$ is the reciprocal of that accumulation factor:

D(t)=\frac{1}{e^{\delta_t t}}=e^{-\delta_t t}.

Multiplying that present value per unit by a redemption amount $F$ gives $P(0,t)=F e^{-\delta_t t}$. This justification uses the continuous-compounding convention. If a rate is quoted using a different compounding convention, convert it or use its matching discount formula; do not insert it directly into this exponential formula.

Worked Calculations

Worked Example: A Positive Spot Rate

A hypothetical zero-coupon security pays $12,000 in 4 years. Its 4-year continuously compounded spot rate is 3% per year. Find the discount factor and the security’s price. State whether the price is above or below redemption.

Method and assumptions. The payment is a single known amount at year 4, and the stated rate is continuously compounded and applies to that maturity. Use $D(4)=e^{-\delta_4(4)}$, then multiply by the redemption value.

The rate in decimal form is $\delta_4=0.03$. The exponent is $ -\delta_4 t=-(0.03)(4)=-0.12$.

D(4)=e^{-0.12}\approx 0.8869204.

The price is $P(0,4)=12{,}000(0.8869204)\approx 10{,}643.0452$. Rounded to the nearest cent, the price is $10,643.05. The discount factor is below 1, so the price is below the $12,000 redemption value.

Check. The positive rate produces an exponent below zero, so the discount factor must be less than 1. The calculated price is indeed less than redemption. The common mistake to avoid is treating 3% as an effective annual rate and using a different compounding formula.

Worked Example: Negative and Zero Spot Rates

A hypothetical portfolio contains two zero-coupon securities. One pays $8,000 in 2 years and has a 2-year continuously compounded spot rate of $-1\%$ per year. The other pays $2,500 in 3 years and has a 3-year spot rate of 0%. Find both prices and their combined present value.

Method and assumptions. Price each payment using its own maturity-specific rate. The first rate is negative, while the second rate is zero. Both rates use continuous compounding and time is measured in years.

For the 2-year payment, $\delta_2=-0.01$ and $t=2$. Thus the exponent is $-\delta_2t=-(-0.01)(2)=0.02$, and

D(2)=e^{0.02}\approx 1.0202013.

The first security’s price is $8{,}000(1.0202013)\approx \$8{,}161.6104$, or $8,161.61 to the nearest cent. Its price exceeds redemption because its discount factor is greater than 1.

For the 3-year payment, $\delta_3=0$, so $-\delta_3t=-(0)(3)=0$. Therefore $D(3)=e^0=1$, and its price is $2{,}500(1)=$ $2,500.00.

The combined present value is $\$8{,}161.61+\$2{,}500.00=$ $10,661.61. The result is consistent with the signs of the rates: the negative-rate security is priced above redemption, and the zero-rate security at redemption.

Common mistake to avoid. A negative rate does not mean a negative discount factor. For the first payment, using $e^{-0.02}$ would reverse the sign incorrectly. The exponent is $-\delta_2t$; because $\delta_2$ itself is negative, that exponent is positive.

Why Discount Factors Need Not Decline with Maturity

A discount factor’s value depends on the product $\delta_t t$, not just on the sign or size of $\delta_t$. Consequently, knowing that the spot rate is positive at two maturities does not establish which discount factor is larger. For example, the longer-maturity rate may be sufficiently lower that its product with the longer maturity is smaller.

This is different from saying that a positive spot rate gives a discount factor above 1: it does not. At each maturity with a positive rate, the factor remains below 1. The point is that two numbers can both be below 1 while the one associated with the longer maturity is larger.

Worked Example: Positive Rates but a Larger Longer-Maturity Factor

A hypothetical liability portfolio has a $4,000 payment at year 1 and a $6,000 payment at year 3. The continuously compounded spot rates are 5% for year 1 and 1.5% for year 3. Calculate the two discount factors, compare them, and find the portfolio’s present value.

Method and assumptions. Apply the spot rate corresponding to each payment’s maturity. Do not apply one rate to both cash flows.

At year 1, $\delta_1=0.05$, so the exponent is $-(0.05)(1)=-0.05$. Hence $D(1)=e^{-0.05}\approx0.9512294$. At year 3, $\delta_3=0.015$, so the exponent is $-(0.015)(3)=-0.045$. Hence $D(3)=e^{-0.045}\approx0.9559975$.

Both discount factors are below 1 because both spot rates are positive. However, $0.9559975>0.9512294$, so the year-3 discount factor is larger. The longer-maturity spot rate is low enough that $\delta_3(3)=0.045$ is less than $\delta_1(1)=0.05$.

The present value of the year-1 payment is $4{,}000(0.9512294)\approx \$3{,}804.9177$. The present value of the year-3 payment is $6{,}000(0.9559975)\approx \$5{,}735.9849$. Adding the unrounded values gives $\$3{,}804.9177+\$5{,}735.9849=\$9{,}540.9026$, which rounds to $9,540.90.

Check. Each payment is below its own nominal amount because its spot rate is positive. The total present value is below the total nominal payments of $10,000, as expected. The fact that the 3-year discount factor is larger than the 1-year factor does not contradict either check.

Common Exam Traps

Using the wrong sign in the exponent. The formula is $e^{-\delta_t t}$. With a negative spot rate, the exponent becomes positive. For example, if $\delta_2=-0.01$, the factor is $e^{0.02}$, not $e^{-0.02}$.
Assuming every discount factor must be below 1. That is true only when the relevant spot rate is positive. A zero rate gives a factor of 1; a negative rate gives a factor above 1.
Assuming positive rates force discount factors to decline with maturity. A positive rate makes each maturity’s factor less than 1, but factors at different maturities depend on different rates and different time periods.
Using one maturity’s rate for every payment. For a set of dated cash flows, use each payment’s own spot rate and maturity.
Mixing rate conventions. The exponential formula here uses a continuously compounded spot rate. A nominal or effective rate requires the corresponding convention and formula.

Key takeaway: For a continuously compounded spot rate, $D(t)=e^{-\delta_t t}$. A positive rate gives a factor below 1, a zero rate gives 1, and a negative rate gives a factor above 1. Compare maturities using the full product $\delta_t t$, not the rate sign alone.

Check Your Understanding

For each problem, use continuously compounded annual spot rates and round money to the nearest cent where needed.

A hypothetical zero-coupon payment of $2,400 is due in 2 years. Its 2-year spot rate is 0%. Find its discount factor and price. (Answer: discount factor $1$; price $2,400.00)
A hypothetical zero-coupon security pays $3,000 in 1.5 years. Its maturity-specific spot rate is $-2\%$ per year. Find the price and state how it compares with redemption. (Answer: $D(1.5)=e^{0.03}\approx1.0304545$; price $3,091.36, above redemption)
A $10,000 zero-coupon payment is due in 4 years, with a 4-year spot rate of 3.5%. Find its price. (Answer: $10{,}000e^{-0.14}\approx$$8,693.58)
The 1-year spot rate is 5%, and the 4-year spot rate is 1%. Calculate and compare the two discount factors. Are both below 1? (Answer: $D(1)=e^{-0.05}\approx0.9512$, $D(4)=e^{-0.04}\approx0.9608$; the 4-year factor is larger, and both are below 1)
A hypothetical portfolio pays $6,000 in 2 years at a 2-year spot rate of $-1\%$, and $4,000 in 5 years at a 5-year spot rate of 2%. Find its present value. (Answer: $6{,}000e^{0.02}+4{,}000e^{-0.10}\approx$$9,740.56)

What You'll Learn