From a Zero-Coupon Price to a Spot Rate
A zero-coupon bond has one promised payment: its redemption value at maturity. Because there are no interim coupons to separate or discount, its price directly records the present value of that single future payment. The key is to interpret the price relative to the redemption value, not as a rate by itself.
This tutorial uses annual effective spot rates. The maturity \(t\) is measured in years from the valuation date, and the bond’s price is the amount paid at that date for the stated future redemption. A market quotation such as “92.50 per 100 of face value” is a price of 92.50% of the stated face value, not a rate of 92.50%.
The price-to-discount-factor conversion requires the price and redemption value to refer to the same bond and currency. It assumes the bond has no interim payments and that the quoted price is the price at the valuation date for the stated maturity. The spot rate calculated from that price is specific to that maturity; it should not automatically be used to discount payments at other maturities.
Why the Price Ratio Is the Discount Factor
Under annual effective compounding, a present amount \(P(0,t)\) accumulates for \(t\) years at the maturity-specific spot rate \(i_t\). For the bond to pay \(R\) at maturity, that accumulated amount must equal \(R\). This relationship gives both the discount-factor conversion and the spot-rate conversion.
Divide both sides by \(R(1+i_t)^t\). The ratio of price to redemption value is the amount today per unit of redemption value, so it is the discount factor. Rearranging the same equation gives the spot rate.
The first equality is the main interpretation: a zero-coupon price per unit of redemption is a discount factor. The second formula converts that factor into an annual effective spot rate for a maturity of \(t\) years. If the price is quoted per 100 of face value, first divide the quoted price by 100 to obtain the discount factor. If the bond’s redemption value differs from its face value, use the actual redemption value in the ratio.
Worked Example: Converting a Quote per 100 of Face Value
A hypothetical zero-coupon bond has a face value and redemption value of $100 per bond. It matures in 2 years and is quoted at 92.50 per 100 of face value. Find the discount factor and the annual effective 2-year spot rate. Check the result by reaccumulating the price.
Method. Divide the price per 100 by 100 to get the discount factor, then convert that factor to a rate using the 2-year maturity.
The redemption value is \(R=100\), and the price is \(P(0,2)=92.50\). Thus,
The annual effective spot rate is
As a percentage, the rate is approximately 3.9750% per year, rounded to four decimal places. The maturity exponent is 2 because the redemption occurs in 2 years; the calculated rate is annual effective, not a 2-year total return.
Check. Using the unrounded rate, \((1+i_2)^2=1/0.925\), so \(92.50(1+i_2)^2=92.50/0.925=100\). The price therefore accumulates to the stated redemption value.
Common mistake avoided: treating 92.50 as the discount factor or as a rate. The price is quoted per 100, so the discount factor is \(92.50/100=0.925\).
Reading Several Maturities
A set of zero-coupon bonds with different maturities supplies a set of discount factors. Each price must be paired with its own redemption value and maturity. The resulting rates may differ: the 1-year spot rate describes discounting to year 1, while the 3-year spot rate describes discounting to year 3.
| Maturity | Price per 100 of redemption | Discount factor | Annual effective spot rate |
|---|---|---|---|
| 1 year | 96.153846 | 0.96153846 | Approximately 4.0000% |
| 2 years | 90.702948 | 0.90702948 | Approximately 5.0000% |
| 3 years | 83.961925 | 0.83961925 | Approximately 6.0000% |
Each row is interpreted separately. For example, the 2-year discount factor is \(90.702948/100=0.90702948\). Its annual effective spot rate is approximately \((1/0.90702948)^{1/2}-1=0.0500000\), or 5.0000%. It is not appropriate to use the 1-year rate for this 2-year payment merely because both bonds are priced on the same date.
Worked Example: Reading a Maturity-Specific Set of Prices
A hypothetical market quotes three zero-coupon bonds, each with redemption value $1,000. Their prices are $961.53846 for maturity in 1 year, $907.02948 for maturity in 2 years, and $839.61925 for maturity in 3 years. Find the discount factor and annual effective spot rate for each maturity. State which rate applies to a payment due in 2 years.
Method. For each bond, divide its price by $1,000. Then use that bond’s own maturity in the rate conversion.
For the 1-year bond,
Its annual effective 1-year spot rate is approximately 4.0000%.
For the 2-year bond,
Its annual effective 2-year spot rate is approximately 5.0000%.
For the 3-year bond,
Its annual effective 3-year spot rate is approximately 6.0000%. The prices are quoted to five decimal places, so the rates computed from them are approximate. A payment due in 2 years is associated with the 2-year spot rate, approximately 5.0000%, rather than the 1-year or 3-year rate.
Check. The corresponding accumulation factors are approximately \(1.04\), \(1.05^2=1.1025\), and \(1.06^3=1.191016\). Multiplying each price by its matching accumulation factor gives approximately $1,000. This confirms that each price, maturity, and rate has been paired consistently.
Common mistake avoided: using one rate for all three bonds. A price at one maturity identifies a discount factor and spot rate for that maturity; the three rates need not be equal.
Converting a Dollar Price and Recognizing Negative Rates
A quote per 100 is a convenient market convention, but a bond may also be described by its total dollar price and redemption value. The conversion is unchanged: divide the dollar price by the dollar redemption value. If the price is above redemption, the discount factor exceeds 1 and the annual effective spot rate is negative. This is mathematically consistent: a negative rate means the present value of the future payment is greater than that payment.
Worked Example: A Price Quoted Above Redemption
A hypothetical zero-coupon bond will pay $250,000 in 2 years. Its current price is $251,250. Find its discount factor and annual effective 2-year spot rate, and check the result. Explain the sign of the rate.
Method. Form the price-to-redemption ratio using the dollar amounts, then convert the discount factor using the 2-year maturity.
The price is greater than the redemption value, so the discount factor is greater than 1. The spot rate is
Thus the annual effective 2-year spot rate is approximately -0.2491%, rounded to four decimal places as a percentage. The negative sign is appropriate: under this rate, the amount invested today grows slightly less than its initial value over the 2-year period.
Check. By construction, \((1+i_2)^2=1/1.005\). Therefore \(251{,}250(1+i_2)^2=251{,}250/1.005=250{,}000\), as required.
Common mistake avoided: assuming every zero-coupon bond must trade below redemption and forcing the rate to be positive. The price-to-redemption ratio determines the discount factor; a ratio above 1 produces a negative spot rate.
Quotation and Calculation Checks
The most reliable approach is to keep three quantities distinct: the quoted price, the discount factor, and the spot rate. A quote may be expressed per 100 of face value, while the discount factor is a ratio per unit of redemption value. The spot rate then depends on that discount factor and the time to maturity.
Use the amount actually paid at maturity and measure the time from the valuation date in years.
For a quote per 100, divide by 100 if redemption is 100. For a dollar price, divide by the dollar redemption value.
The price divided by redemption is the discount factor for that specific maturity.
Take the reciprocal of the discount factor to the power \(1/t\), then subtract 1.
The price multiplied by the maturity-specific accumulation factor should equal the redemption value, subject to the precision of the quoted price.
The check is especially useful when a quotation has been converted from “per 100” or when the maturity is not one year. A price of 90 per 100 corresponds to a discount factor of 0.90, not a 10% annual rate by itself. The annual effective rate depends on the maturity: the same discount factor over 1 year and over 2 years implies different annual effective rates.
A zero-coupon price can identify a discount factor directly because the bond has only one payment date. For a bond with several coupon payments, its total price combines the present values of payments at several dates; the total price alone is not the discount factor for any one maturity. The distinction matters whenever prices are used to interpret a term structure.
Check Your Understanding
For each problem, use the stated maturity and redemption value. Rates may be reported to four decimal places as percentages.
- A zero-coupon bond with redemption value $100 is quoted at 94 per 100 and matures in 2 years. Find its discount factor and annual effective spot rate. (Answer: discount factor \(0.94\); rate \((1/0.94)^{1/2}-1\approx 3.1421\%\).)
- A hypothetical 3-year zero-coupon bond has redemption value $1,000 and price $850. Find the discount factor and annual effective spot rate, then state the amount to which the price accumulates at that rate. (Answer: discount factor \(0.85\); rate \((1/0.85)^{1/3}-1\approx 5.5667\%\); accumulated amount $1,000.)
- A zero-coupon bond with redemption value $5,000 is quoted at 92.40 per 100 and matures in 4 years. Convert the quote to a dollar price, then find the discount factor and annual effective spot rate. (Answer: price $4,620; discount factor \(0.924\); rate \((1/0.924)^{1/4}-1\approx 1.9957\%\).)
- Two zero-coupon bonds each have redemption value $1,000. The 1-year bond costs $970 and the 2-year bond costs $900. Find both discount factors and spot rates, and identify the rate associated with a payment due in 2 years. (Answer: discount factors \(0.97\) and \(0.90\); rates \(1/0.97-1\approx 3.0928\%\) and \((1/0.90)^{1/2}-1\approx 5.4093\%\); use the 2-year rate, approximately \(5.4093\%\).)
- A 2-year zero-coupon bond will redeem for $100,000 and has a current price of $101,500. Find its discount factor and annual effective spot rate, and explain whether the rate is positive or negative. (Answer: discount factor \(1.015\); rate \((1/1.015)^{1/2}-1\approx -0.7417\%\), which is negative because price exceeds redemption.)