Why Face Value and Maturity Matter
A zero-coupon bond promises a single payment at maturity and makes no intervening coupon payments. Its price therefore depends directly on the amount promised and on how far in the future that amount will be received. Comparing bonds at the same required yield makes these effects especially clear: changing face value scales price proportionally, while changing maturity changes the amount of discounting.
The comparisons in this tutorial assume a common effective annual required yield and annual compounding. The yield is the return investors require for the bond’s promised cash flow, and the price is its present value under that yield. These comparisons do not by themselves account for taxes, transaction costs, or the possibility that the promised payment will not be made.
How Price Scales with Face Value
Hold the yield and maturity fixed. The discount factor \(v^n\) is then the same for every bond being compared, so price is face value multiplied by a constant. If one bond has \(k\) times the face value of another, its price is also \(k\) times as large. This result does not require calculating both prices separately.
Worked Example: Scaling Price with Face Value
Two hypothetical zero-coupon bonds mature in two years and have a common effective annual required yield of 4%. One has face value $600 and the other has face value $1,500. Find both prices and the ratio of the larger price to the smaller price. Assume each bond pays its stated face value at maturity and makes no other payments.
Method. Apply the zero-coupon formula with \(n=2\). The two-year accumulation factor is $$ (1.04)^2=1.0816, $$ so the two-year discount factor is $$ v^2=\frac{1}{1.0816}=0.924556213\ldots. $$
For the $600 bond, $$ P_1=\frac{600}{1.0816}=554.7337\ldots. $$ Rounded to the nearest cent, its price is $554.73.
For the $1,500 bond, $$ P_2=\frac{1,500}{1.0816}=1,386.8343\ldots. $$ Rounded to the nearest cent, its price is $1,386.83.
The price ratio, larger bond to smaller bond, is $$ \frac{P_2}{P_1}=\frac{1,500}{600}=2.5. $$ The ratio of the calculated prices is 2.5 before rounding. As a check, the larger face value is 2.5 times the smaller one, and the common two-year discount factor leaves that ratio unchanged. The common mistake avoided here is rounding the discount factor or an intermediate price too early.
How Price Changes with Maturity
Now hold face value and required yield fixed, and compare two maturities \(n_1\) and \(n_2\). Dividing the two price formulas cancels face value: $$ \frac{P(F,n_2)}{P(F,n_1)}=(1+i)^{-(n_2-n_1)}. $$ The ratio depends on the difference in maturities, not on the face value. If \(i>0\) and \(n_2>n_1\), then the ratio is less than 1, so the longer-maturity bond has the lower price. At a zero yield, the prices are equal. At a negative yield, the price of a fixed promised payment increases with maturity under this formula.
The percentage decline from the shorter-maturity price to the longer-maturity price is $$ \frac{P(F,n_1)-P(F,n_2)}{P(F,n_1)} =1-\frac{P(F,n_2)}{P(F,n_1)} =1-(1+i)^{-(n_2-n_1)}. $$ The shorter-maturity price is the base in this calculation because the question is asking how much the price falls from that price.
Worked Example: Measuring the Price Fall from a Longer Term
A hypothetical zero-coupon bond has face value $1,000 and a fixed effective annual required yield of 4%. Compare its prices at two and six years, then find the percentage by which the six-year price is below the two-year price. Assume annual compounding and a single payment of $1,000 at the selected maturity.
Method. First calculate both prices using the same face value and yield. For two years, $$ P(F,2)=\frac{1,000}{(1.04)^2} =\frac{1,000}{1.0816} =924.5562\ldots, $$ or $924.56 to the nearest cent.
For six years, $$ (1.04)^6=1.265319018496, $$ and therefore $$ P(F,6)=\frac{1,000}{1.265319018496} =790.3145\ldots, $$ or $790.31 to the nearest cent.
The maturity difference is \(6-2=4\) years. The price ratio and percentage decline are $$ \frac{P(F,6)}{P(F,2)}=(1.04)^{-4}=0.8548042\ldots, $$ $$ 1-0.8548042\ldots=0.1451958\ldots=14.52\%. $$ The six-year price is therefore 14.52% below the two-year price, rounded to two decimal places. The ratio method gives the same comparison as dividing the two unrounded prices. The common mistake avoided is using the six-year price as the denominator: that would answer a different percentage-change question.
Why the Scaling Formulas Hold
The face-value and maturity comparisons follow directly from discounting the single promised payment. A payment \(F\) received at the end of year \(n\) is worth \(F/(1+i)^n\) today under the stated yield. For two bonds with face values \(F_1\) and \(F_2\) but the same maturity, take the ratio of their prices:
The common discount factor cancels, which proves proportionality to face value. For two maturities and a common face value, taking the ratio instead gives
Here face value cancels, leaving only the extra years of discounting. This is why a longer bond’s price can be compared with a shorter bond’s price using the maturity difference alone, provided the face value and yield are the same.
Combining Face Value and Maturity Changes
When both face value and maturity differ, neither scaling rule alone is enough. The general ratio keeps both effects visible:
A larger face value raises price, while a longer maturity at a positive yield lowers it. The two changes may offset one another exactly; a larger face value does not automatically mean a higher price if its payment is sufficiently farther away.
Worked Example: When a Larger Face Value Offsets a Longer Term
Bond A is a hypothetical zero-coupon bond with face value $1,000 and maturity in two years. Bond B has face value $1,210 and maturity in four years. Both have an effective annual required yield of 10%, make no interim payments, and pay face value at maturity. Compare their prices.
Method. First calculate the two-year price of A: $$ P_A=\frac{1,000}{(1.10)^2} =\frac{1,000}{1.21} =826.4462\ldots. $$ Rounded to the nearest cent, \(P_A\) is $826.45.
For B, the four-year accumulation factor is $$ (1.10)^4=1.4641. $$ Thus $$ P_B=\frac{1,210}{1.4641} =826.4462\ldots, $$ which also rounds to $826.45.
The ratio confirms the equality: $$ \frac{P_B}{P_A} =\frac{1,210}{1,000}(1.10)^{-(4-2)} =1.21\left(\frac{1}{1.21}\right)=1. $$ The larger face value is exactly offset by two additional years of discounting at 10%. The common mistake avoided is to compare face values alone and conclude that B must be more valuable.
Worked Example: Finding a Face Value for an Equal Price
A hypothetical zero-coupon bond with face value $1,000 matures in three years. At an effective annual required yield of 4%, find the face value a four-year zero-coupon bond would need to have to have the same price. Both bonds pay only at maturity, and the yield is the same for both.
Method. Let \(F\) be the face value of the four-year bond. Equality of prices requires $$ \frac{1,000}{(1.04)^3}=\frac{F}{(1.04)^4}. $$ Multiply both sides by \((1.04)^4\): $$ F=1,000(1.04)=1,040. $$ Thus, the four-year bond must have a face value of $1,040. The calculation can be checked by noting that one extra year of discounting at 4% requires 4% more face value to preserve the same present value. The common mistake avoided is adding 4% of the original price rather than scaling the future face value by the additional accumulation factor.
Common Errors and Practical Checks
A zero-coupon comparison is simple only after the cash-flow timing and rate units are aligned. Before calculating, identify the single payment date, the effective rate period, and the base price for any percentage change.
- Using the wrong number of periods. With an effective annual yield, a payment in \(n\) years is discounted by \((1+i)^n\). Do not discount a two-year payment using only one annual period.
- Mixing rate conventions. The formula here takes \(i\) as an effective annual rate. A quoted rate with a different compounding convention must be converted or matched to the payment periods before it is used.
- Reversing a price ratio. State which bond is in the numerator. For example, \(P(F,n_2)/P(F,n_1)\) describes the longer-maturity price relative to the shorter-maturity price.
- Using the wrong percentage base. A percentage fall from the shorter-term price uses that shorter-term price as denominator. A percentage rise from the longer-term price uses the longer-term price instead; the two percentages are not equal.
- Assuming face value alone determines price. Face value scales price only when yield and maturity are held fixed. When the term changes too, use the combined price-ratio formula.
For instance, a common incorrect calculation for the $1,500 bond in the first example is \(1,500/1.04=1,442.31\), which discounts only one year. The bond pays in two years, so the correct calculation is \(1,500/(1.04)^2=1,386.83\), rounded to the nearest cent. Checking the payment date against the exponent catches this error.
Check Your Understanding
Use annual compounding and the stated effective annual required yield in each question. Assume every bond is a hypothetical zero-coupon bond that pays its face value once, at maturity.
- A bond with face value $750 matures in three years. At a 5% effective annual yield, calculate its price to the nearest cent. (Answer: $647.88)
- Two zero-coupon bonds have the same maturity and the same required yield. The second bond’s face value is 1.6 times the first bond’s face value. What is the ratio of the second price to the first price? (Answer: 1.6)
- At a 5% effective annual yield, a fixed-face-value bond’s maturity changes from three years to five years. By what percentage does its price fall relative to its three-year price? Round to two decimal places. (Answer: 9.30%)
- A $1,000-face-value zero-coupon bond matures in three years. At a 4% effective annual yield, what face value must a four-year zero-coupon bond have to have the same price? (Answer: $1,040)
- At a 5% effective annual yield, compare a $900-face-value bond maturing in two years with a $1,000-face-value bond maturing in four years. Find the ratio of the first price to the second price and state which price is higher. (Answer: 0.99225; the $1,000 four-year bond has the higher price)