Why Comparisons Need a Common Basis
A zero-coupon bond makes one payment at redemption and no coupon payments before then. Comparing two such bonds may appear straightforward, but a raw price reflects several things at once: the redemption amount, the time until payment, and the yield used to discount it. A bond with the higher price is not necessarily the more expensive bond in a meaningful per-dollar sense, and it does not necessarily have the lower yield.
A useful comparison starts by fixing the valuation date, identifying each redemption amount and term, and expressing the yield on a consistent basis. In particular, distinguish price, which is an amount of money, from price per dollar of redemption, which isolates the discounting effect. This distinction is especially important when the bonds have different face values.
This tutorial focuses on comparing bonds across face values, terms, and quoted compounding conventions. It does not assume that the bonds have identical risk or contractual features; the comparisons below are mathematical comparisons under the stated discounting assumptions.
Compare Face Values by Normalizing Price
Under a common effective annual yield and common term, price is directly proportional to redemption value. If one bond promises twice the redemption amount of another, it costs twice as much, all else equal. Comparing the raw prices alone mostly reports this difference in scale.
Dividing price by redemption value removes that scale effect. The normalized price then depends on the yield and the term, not on the size of the bond’s redemption payment. It can be interpreted as the current cost of one dollar payable at the specified future date.
For example, normalized prices of \(0.91\) and \(0.84\) mean that the first bond costs 91 cents per dollar of redemption and the second costs 84 cents per dollar. The second bond has the lower price per dollar of redemption. That statement alone does not identify which bond has the higher yield unless their terms are also the same.
Worked Example: Different Face Values at the Same Yield
Two bonds are valued on the same date. Bond A redeems for $1,000 in 2 years, and Bond B redeems for $1,500 in 3 years. Both have an effective annual yield of 5%, compounded annually. Find each price and normalized price, then compare the bonds per $1,000 of redemption.
Method. Use the same effective annual yield for each bond, discount each redemption amount for its own term, and divide each price by its redemption amount. This avoids interpreting a larger face value as a more expensive discounting basis.
For Bond A, the 2-year accumulation factor is \(1.05^2=1.1025\). Therefore,
Rounded to the nearest cent, \(P_A=\$907.03\). Its normalized price is \(907.029478\ldots/1000=0.907029478\ldots\), or $907.03 per $1,000 of redemption.
For Bond B, the 3-year accumulation factor is \(1.05^3=1.157625\). Thus,
Rounded to the nearest cent, \(P_B=\$1,295.76\). Its normalized price is \(1295.756\ldots/1500=0.863837599\ldots\), or $863.84 per $1,000 of redemption.
Bond B has the higher raw price because it promises a larger redemption amount. On a per-$1,000 basis, Bond B has the lower price: $863.84 versus $907.03. The different terms matter: the comparison is not evidence that the bonds have different yields, since both yields were specified as 5%.
Check. Multiplying each normalized price by its face value recovers its unrounded price: \(0.907029478\ldots(1000)=907.029478\ldots\), and \(0.863837599\ldots(1500)=1295.756\ldots\). Money is rounded only at the final step to two decimal places.
Compare Rate Quotes on a Consistent Basis
A rate quote must be interpreted with its compounding convention. An effective annual rate \(i\) applies once per year. A nominal annual rate \(i^{(m)}\), convertible \(m\) times per year, applies at a periodic rate of \(i^{(m)}/m\) for each of \(m\) periods per year. A force of interest \(\delta\) is a continuously compounded rate.
To compare quoted rates, convert them to equivalent accumulation or discount factors over the period being compared. For example, a nominal rate \(i^{(m)}\) has equivalent effective annual rate \(\left(1+i^{(m)}/m\right)^m-1\). A quoted number is not comparable merely because its percentage looks larger or smaller; the compounding basis must be included.
The pricing formula follows directly from the meaning of an accumulation factor. Under effective annual yield \(i\), one dollar invested at time 0 accumulates to \(1+i\) after 1 year, and to \((1+i)^n\) after \(n\) years. Therefore, a present amount \(P\) that grows to redemption \(F\) must satisfy \(P(1+i)^n=F\). Solving for \(P\) gives \(P=F(1+i)^{-n}\). Dividing this equation by \(F\) gives the normalized price formula. The same reasoning with periodic or continuous accumulation produces the other discount factors above.
Worked Example: Equivalent Nominal and Effective Quotes
A bond redeems for $1,800 in 2 years. Its yield is quoted as 10% nominal, convertible semiannually. A second bond redeems for $2,500 in 2 years and has an effective annual yield of 10.25%. Find both prices and compare their prices per $1,000 of redemption.
Method. Convert the nominal quote into its periodic accumulation factor, then compare that factor with the effective annual quote over the same 2-year term. The trap to avoid is treating 10% nominal convertible semiannually as a 10% effective annual yield.
For the first bond, the semiannual rate is \(0.10/2=0.05\), and 2 years contain \(2(2)=4\) half-year periods. Its accumulation factor is
The price is therefore
Its normalized price is \(1/1.21550625=0.822702\ldots\), or $822.70 per $1,000 of redemption.
For the second bond, the effective annual accumulation factor over 2 years is
This is exactly the same factor. Thus,
Its normalized price is also \(0.822702\ldots\), or $822.70 per $1,000 of redemption. The raw prices differ because the redemption values differ; the prices per dollar of redemption match because the discount factors match.
Check. The nominal quote’s equivalent annual rate is \((1.05)^2-1=0.1025\), or 10.25%. It therefore agrees with the second bond’s effective annual yield. Prices are rounded only at the final step to two decimal places.
Different Maturities: Price Is Not a Yield Ranking
When two bonds have the same redemption value and valuation date but different terms, their prices reflect both their yields and the amount of time for which redemption is discounted. Even if their yields are known, the raw price ordering does not necessarily follow the yield ordering.
At a fixed positive effective annual yield, extending the term lowers the normalized price because \((1+i)^{-n}\) decreases as \(n\) increases. But if yields differ as well, the combined effects must be calculated. A shorter-term bond can have a higher price even when it has a higher yield.
Worked Example: A Shorter Bond Can Cost More Despite a Higher Yield
Bond A redeems for $1,000 in 2 years at an effective annual yield of 10%. Bond B redeems for $1,000 in 3 years at an effective annual yield of 8%. Both are valued today, with annual compounding. Calculate the prices and explain the ordering.
Method. Price each redemption using its own term and effective annual yield. This avoids assuming that price alone ranks yield when the maturities differ.
For Bond A, the accumulation factor is \(1.10^2=1.21\). Therefore,
For Bond B, the accumulation factor is \(1.08^3=1.259712\). Therefore,
Bond A costs $32.61 more, rounded to cents, even though its yield is higher. Its redemption payment is discounted for only 2 years, while Bond B’s is discounted for 3 years. The shorter term more than offsets the yield difference in this price comparison.
Check. The normalized prices are \(826.446280\ldots/1000=0.826446280\ldots\) for A and \(793.832\ldots/1000=0.793832\ldots\) for B. These agree with the respective discount factors. Money is rounded only at the final step to two decimal places.
Common Mistakes and Exam Traps
The following errors can produce plausible-looking results while answering a different question from the one asked.
- Comparing raw prices across different face values. A bond with a larger redemption value will generally have a larger price under the same discount factor. Divide by the redemption value, or scale both bonds to the same redemption amount, before comparing discounting.
- Ignoring the term difference. A lower price can result from a longer term, a higher yield, or both. Do not infer which yield is higher from prices alone when terms differ.
- Using a nominal quote as an effective annual rate. For a nominal quote, use the periodic rate and the correct number of periods. A nominal 10% convertible semiannually uses 5% per half-year, not 10% per year as an effective rate.
- Mixing valuation dates. A price from one date and a price from another date do not form a direct comparison without first moving both cash-flow values to a common date using a stated rate basis.
- Rounding too early. Keep the discount factor and unrounded price through the calculation. Round money to cents only at the end unless the problem specifies another precision.
For instance, the incorrect calculation \(1800/(1.10)^2=\$1,487.60\) treats a 10% nominal rate convertible semiannually as a 10% effective annual rate. The correct calculation uses four half-year periods: \(1800/(1.05)^4=\$1,480.86\). The difference arises from the quote’s compounding convention, not from a change in the redemption amount or term.
A Consistent Comparison Procedure
A disciplined sequence keeps the distinct effects visible. State the valuation date first, then record each bond’s redemption amount, time to redemption, and rate quote. Convert the quotes to discount factors over the relevant terms. Finally, compare either actual prices when the dollar amounts matter or normalized prices when the goal is to compare the cost per dollar of redemption.
Make sure each price is measured at the same date, or move the values to a common date under an explicitly stated rate assumption.
Record the redemption amount and the time until it is paid. Do not assume the bonds have the same face value or maturity.
Use the effective, nominal, or continuous discount factor that matches the stated convention and term.
Use price for total dollars invested; use price divided by redemption value to compare discounting per dollar of redemption.
If terms or rate bases differ, a price ranking by itself is not a yield ranking.
These distinctions are useful in bond pricing and yield problems aligned with the SOA Exam FM (Financial Mathematics) syllabus. The essential habit is to separate scale, time, and rate: normalize for different redemption amounts, account for each maturity, and translate quoted rates into consistent accumulation factors before comparing.
Check Your Understanding
For each problem, use the stated valuation date and compounding convention. Round monetary answers to cents where requested.
- A zero-coupon bond is valued today and redeems for $2,400 in 4 years at an effective annual yield of 5%. Find its price and its price per $1,000 of redemption. (Answer: $1,974.49; $822.70 per $1,000.)
- A bond redeems for $1,000 in 3 years and yields 6% nominal, convertible semiannually. Find the discount factor and price. (Answer: discount factor \(1/(1.03)^6\); price $837.48.)
- Two bonds have the same effective annual yield of 4% and are valued today. Bond A redeems for $100 in 2 years; Bond B redeems for $200 in 3 years. Find both prices and compare per $100 of redemption. (Answer: A costs $92.46, or $92.46 per $100; B costs $177.80, or $88.90 per $100.)
- A $1,000 redemption is due in 2 years for Bond A at 10% effective annually, and in 3 years for Bond B at 8% effective annually. Which price is higher, and by how much? (Answer: Bond A; its price is higher by $32.61.)
- A $3,000 redemption is due in 2 years. Compare a 8% nominal yield convertible quarterly with an 8% effective annual yield. State the nominal quote’s equivalent effective annual rate and which bond has the higher price. (Answer: \((1.02)^4-1=8.2432\%\); the 8% effective annual yield gives the higher price.)