FIN521 Chap.9 Bond Prices, Yields and the Term Structure
Bond Prices, Yields and the Term Structure
One equation, read in two directions
The teaching schedule gives bond prices and yields the eighth week and the term structure the ninth, and they belong together because the second asks the first question of many maturities at once. A bond is a set of dated payments and its price is what those payments are worth today. Given the market's required return, solve for the price, which is arithmetic.
Given the price, solve for the return, which is the yield to maturity and has no closed form, so it is found by iteration. Work in the bond's own period throughout: a semi-annual bond has half-year periods, half the annual coupon and half the quoted annual yield.
Why price and yield must move in opposite directions
The coupon and the face value are fixed at issue and never change.
If the return the market demands rises, the only quantity that can adjust to deliver that higher return on a fixed set of payments is the price paid for them. This is a consequence of the definition rather than an empirical tendency, and an answer that reports it as an observation has missed that it could not have been otherwise.
The relationship is also curved rather than straight, and the curvature is convexity: a fall in yield gains the holder more than an equal rise loses, which is a genuine advantage and is priced.
Three yields with the same name
The coupon rate is the promised annual payment as a percentage of face value and never changes.
The current yield is the annual coupon over the current price, capturing income but ignoring the gain or loss to redemption. The yield to maturity is the single discount rate equating the present value of all payments with the price, so it captures both. Only the third is comparable across bonds.
For a discount bond the three rise in that order, and for a premium bond they fall in it, which gives a free consistency check on any answer.
Duration, the curve, and the spread
Two bonds maturing on the same day can respond very differently to a rate move, because one returns most of its value early in coupons and the other almost all at the end.
Duration is the weighted average time to receipt and is what a rate change actually acts on: a longer maturity raises it, a higher coupon lowers it, and a higher yield lowers it. Plotting yields against maturity across government issues gives the term structure, normally upward sloping.
Two accounts compete, one attributing the slope to expected rate changes and one to a premium for tying money up, and the curve alone cannot separate them. Every corporate yield is then that curve plus a credit spread, and keeping the two components apart is what lets an analyst say whether a bond cheapened because rates moved or because the issuer deteriorated.
What this chapter covers
- 01
Present value as the definition of a bond price
- 02
Working in half-years and converting only at the end
- 03
Why the price must move opposite to the yield
- 04
Convexity and why the relationship is curved
- 05
Coupon rate, current yield and yield to maturity distinguished
- 06
The ordering check that catches period errors
- 07
Duration as weighted average time to receipt
- 08
What raises and lowers duration, and why the estimate is a straight line
- 09
Curve shapes, the two competing accounts, and the credit spread
Price a bond and check the three yields against each other
- 3Set up the calculation in half-years and state the three inputs.
- 4Value the coupon stream and the principal, then add them.
- 3Compute the current yield and order the three yields with a reason.
Key terms
- Yield To Maturity
- The single discount rate that makes the present value of a bond's remaining payments equal its price, counting both coupons and redemption.
- Current Yield
- The annual coupon divided by the current price, which measures income alone and ignores the gain or loss at redemption.
- Convexity
- The curvature of the price and yield relationship, which makes a fall in yield gain the holder more than an equal rise loses.
- Duration
- The weighted average time at which a bondholder receives cash, each date weighted by the present value of the payment arriving then.
- Modified Duration
- The proportional price sensitivity of a bond to a small change in yield, equal to the slope of the price and yield curve at one point.
- Term Structure
- The set of yields on otherwise identical government securities plotted against their maturities, which is the risk-free curve.
- Forward Rate
- The break-even future short rate implied by today's curve, which makes a long investment and a rolled sequence of short ones equally good.
- Credit Spread
- The excess of a corporate yield over the government yield of the same maturity, which is the price of the issuer's default risk.
- Liquidity Preference
- The account of an upward-sloping curve that attributes the slope to compensation demanded for tying money up for longer.
Bond Prices, Yields and the Term Structure FAQ
Why does duration matter when I already know the maturity?
Because maturity says only when the bond ends, while duration says how much of your money is still waiting. Two bonds redeeming on the same day can behave nothing alike if one pays large coupons along the way and the other pays almost everything at the end.
A rate change acts on the payments still outstanding, so the weighted average time to receipt is the right ruler, and questions in this area routinely fix maturity across both bonds precisely so that it cannot be the answer.
Does an inverted yield curve predict a recession?
It is an empirical regularity rather than a deduction, and stating it that way is what a careful answer does. The curve is observed, and the two accounts of its shape assign the same slope to different causes: expected falls in short rates under one, a shrinking term premium under the other. Since the curve alone cannot separate them, a prediction drawn from the shape is an inference from past correlation.
What the curve does establish beyond dispute is a set of prices at which money can be borrowed and lent today for each term.
Why is my calculated price wrong when the formula is right?
Almost always because annual and semi-annual quantities have been mixed. The rule is mechanical: for a bond paying twice a year use half-year periods, half the annual coupon and half the quoted annual yield, and convert nothing until the price is on the page. The quoted yield is a doubled semi-annual figure by convention, so doubling at the end reproduces the market convention while compounding does not.
Checking the coupon and yield ordering catches the error immediately.
Exam move
Price the same bond three times, at a yield above the coupon, equal to it and below it, and watch the price cross par. That single exercise makes the ordering check automatic and removes the most common period error at the same time.
Then take two bonds with the same maturity and very different coupons and reason out which has the longer duration before computing anything, because that qualitative comparison is the form the question usually takes.
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