BFC2140 · Corporate Finance
Bond Valuation
A bond's price is just the present value of its promised cash flows — the coupons (an annuity) plus the face value at maturity — discounted at the yield to maturity (YTM). This chapter covers pricing coupon and zero-coupon bonds, the YTM, semi-annual coupons, the premium/discount/par logic and the four bond-price laws, plus interest-rate and credit risk. It is a high-yield exam topic: expect Section B numerical pricing and YTM questions and Section C parts explaining why a bond trades at a premium or discount or how its price reacts to a yield change.
What this chapter covers
- 01The coupon-bond price formula: PV of the coupon annuity plus PV of the face value
- 02Pricing zero-coupon bonds: P = FV/(1 + y)ⁿ
- 03Yield to maturity (YTM) as the bond's internal rate of return, found by trial/interpolation
- 04Semi-annual coupons: halve the coupon and yield, double the number of periods
- 05Premium / par / discount: coupon vs YTM determines whether price is above, at or below face
- 06The four bond-price laws (price-yield inverse, par logic, maturity effect, coupon effect)
- 07Interest-rate risk: longer maturity and lower coupon mean larger price swings
- 08Credit/default risk and its effect on the required return (YTM)
Pricing a coupon bond and identifying premium vs discount
- 1 markFind the coupon dollar amount: CPN = 6% × $1,000 = $60 per year, paid for n = 8 years; the face value FV = $1,000 is repaid at year 8; the per-period yield is y = 7.5%.
- 1 markWrite the price as the PV of the coupon annuity plus the PV of the face value: P = 60 × [1 − (1.075)^(−8)]/0.075 + 1,000/(1.075)^8.
- 1 markValue the coupon annuity: 60 × 5.8573 = $351.44.
- 1 markValue the face-value repayment: 1,000 × 0.56070 = $560.70, so P = 351.44 + 560.70 = $912.14.
- 1 markClassify: the coupon rate (6%) is below the YTM (7.5%), so the bond sells at a discount (price $912.14 < $1,000 face) — investors pay less than par to earn the higher market yield.
Key terms
- Coupon bond
- A bond that pays a fixed periodic coupon plus the face value at maturity. Its price is P = CPN·[1 − (1 + y)^(−n)]/y + FV/(1 + y)ⁿ — the PV of the coupon annuity plus the PV of the face value, discounted at the yield to maturity.
- Zero-coupon bond
- A bond with no coupons that pays only the face value at maturity, so P = FV/(1 + y)ⁿ. It always trades at a discount to face and its entire return comes from the price appreciation toward par.
- Yield to maturity (YTM)
- The single discount rate that sets the PV of a bond's cash flows equal to its market price — the bond's internal rate of return if held to maturity. It is found by trial-and-error/interpolation and moves inversely with price.
- Premium / par / discount
- A bond sells at a premium (price > face) when its coupon rate exceeds the YTM, at par when they are equal, and at a discount (price < face) when the coupon is below the YTM. Price and required yield always move in opposite directions.
- Interest-rate risk
- The sensitivity of a bond's price to changes in market yields. For a given yield change, prices of longer-maturity and lower-coupon bonds move by a larger percentage, so those bonds carry more interest-rate risk.
- Credit (default) risk
- The risk that the issuer fails to pay coupons or face value as promised. Higher default risk raises the yield investors demand (lowering the price) and widens the gap between promised and expected return.
Bond Valuation FAQ
Why does a bond's price move opposite to its yield?
The bond's cash flows (coupons and face value) are fixed at issue, so the only way the market can re-price the bond to deliver a higher required return is to lower its price today — and to deliver a lower required return, to raise the price. That inverse relationship is the first of the four bond-price laws, and it is why rising market interest rates push existing bond prices down.
How do I handle semi-annual coupons?
Make three adjustments before pricing: halve the annual coupon (so a 6% coupon on $1,000 becomes $30 per half-year), halve the quoted nominal yield (a 7% nominal YTM becomes 3.5% per period), and double the number of periods (8 years becomes 16). Price exactly as before with these per-period inputs. To convert a per-period yield to an effective annual yield, use (1 + y_semi)² − 1.
What are the four bond-price laws?
(1) Bond prices and yields move in opposite directions. (2) Coupon vs YTM sets the price level — coupon > YTM gives a premium, coupon = YTM gives par, coupon < YTM gives a discount. (3) The maturity effect: for a given yield change, longer-maturity bonds have larger percentage price changes. (4) The coupon effect: for a given yield change, lower-coupon bonds have larger percentage price changes. Laws 3 and 4 together describe interest-rate risk and are common Section C explanation questions.
How is YTM different from the coupon rate?
The coupon rate is fixed and just sets the dollar coupon (coupon rate × face value). The YTM is the market's current required return — the discount rate that makes the present value of all the bond's cash flows equal to its price. They are equal only when the bond trades at par; otherwise the price adjusts so the buyer earns the YTM, not the coupon rate. Importantly, when this bond's issuer's cost of debt is needed (Cost of Capital chapter), it is the YTM, not the coupon, that matters.
How is bond valuation examined in BFC2140?
It is a recurring computational skill flagged in the unit's exam guidance. Expect Section B numerical questions that ask you to price a coupon or zero-coupon bond (often semi-annual) or to solve for the YTM given a price, and Section C parts that ask you to explain premium/discount status, apply the four bond-price laws, or discuss interest-rate and credit risk. It also feeds directly into the cost-of-debt component of WACC later in the unit.
Exam move
Treat every bond as 'an annuity plus a lump sum' and the rest follows. Drill the coupon-bond formula until you can set it up from a worded scenario in seconds, and always run the premium/discount sanity check (compare coupon rate to YTM) before trusting your price. Practise the semi-annual conversion as a fixed habit — halve the coupon, halve the yield, double the periods — because forgetting one of the three is the most common pricing error. Learn to find YTM by interpolation on your financial calculator and verify it by re-pricing the bond at that yield. Be ready to state and apply the four bond-price laws in words for Section C, and connect the topic forward: the bond's YTM is the pre-tax cost of debt you will reuse when you build a firm's WACC.
