FNCE90060 Chap.5 Valuing Bonds
Valuing Bonds
Week 5 values a bond from its promised payments. A zero-coupon bond pays face value once at maturity. A coupon bond pays regular coupons and returns face value with the last coupon. The coupon payment equals coupon rate times face value divided by the number of payments per year. Yield to maturity is the single rate that equates current price with the present value of remaining promised cash flows.
For semiannual contracts, coupon, periodic yield and period count must all use half-year units. Bond price moves inversely with yield, and the relationship is curved. Longer maturity and lower coupons generally increase sensitivity. A risky bond’s promised yield is not automatically its expected return because default can reduce payments.
A holding-period return also differs from the initial YTM when the bond is sold early or coupons are reinvested at different rates.
What this chapter covers
- 01
Zero-coupon bonds
- 02
Coupon frequency and payment amount
- 03
Yield to maturity
- 04
Spot rates and the Law of One Price
- 05
Premium and discount bonds
- 06
Duration, convexity and holding returns
Worked example · free
Price a semiannual coupon bond
- 1Each coupon is $20 because $1,000 times 4% divided by two equals $20.
- 1The periodic yield is 3% and there are four half-year periods.
- 1Present value of coupons is $20 times the four-period annuity factor at 3%.
- 1Present value of face value is $1,000 divided by 1.03 to the fourth power.
- 1Total price is approximately $962.83, a discount because coupon rate is below yield.
Key terms
- Face Value
- The principal amount promised at bond maturity.
- Coupon Rate
- The annual contractual rate used with face value to determine coupons.
- Zero Coupon Bond
- A bond promising one face-value payment at maturity and no coupons.
- Spot Rate
- The yield on a default-free zero-coupon payment at a specified maturity.
- Duration
- A measure summarising bond-price sensitivity to a small yield change.
- Convexity
- The curvature in the inverse relationship between bond price and yield.
Valuing Bonds FAQ
Why does a bond price fall when yield rises?
The bond’s promised payments do not change, but a higher required return discounts each payment more heavily. Their total present value therefore falls until it offers the market yield.
When does a bond trade at a premium?
A coupon bond trades above face value when its coupon rate exceeds the market yield for its risk and maturity. Its richer contractual coupons require a higher current price.
Why is YTM not always the realised return?
Selling before maturity exposes the investor to the later market price, coupons may be reinvested at different rates and default can reduce payments. Those actual cash flows can differ from the original YTM assumptions.
How does semiannual compounding change the setup?
Divide the annual coupon and quoted APR into half-year amounts, then double the number of years to count periods. Every cash-flow and rate input must use the same half-year clock.
Why are coupons and face value priced separately?
Coupons form a repeated stream, while face value is one payment at maturity. Separating them makes the payment count and timing visible, then their present values can be added at today's date. The final period contains both the last coupon and principal, so omitting either understates the bond price.
When should spot rates replace one YTM?
Use maturity-specific spot rates when the question supplies or requires a term structure. Each promised payment is then discounted at the zero-coupon rate for its own date. YTM is a single price-equating summary rate; it need not describe the return available for every maturity along a non-flat yield curve.
How do duration and convexity divide the sensitivity task?
Duration gives a first-order estimate of the percentage price change for a small yield move. Convexity accounts for the curvature that makes equal yield rises and falls produce asymmetric price changes. State the inverse price-yield direction first, then use duration and convexity to compare the magnitude across bonds.
What cash flows belong in a holding-period return?
Use the purchase price, coupons actually received during the chosen horizon and the sale price or terminal value at the horizon. The result is tied to that interval. It differs from purchase-date YTM when the bond is sold early, market yields move, coupons are reinvested differently or promised payments are not made.
Exam move
Draw the bond timeline before using a calculator. Convert face value and coupon rate into the actual periodic payment, translate the quoted yield into a periodic rate and count matching periods. Price coupons and face value separately, then use premium or discount direction as a check. For conceptual questions, state price and yield move oppositely and explain how maturity and coupon timing change sensitivity.
Keep promised YTM, expected return and realised holding-period return as distinct definitions. Reprice the same contract after a small yield rise and fall to verify the inverse relation. Then compare a short and long maturity at the same coupon to see why duration changes, while keeping the contractual cash flows separate from market-required return.
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