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CFA® Level I · Fixed Income · free lesson
Fixed-Income Bond Valuation: Prices and Yields
A bond's price is simply the present value of the cash flows it promises, and its yield-to-maturity is the single discount rate that links the two. This module shows how to price a bond on and between coupon dates, how to back out the yield-to-maturity from a price, how prices respond to changes in yield, coupon and maturity, and how analysts price bonds that rarely trade.
What you will learn
1. Calculate the price of a fixed-rate bond on a coupon date given its market discount rate, and explain when a bond trades at a premium, at a discount or at par.
- Price = present value of coupons and principal discounted at the market discount rate (YTM).
- Match periodicity: coupon per period, yield per period and number of periods must all use the same frequency.
- Coupon = yield: par; coupon < yield: discount; coupon > yield: premium.
- Zero-coupon price = FV / (1 + r)^N.
- The market discount rate = benchmark rate + spread for credit and liquidity risk.
2. Calculate a bond's yield-to-maturity from its price and explain the assumptions behind the yield-to-maturity.
- YTM is the internal rate of return that equates the present value of promised cash flows with the price.
- For semiannual bonds, YTM = 2 x the periodic rate (semiannual bond basis).
- Par bond: YTM = coupon; discount bond: YTM > current yield > coupon; premium bond: YTM < current yield < coupon.
- YTM assumes holding to maturity, no default and reinvestment of coupons at the YTM.
- Realized returns differ from YTM when bonds are sold early, rates change or the issuer defaults.
3. Identify and explain the relationships among a bond's price, coupon rate, maturity and market discount rate, including convexity and the movement of price toward par over time.
- Bond prices and yields move in opposite directions.
- Convexity: a yield fall raises the price by more than an equal yield rise lowers it.
- Lower coupon and longer maturity mean greater price sensitivity to yield changes.
- At a constant yield, discount bonds rise and premium bonds fall toward par (pull to par).
- Most sensitive: long-dated, low-coupon bonds; least sensitive: short-dated, high-coupon bonds.
4. Calculate the full price, accrued interest and flat price of a bond between coupon dates using the appropriate day-count convention.
- Full price = PV at last coupon date x (1 + r)^(t/T); the buyer pays the full price.
- Accrued interest = (t/T) x coupon per period.
- Flat (quoted) price = full price - accrued interest; quoting flat avoids the sawtooth pattern.
- Actual/actual for government bonds; 30/360 for many corporate bonds.
- t = days from the last coupon to settlement; T = days in the coupon period.
5. Describe and apply matrix pricing to estimate the price or required yield of a bond that is illiquid or not yet issued.
- Matrix pricing values illiquid or unissued bonds from the yields of comparable traded bonds.
- Linear interpolation by maturity: y = y_short + (N - N_short) / (N_long - N_short) x (y_long - y_short).
- Discount the target bond's own cash flows at the interpolated yield.
- The spread version interpolates credit spreads and adds them to the government yield; it is used to price new issues.
- Matrix prices are estimates and can lag the market in stressed conditions.
Module at a glance
- Bond price = present value of promised cash flows at the market discount rate, using the bond's periodicity.
- Coupon > yield: premium; coupon < yield: discount; coupon = yield: par.
- YTM is the bond's internal rate of return; it assumes holding to maturity, no default and reinvestment at the YTM.
- Prices fall when yields rise; longer, lower-coupon bonds move most; convexity makes gains exceed losses.
- At a constant yield, prices converge to par (pull to par).
- Full price = PV x (1 + r)^(t/T); flat price = full price - accrued interest; check the day count.
- Matrix pricing interpolates yields of comparable bonds to value illiquid or new bonds.
This free module has 5 lessons, 14 practice questions with full explanations.