What is bond valuation (short answer)
– Bond valuation is the process of estimating what a bond is worth
Why bond valuation matters (short continuation)
– Valuation tells you whether the current market price fairly reflects the bond’s expected cash flows, given the market’s required return. It is the foundation for comparing bonds, assessing interest-rate risk, and making buy/sell decisions.
Core concept and formula
– Bond valuation is just present-value math: a bond’s price equals the present value (PV) of all future coupon payments plus the PV of the principal (face value) returned at maturity.
– Notation:
– C = coupon payment per period (coupon rate × face value / periods per year)
– M = maturity (face or par value, usually 100 or 1,000)
– r = required yield per period (market yield divided by periods per year)
– N = total number of periods (years × periods per year)
– Price formula:
Price = Σ_{t=1 to N} [C / (1 + r)^t] + M / (1 + r)^N
Worked numeric example — plain annual coupon bond
– Inputs: 5-year bond, face value M = $1,000, annual coupon rate = 6% → C = $60, required yield (market rate) = 4% → r = 0.04, N = 5.
– Compute PV of coupons: PV_coupons = 60/(1.04)^1 + 60/(1.04)^2 + … + 60/(1.04)^5.
You can use the annuity formula: PV_coupons = C × [1 − (1 + r)^−N] / r.
PV_coupons = 60 × [1 − (1.04)^−5] / 0.04 ≈ 60 × 4.4518 = 267.11
– PV of principal: PV_principal = 1,000 / (1.04)^5 ≈ 1,000 / 1.21665 = 821.93
– Price = 267.11 + 821.93 = $1,089.04
– Interpretation: coupon > yield → bond sells at a premium (> $1,000).
Worked numeric example — zero-coupon bond
– Zero-coupon bond pays no coupons; price = M / (1 + r)^N.
– Example: $1,000 face, 10-year, r = 3% → price = 1,000 / (1.03)^10 ≈ $744.09.
Yield measures and what they mean
– Current yield = annual coupon / market price. Quick snapshot of income, ignores capital gains/losses and time value.
– Yield to maturity (YTM) = the single discount rate that equates the present value of future cash flows to the bond’s price. It assumes all coupons are reinvested at the YTM (reinvestment assumption). YTM is solved numerically (no closed-form when coupons exist); financial calculators or spreadsheet functions (Excel RATE or IRR) are used.
– Yield to call (YTC) applies when a bond is callable; compute like YTM but treat the call date and call price as maturity.
Price–yield relationship and signs of risk
– Price and yield move inversely: when yields rise, prices fall and vice‑versa.
– Duration is a measure of interest-rate sensitivity: it approximates the percent price change for a small change in yield.
– Macaulay duration = weighted-average time to receive cash flows (weights are PV of each cash flow / bond price), in years.
– Modified duration ≈ Macaulay duration / (1 + r) and gives the approximate percentage price change per 1 percentage point change in yield.
– Convexity is the next-order adjustment that improves the duration approximation for larger yield changes.
Checklist: inputs you need to value a bond
– Face value (par), coupon rate and payment frequency (annual, semiannual, etc.), maturity date, current market price (if solving for yield), required market yield (if solving for price), call or sinking-fund features (if any), and tax or credit considerations.
– Decide on the appropriate discount rate: for a corporate bond this is typically a yield reflecting credit risk and comparable maturities; for government bonds use prevailing Treasury yields.
Step-by-step to compute bond price (practical)
1. Convert all rates to the same period (e.g., semiannual coupons → divide annual rates by 2).
2. Compute C, r, and N in period terms.
3. Use the price formula or an annuity formula for coupons + PV of principal.
4. If you need yield from price, use numeric methods: financial calculator, Excel’s RATE or IRR, or a solver.
5. Check for special features (callable, convertible) and adjust cash flows accordingly.
Common assumptions and limitations
– Cash flows are certain (no defaults). For corporate bonds, use credit spreads and adjust discount rates to reflect default risk.
– Coupons are reinvested at the calculated YTM when interpreting YTM as a realized return.
– Market liquidity, taxes, transaction costs and embedded options change real-world outcomes and may require additional modeling.
Quick example: semiannual bond calculation (illustrative)
– Bond: 10-year, 8% annual coupon, semiannual payments → C = (0.08 × 1,000)/2 = $40 per period. N = 20 periods. Market yield = 6% annual → r = 0.03 per period.
– PV_coupons = 40 × [1 − (1.03)^−20] / 0.03 ≈ 40 × 14.8775 = 595.10
– PV_principal = 1,000 / (1.03)^20 ≈ 1,000 / 1.8061 = 553.68
– Price ≈ 595.10 + 553.68 = $1,148.78
Where to learn more (reputable references)
– U.S. Securities and Exchange Commission — Introduction to Bonds: https://www.sec.gov/investor/pubs/bonds.htm
– U.S. Department of the Treasury — Treasury Securities and Rates: https://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/default.aspx
– CFA Institute — Fixed Income Valuation (overview and curriculum references): https://www.cfainstitute.org
– Federal Reserve Education — Bond basics and yields: https://www.federalreserveeducation.org
– Investopedia — Bond valuation (background article): https://www.investopedia.com/terms/b/bond-valuation.asp
Educational disclaimer
– This is educational material, not individualized investment advice. Valuation requires judgment and up-to-date market inputs; consult a qualified financial professional before making investment decisions.