Bond Pricing Formulas Cheat Sheet
Core Bond Pricing Formula
Where: P = bond price, C = coupon payment per period, r = yield per period, n = total number of periods, FV = face value (usually $1,000).
Key Bond Pricing Formulas
| Formula | Equation | Use Case |
|---|---|---|
| Current Yield | Annual Coupon ÷ Current Market Price | Quick income measure; ignores time value and capital gains/losses |
| Yield to Maturity (YTM) | Solve for r in: P = C × [1 − (1+r)⁻ⁿ] / r + FV / (1+r)ⁿ | Total return if held to maturity — the bond market’s standard yield measure |
| Yield to Call (YTC) | Same formula as YTM but n = periods to call date, FV = call price | Expected return if issuer redeems a callable bond early |
| Yield to Worst (YTW) | min(YTM, YTC₁, YTC₂, …) | Most conservative yield — the lowest possible return across all call scenarios |
| Zero-Coupon Bond Price | P = FV / (1 + r)ⁿ | No coupons — price is purely the discounted face value |
| Clean Price | Dirty Price − Accrued Interest | The quoted market price — excludes interest earned since last coupon |
| Dirty Price | Clean Price + Accrued Interest | The actual settlement price — what the buyer pays |
Accrued Interest
| Day Count Convention | Used For | How It Works |
|---|---|---|
| 30/360 | Corporate bonds, municipal bonds | Assumes 30-day months, 360-day year — simplifies calculations |
| Actual/Actual | Treasury bonds | Uses actual calendar days — most precise method |
| Actual/360 | Money market instruments, T-Bills | Actual days elapsed over a 360-day year |
Duration and Convexity
Where k = number of coupon periods per year.
| Concept | Formula / Rule | What It Tells You |
|---|---|---|
| Macaulay Duration | Weighted average time to receive all cash flows | Measured in years — the “payback period” of a bond’s cash flows |
| Modified Duration | Macaulay Duration ÷ (1 + YTM/k) | % price change for a 1% change in yield — the primary sensitivity measure |
| Effective Duration | (P₋ − P₊) ÷ (2 × P₀ × Δy) | Used for bonds with embedded options where cash flows change with rates |
| Dollar Duration (DV01) | Modified Duration × Price × 0.01 | Dollar change in price for a 1 basis point yield move |
| Convexity | Σ [t(t+1) × PV(CFₜ)] / [P × (1+r)²] | Measures how duration changes as yields change — captures the curvature |
| Price Change Estimate | ΔP ≈ −D_mod × Δy + ½ × Convexity × (Δy)² | Combines duration and convexity for accurate price change estimates |
Price-Yield Relationship Rules
| Rule | Explanation |
|---|---|
| Inverse relationship | When yields rise, bond prices fall — and vice versa |
| Higher coupon = lower duration | More cash flow arrives sooner, reducing sensitivity to rate changes |
| Longer maturity = higher duration | Cash flows are further in the future, amplifying rate sensitivity |
| Lower yield = higher duration | Present value of distant cash flows increases at lower discount rates |
| Convexity benefit | A bond gains more from a rate drop than it loses from an equal rate rise |
| Premium vs. discount | Coupon > YTM → premium (price > par); Coupon < YTM → discount (price < par) |
Key Takeaways
- Bond price = PV of all future coupons + PV of face value at maturity
- YTM is the market’s standard yield metric — assumes reinvestment at the same rate
- Modified duration tells you the % price change per 1% yield change
- Convexity captures the curvature that duration misses — essential for large rate moves
- Know your day count conventions: 30/360 for corporates, Actual/Actual for Treasuries
Frequently Asked Questions
What is the difference between clean price and dirty price?
Clean price is the quoted market price of a bond that excludes accrued interest. Dirty price (also called the full price or invoice price) adds accrued interest to the clean price and represents what the buyer actually pays at settlement. Bonds are quoted clean but settled dirty.
Why can’t YTM be solved algebraically?
The bond pricing equation is a polynomial with no closed-form solution when there are multiple coupon payments. YTM must be found through iterative methods (trial and error or Newton-Raphson). Financial calculators and Excel’s RATE function handle this automatically.
What is the difference between Macaulay and modified duration?
Macaulay duration is the weighted average time (in years) until a bond’s cash flows are received. Modified duration divides Macaulay duration by (1 + YTM/k) to convert it into a price sensitivity measure — telling you how much the price changes for a 1% yield move.
When should I use effective duration instead of modified duration?
Use effective duration for bonds with embedded options — callable bonds, putable bonds, and mortgage-backed securities. Modified duration assumes cash flows don’t change when yields move, but embedded options alter cash flow patterns. Effective duration accounts for this by re-pricing the bond at yields above and below the current level.
How does convexity affect bond portfolio management?
Positive convexity is desirable — it means your portfolio gains more when rates fall than it loses when rates rise by the same amount. Portfolio managers often seek to maximize convexity at a given duration target. Callable bonds have negative convexity at low yields because the issuer is likely to call the bond, capping your upside.