Bond Calculator
Price bonds, compute yield to maturity, and analyze coupon payments to evaluate fixed-income investment opportunities
Formula
Price = Σ[C/(1+r)^t] + F/(1+r)^n
Sum of present value of all coupon payments plus present value of face value at maturity, discounted at market rate.
Worked Examples
Example 1: Bond Pricing - Discount Example
Problem:$1,000 face value bond, 5% coupon (semi-annual), 10 years to maturity, market rate 6%. Calculate price.
Solution:Semi-annual coupon: $1,000 × 5% ÷ 2 = $25 Semi-annual rate: 6% ÷ 2 = 3% Number of periods: 10 × 2 = 20 PV of coupons: $25 × [1 - (1.03)^-20] / 0.03 = $371.94 PV of face value: $1,000 / (1.03)^20 = $553.68 Bond price = $371.94 + $553.68 = $925.62 Trading at discount because: Coupon (5%) < Market rate (6%) Discount = $1,000 - $925.62 = $74.38
Result:Price: $925.62 (discount)
Example 2: Current Yield vs YTM
Problem:Bond trading at $920, 6% coupon, $1,000 face, 5 years to maturity. Calculate current yield and approximate YTM.
Solution:Current yield: Annual coupon ÷ Current price $60 ÷ $920 = 6.52% YTM (approximate formula): [Coupon + (Face - Price)/Years] ÷ [(Face + Price)/2] [$60 + ($80/5)] ÷ [($1,000 + $920)/2] [$60 + $16] ÷ $960 $76 ÷ $960 = 7.92% Exact YTM (solver, semi-annual coupons): 7.97% YTM > Current yield because: Price appreciation ($80 over 5 years) adds to return
Result:Current yield: 6.52% | YTM: 7.97%
Example 3: Interest Rate Sensitivity
Problem:20-year Treasury at par ($1,000), 4% coupon. What happens if rates rise to 5%?
Solution:Original: 4% coupon, 4% market rate, price = $1,000 (par) If rates rise to 5%: Semi-annual coupon: $20 Semi-annual rate: 2.5% Periods: 40 PV of coupons: $20 × [1 - (1.025)^-40] / 0.025 = $502.07 PV of face: $1,000 / (1.025)^40 = $372.43 New price = $874.50 Price drop: $1,000 - $874.50 = $125.50 (12.6% loss) This illustrates duration risk - 1% rate increase caused ~12.5% price drop for this long-term bond.
Result:1% rate rise → 12.6% price drop
Frequently Asked Questions
How are bonds priced?
Bond price equals the present value of all future cash flows: coupon payments plus face value at maturity, all discounted at the market rate (yield to maturity). If market rate > coupon rate, bond trades at discount. If market rate < coupon rate, bond trades at premium. This inverse relationship is fundamental to bond investing.
What is yield to maturity (YTM)?
YTM is the total annual return if you hold the bond to maturity, assuming all coupons are reinvested at the same rate. It accounts for: purchase price (premium/discount), all coupon payments, and face value at maturity. YTM is the single rate that equates present value of cash flows to current price.
What's the difference between coupon rate and yield?
Coupon rate is the annual interest payment as % of face value (fixed when issued). Current yield is annual coupon ÷ current price. YTM includes price appreciation/depreciation to maturity. Example: 5% coupon, $950 price, $1,000 face: coupon = 5%, current yield = 5.26%, YTM ≈ 5.5% (includes $50 gain).
Why do bond prices fall when interest rates rise?
Existing bonds with lower coupons become less attractive when new bonds offer higher rates. To compete, older bonds must drop in price until their YTM equals market rates. A 20-year bond can lose 15-20% value if rates rise 1%. This interest rate risk is why long-term bonds are more volatile.
What is duration?
Duration measures bond price sensitivity to interest rate changes. A duration of 7 means: if rates rise 1%, price falls ~7%. Longer maturities = higher duration = more rate risk. Zero-coupon bonds have duration equal to maturity. Duration helps compare bonds' interest rate risk.
What are premium vs discount bonds?
Premium: price > face value (coupon > market rate). Discount: price < face value (coupon < market rate). At maturity, both converge to face value. Premium bond buyers get higher coupons but capital loss at maturity. Discount bond buyers get lower coupons but capital gain. Total return (YTM) reflects this.
How do I compare different bonds?
Compare YTM (total return measure), credit rating (risk), duration (rate sensitivity), callability (early redemption risk), and tax treatment (muni bonds may be tax-free). Higher YTM usually means higher risk. Match duration to your investment horizon to reduce interest rate risk.
What is a callable bond?
Issuer can redeem before maturity, typically at a premium to face value. Issuers call when rates fall (refinancing at lower rate). Investors face reinvestment risk - getting money back when rates are lower. Yield-to-call may be more relevant than YTM for callable bonds trading above call price.
How does credit rating affect yield?
Lower credit = higher yield (risk premium). AAA bonds (highest quality) yield less than BBB (investment grade minimum). Below BBB is 'junk' or 'high-yield.' Example: 10-year Treasury 4%, AAA corporate 4.5%, BBB corporate 5.5%, BB (junk) 7%. Spread widens during economic stress.
What's the difference between government and corporate bonds?
Government (Treasury) bonds are backed by US government - essentially zero default risk. Corporate bonds carry credit risk. Munis (state/local) may be tax-exempt. Agency bonds (Fannie Mae) have implicit government backing. Higher risk = higher yield. Treasuries are the benchmark for 'risk-free' rate.
Background & Theory
Bond pricing connects current market interest rates to fixed future cash flows through present value calculations. The inverse relationship between rates and prices - rates rise, prices fall - is fundamental to understanding bonds. This mechanism affects everything from retirement portfolios to mortgage rates to government financing capacity.
Bond Pricing Formula:
Price = Σ [C / (1+r)^t] + [F / (1+r)^n]
Where:
- C = Coupon payment
- r = Market rate per period
- F = Face value
- n = Number of periods
Key Bond Metrics:
| Metric | Formula | Meaning |
|---|---|---|
| Coupon Rate | Annual coupon ÷ Face value | Fixed interest rate |
| Current Yield | Annual coupon ÷ Price | Income return only |
| YTM | IRR of all cash flows | Total return to maturity |
| Duration | Weighted avg time to cash flows | Interest rate sensitivity |
Price-Yield Relationship:
The inverse relationship is fundamental:
- Rates ↑ → Prices ↓
- Rates ↓ → Prices ↑
Why? New bonds offer market rates. Old bonds must adjust price until their return equals market rates.
Premium vs Discount:
| Condition | Bond Trades At | Price Movement |
|---|---|---|
| Coupon > Market rate | Premium (>$1,000) | Declines to par at maturity |
| Coupon = Market rate | Par ($1,000) | Stays at par |
| Coupon < Market rate | Discount (<$1,000) | Rises to par at maturity |
Duration and Convexity:
Duration approximates price sensitivity:
ΔPrice ≈ -Duration × ΔYield × Price
Example: Duration = 8, yield rises 0.5%
Price change ≈ -8 × 0.005 = -4%
Convexity refines this for large rate changes.
Bond Types:
| Type | Issuer | Risk | Features |
|---|---|---|---|
| Treasury | US Government | None | Benchmark, very liquid |
| Agency | GSEs (Fannie, Freddie) | Very low | Slightly higher yield |
| Municipal | State/Local govt | Low-Medium | Often tax-exempt |
| Corporate | Companies | Medium-High | Higher yields |
| High Yield | Lower-rated corps | High | Significant default risk |
Credit Ratings:
| Rating | Grade | Default Risk |
|---|---|---|
| AAA | Investment | Extremely low |
| AA | Investment | Very low |
| A | Investment | Low |
| BBB | Investment | Moderate |
| BB | Speculative | Significant |
| B | Speculative | High |
| CCC | Speculative | Very high |
| D | Default | In default |
Yield Spread:
Difference between corporate bond yield and Treasury (the "risk-free" rate). Wider spreads indicate higher perceived risk. Spreads widen during economic uncertainty.
Total Return Components:
- Coupon income
- Price change (if sold before maturity)
- Reinvestment income (coupons reinvested)
YTM assumes coupons reinvested at YTM rate - often unrealistic.