In plain English
A bond is a loan you can trade. You know the payments in advance, so the only thing that moves is the price people will pay for them today.
The advanced view
Price and yield are two views of the same object, linked by a convex, decreasing function. Duration is the first derivative scaled by price (the weighted average time to cash flow), convexity the second. Spread decomposition — government curve, credit, liquidity, optionality — tells you which risk you are being paid for.
A bond is a contract: fixed coupons for a period, then the face value back. Its price is simply the present value of that promised stream at the yield the market currently demands. Because the cash flows are fixed and only the discount rate moves, bond pricing is the cleanest possible application of time value of money — and it makes the inverse relationship unavoidable: when required yields rise, prices fall.
Price and yield
P = Σ C/(1+y)^t + F/(1+y)^n Current yield = Annual coupon / Price Coupon rate > y → premium bond; coupon rate < y → discount bond Approx. price change = −Modified duration × Δy + ½ × Convexity × (Δy)²
Yield to maturity is the single discount rate that makes the present value of the promised cash flows equal the market price. It is an internal rate of return, so it silently assumes coupons are reinvested at that same rate — which is why realised return rarely equals YTM. Duration then converts yield changes into price sensitivity: Macaulay duration is the weighted average time to cash flow, and modified duration is the percentage price change per one percentage point of yield. Convexity is the curvature correction that matters once moves get large.
The yield curve plots yield against maturity. An upward slope is normal and reflects term premium plus growth expectations; a flat curve signals uncertainty; an inverted curve — short rates above long rates — has preceded most recessions because it means the market expects rate cuts. Spreads over the government curve price credit risk: default probability, loss given default and liquidity. Credit ratings summarise that judgement, and the investment-grade to high-yield boundary drives large forced flows because many mandates cannot hold below it.
Essential vocabulary
- Par / face value
- The principal repaid at maturity, typically 100 or 1,000. Coupons are quoted as a percentage of it.
- Yield to maturity (YTM)
- The IRR of holding the bond to maturity at the current price. The market's required return.
- Modified duration
- Approximate percentage price change for a 1 percentage point change in yield.
- Credit spread
- The yield premium over a comparable government bond, compensating for default and liquidity risk.
- Zero-coupon bond
- No coupons; sold at a discount, repays face value. Duration equals maturity.
- Callable bond
- The issuer may redeem early, usually after rates fall. Caps upside for the investor, so it yields more.
Strategy connection
Debt markets set the price of a company's strategic options. A firm whose spread widens loses the ability to fund acquisitions, refinance cheaply, or outlast a downturn — so credit capacity, not just cost of capital, belongs in any strategy discussion about growth or resilience.
Intuition
A bond is a fixed set of cash flows, so its price moves only because the discount rate moves — that is the whole inverse relationship. Duration is the first derivative of that relationship: price change ≈ −modified duration × yield change, with convexity as the correction when the move is large.
Common pitfalls
- ×Confusing coupon rate, current yield and yield to maturity.
- ×Applying duration alone to a 200bp move, which overstates the loss.
- ×Ignoring reinvestment risk when quoting yield to maturity as a realised return.
Worked example — price move from a rate rise
Step 1 of 4
- 1Price 100, modified duration 7.0, convexity 60
Why it works
Duration works as a risk measure because price is a sum of discounted cash flows, and differentiating that sum yields a weighted average of maturities. Longer cash flows react more to rates simply because they are discounted more times. Convexity is the correction term: it is positive for plain bonds, which is why a duration estimate always overstates the loss from rising rates.
How it is used — price a rate move
Step 1 of 4
- 1Bond price 100, modified duration 7, convexity 60. Rates +50bp.
Deeper
Deeper: duration, convexity and the credit spread
Modified duration approximates the percentage price change for a one-point yield move: ΔP/P ≈ −D_mod × Δy + ½ × convexity × Δy². Duration is a first-order (linear) approximation, and it under-predicts the gain when yields fall and over-predicts the loss when they rise — that asymmetry is convexity, and it is why investors pay for it.
A corporate yield decomposes into the risk-free rate plus a credit spread, and the spread compensates for expected loss (probability of default × loss given default) plus liquidity and risk premia. A rough rule: spread ≈ PD × LGD in basis points, plus a premium usually of similar size.
Must know cold
- ✓Price and yield move in opposite directions.
- ✓ΔP/P ≈ −D_mod × Δy (+ convexity term).
- ✓Longer maturity and lower coupon mean higher duration.
- ✓Yield = risk-free + credit spread; spread ≈ PD × LGD plus premia.
Exercises
Try each one on paper before revealing the worked solution.
Exercise 1
A bond has modified duration 7.2 and convexity 65. Yields rise 100 bp. Estimate the price change.
Exercise 2
A 5-year bond yields 6% while the government curve is at 3.2%. If LGD is 60%, what default probability is priced, roughly?