01Discount a contractual cash-flow stream
A conventional fixed-rate bond promises coupons and repayment of principal. Its price is the present value of these payments using the return investors currently require. The coupon rate is fixed by the contract; current yield divides annual coupon by the current price; yield to maturity is the rate equating all promised payments to the price under the relevant convention. These are different concepts. Yield to maturity assumes contractual payments occur; it does not guarantee the realized return if default, early sale or reinvestment conditions change.
02Why price falls when yield rises
Imagine receiving $100 exactly one year from now with no coupon and no credit loss. At a 4% required return, the price is 100 / 1.04 = $96.15. At 5%, it is 100 / 1.05 = $95.24. The promised $100 did not change; the price did. More distant payments are generally more sensitive to a discount-rate change. A government guarantee on contractual repayment does not protect a bond’s resale price before maturity.
03Duration is a sensitivity, not just maturity
Modified duration estimates proportional price sensitivity to a small yield change. A duration of 6 implies roughly −6% for a +1 percentage-point yield shift, holding other factors constant. Convexity captures curvature and improves the approximation for larger moves. Different curve points can move differently: one duration number assumes a simplified parallel shift. Credit spreads, embedded options and liquidity can also move the price, so a duration estimate is not a complete risk model.
Formula and worked example
ΔP / P ≈ −Dmod × Δy, where Δy is a decimal rate change (50 bps = 0.005).Hypothetical $10,000 position, duration 6, yield +50 bps: proportional change ≈ −6 × 0.005 = −0.03, or −3%. Estimated value $9,700, before convexity, coupon accrual, spread moves and costs.