Yield Curve

A yield curve begins with market prices, not yields. For each maturity, the price of a bond implies a single discount rate that equates the present value of its cash flows to that price, and that internal rate of return is the point plotted. Because a coupon bond's yield to maturity blends discount rates across all its payment dates, the resulting curve is only an approximation of the true term structure.

Analysts therefore work with several related curves. The spot (zero) curve gives the yield on a single payment at each horizon and is derived from coupon bonds by bootstrapping: solve the shortest maturity first, then use that rate to strip the known coupons out of the next bond and back out its final discount rate, repeating along the maturity spectrum. The par curve gives the coupon rate that would price each maturity at par, and the forward curve gives rates for periods beginning in the future. All four describe the same information set from different angles.

Curve movements are conventionally decomposed into three factors. A level shift moves all maturities in the same direction and explains most of the variance in yields. A slope change alters the spread between long and short maturities, producing steepening or flattening; when short yields exceed long yields the curve is inverted. Curvature changes move intermediate maturities relative to the wings, the butterfly shape.

Competing explanations of shape matter for interpretation. Pure expectations attributes the slope entirely to anticipated future short rates. Liquidity preference adds a positive term premium that grows with maturity, so an upward slope can exist with flat rate expectations. Segmented markets and preferred habitat emphasise that supply and demand at particular maturities, driven by pension funds, insurers, banks and central bank purchases, can bend the curve independently of expectations.

Government curves anchor pricing across a currency. Corporate and sovereign risk is quoted as a spread over a benchmark point, so a G-spread or interpolated spread is meaningless without the underlying curve. Swap and overnight index swap curves serve the same role for derivatives and increasingly for discounting collateralised exposures. Mortgage, municipal and inflation-linked instruments each have their own curve constructed on identical principles.

In valuation, discounting each cash flow at its own spot rate rather than a single yield to maturity eliminates the reinvestment assumption embedded in yield to maturity and produces the arbitrage-free price. Practitioners also position for curve views directly: barbell versus bullet structures, steepener and flattener trades, and carry-and-roll-down strategies that earn a bond's yield plus the price gain from rolling toward lower yields on an upward-sloping curve.

Be able to move between curves. Given two spot rates, extract the implied forward rate by setting the proceeds of investing to the longer horizon equal to investing to the shorter horizon and then reinvesting at the forward rate, and solving for that unknown. Given spot rates, price a coupon bond by discounting each payment at its matching rate, then confirm that the single yield to maturity that reproduces that price is a weighted average of the spot rates.

You should also classify shapes correctly, state which term-structure theory each shape is consistent with, and explain why yield to maturity is realised only under specific reinvestment and holding-period conditions. Finally, be precise about spreads: know which benchmark a quoted spread is measured against, and recognise that a spread change and a curve change are separate sources of return.