Flip pricing off a discount curve around: you are handed a price, not a curve, and asked for the single rate that rationalizes it. That single rate is yield to maturity (YTM), the flat, constant discount rate that, applied to every promised cash flow, reproduces the observed price exactly. Formally, YTM is the internal rate of return (IRR) of the bond's cash-flow stream from the buyer's standpoint: solve
Price = Σ CFᵢ / (1 + YTM/k)^(k·tᵢ)
for YTM, where k is the number of compounding periods per year (2 for a semiannual-pay bond, the US Treasury and G-Sec convention; 1 for an annual payer, common among Indian corporate NCDs). There is, on purpose, no algebraic step you can perform to isolate YTM here: with more than two cash flows the equation is a high-degree polynomial in the discount rate, and general polynomials of that degree have no closed-form root. YTM is therefore always found iteratively, by bisection (bracket a rate you know is too low and one too high, and repeatedly halve the interval, the method used throughout this module's Python verification), by Newton-Raphson (using the price function's derivative to jump toward the root faster), or by whatever numerical solver sits behind your spreadsheet's YIELD/IRR function or a Python scipy.optimize call. Every one of those methods is doing the same thing: guessing a rate, pricing the bond at that rate, comparing the result to the actual price, and adjusting, exactly the muscle you will build by hand in the worked examples and practice set.
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How does yield to maturity relate to spot rates?
One subtlety worth internalizing early, because it explains why YTM and curve-pricing can coexist without contradiction: a bond's YTM is not any single point on the curve, it is a complicated, cash-flow-weighted average of every spot rate the bond touches. A bond's early coupons are more heavily influenced by the short end of the curve; its final (largest) redemption cash flow pulls YTM toward the long end. On an upward-sloping curve, YTM for a coupon bond will generally sit below the long-end spot rate and above the short end, pulled toward, but not equal to, the maturity-matched spot rate, and the gap is exactly what Worked Example 1 computes and displays.