Price & Yield

Yield to Maturity

YTM is the yield that equates bond value to price by discounting all future cash flows.

highlighted = computed this step

Solve for YTM

Yield to maturity is the discount rate that makes discounted contract value equal the observed price. At par, this worked bond has price $1,000.00 and yield 10%.

P=∑t=1nCt(1+YTM)t\text{P}=\sum_{t=1}^{n}\frac{C_t}{(1+\text{YTM})^t}

Off-par price

Instead of pretending to algebraically invert price to yield, choose an off-par yield of 12.5% and price the same cash flows exactly. The result is rounded $940.47, exact 685600/729 dollars.

y=12.5%⇒P≈$940.47y=12.5\%\Rightarrow P\approx \$940.47
YTM by constructionThe chosen yield reprices the bond exactly.Bond cash flowsPeriodCouponPrincipalCash flowPV1$100.00$0.00$100.00$88.892$100.00$0.00$100.00$79.013$100.00$1,000.00$1,100.00$772.57Price = Σ PV$940.47

Substitution check

Substituting that same yield gives exact dollar terms 800/9, 6400/81, and 563200/729. They sum to 685600/729 dollars, the exact constructed price.

$100.001+1/8+$100.00(1+1/8)2+$1,100.00(1+1/8)3=685600/729\frac{\$100.00}{1+1/8}+\frac{\$100.00}{(1+1/8)^2}+\frac{\$1,100.00}{(1+1/8)^3}=685600/729

Root-finding note

That substitution check confirms the chosen rate prices the observed off-par bond. In general, solving price to yield is a numerical root-finding problem; here we demonstrate YTM by construction, not by claiming a closed-form inverse.

Pobserved=685600/729 dollars⇒YTM=12.5%P_{\text{observed}}=685600/729\text{ dollars}\Rightarrow \text{YTM}=12.5\%