15  Interactive: bond pricer

Two modes: compute PV given r, or back out r given PV.

15.1 Mode 1: price the bond

viewof FV1 = Inputs.range([100, 10000], {value: 1000, step: 50, label: "Face value FV"})
viewof r1 = Inputs.range([0.001, 0.20], {value: 0.05, step: 0.005, label: "Interest rate r"})
viewof T1 = Inputs.range([1, 30], {value: 5, step: 1, label: "Years to maturity T"})

PV1 = FV1 / Math.pow(1 + r1, T1)

md`**Present value:** \`PV = \$${FV1} / (1 + ${r1.toFixed(3)})^${T1} = \$${PV1.toFixed(2)}\``

Plot.plot({
  width: 600,
  height: 320,
  marginLeft: 70,
  marginBottom: 50,
  x: {label: "Interest rate r (%)", domain: [0, 20]},
  y: {label: "Bond PV ($)", domain: [0, FV1 * 1.05]},
  marks: [
    Plot.line(d3.range(0.001, 0.20, 0.002).map(rr => ({rr: rr * 100, pv: FV1 / Math.pow(1 + rr, T1)})), {x: "rr", y: "pv", stroke: "#0e7c7b", strokeWidth: 2.2}),
    Plot.dot([{rr: r1 * 100, pv: PV1}], {x: "rr", y: "pv", fill: "#b7410e", r: 6}),
    Plot.text([{rr: r1 * 100, pv: PV1, l: `PV = $${PV1.toFixed(0)}`}], {x: "rr", y: "pv", text: "l", dx: 12, dy: -8, fill: "#b7410e", fontSize: 11, fontWeight: "bold"})
  ]
})

15.2 Mode 2: back out the implied yield

Given the market price, what yield is the bond paying?

viewof FV2 = Inputs.range([100, 10000], {value: 1000, step: 50, label: "Face value FV"})
viewof PV2 = Inputs.range([10, 9999], {value: 800, step: 10, label: "Market price PV"})
viewof T2 = Inputs.range([1, 30], {value: 5, step: 1, label: "Years to maturity T"})

implied_r = Math.pow(FV2 / PV2, 1 / T2) - 1

md`**Implied yield:** \`r = (${FV2}/${PV2})^(1/${T2}) − 1 = ${(implied_r * 100).toFixed(3)}%\``

15.3 What to play with

• Set FV = 1{,}000, T = 5. Move r from 1% to 20%. Watch PV drop from $951 to $402. The shorter the maturity, the less price-sensitive to r.
• Set T = 30. Same r swing now moves PV from $742 down to $4. Long bonds are extremely interest-rate-sensitive. This is why the long end of the yield curve is volatile.
• Mode 2 with PV = 800, FV = 1{,}000, T = 5. Implied yield \approx 4.56\%. Useful for “given this discount, what return?”

15.4 The intuition

A bond is a fixed promise of future dollars. If the discount rate rises, those future dollars are worth less today. Equivalently: if new bonds pay 8% and you hold a 7% bond, no one will pay full face value for yours; its market price falls until its yield-to-maturity matches 8%.