Bond pricing = PV of cash flows (integral callback)

Bond price = sum of discounted cash flows

A bond is just a sequence of future cash flows. The price you should pay today is the present value of those cash flows. Track 1's integral lesson — area under the curve, summed up — is exactly this picture in discrete form.

For a coupon bond paying coupon C annually for n years and returning face value F at the end, with discount rate r:

P = \frac{C}{1 + r} + \frac{C}{( 1 + r ) ^{2}} + \dots + \frac{C}{( 1 + r ) ^{n}} + \frac{F}{( 1 + r ) ^{n}}

Each term is one future payment, discounted to today. The first n terms are the coupons. The last term is the face value at maturity. Sum them and you get the price.

Why this is just integral made discrete

Track 1 lesson 9 said integrals are area under a curve. Bond pricing is the discrete version: each payment is a "slice" with height = (cash flow's PV) and width = 1 year. Add the slices, you get the total area, which equals the bond's price. Same picture, different rendering.

For continuous-cash-flow products (some derivatives, swaps), the math becomes a real integral. For discrete coupon bonds, it's a sum. The shape doesn't change.

Worked example

5% 3-year ₩10,000 bond. Assume YTM = 6%. Cash flows: ₩500 at year 1, ₩500 at year 2, ₩500 + ₩10,000 = ₩10,500 at year 3.

Price = ₩500/(1.06) + ₩500/(1.06)² + ₩10,500/(1.06)³

= ₩471.70 + ₩444.99 + ₩8,815.62 ≈ ₩9,732

So this bond's market price is ₩9,732 — below ₩10,000 face value because YTM (6%) exceeds coupon (5%). The discount captures the "extra return" the buyer gets from buying below par.

What changes the price

The fixed cash flows (C and F) don't change for a given bond. So price changes are driven by changes in the discount rate r. From last lesson:

r rises → each term's denominator grows → price falls
r falls → each term's denominator shrinks → price rises

This is the price-yield inverse relationship made explicit at the formula level. Every cash flow gets re-discounted when r moves; the longer-dated cash flows get the biggest impact (because their denominators have the highest powers of (1+r)). Lesson 7-4 (duration) makes this precise.

The takeaway

Bond price = sum of discounted future cash flows. Coupons each year + face value at maturity, all discounted back to today using yield r. Same idea as DCF for stocks (Track 6 lesson 2), but with a known cash flow schedule and (usually) a clear maturity. The shape is the integral picture from Track 1, made discrete. Lesson 7-4 explores how sensitive the price is to changes in r.