Answer First

A bond’s price equals the present value of its future cash flows because no investor will pay more than the discounted value of what the bond will pay them, and no seller will accept less.

The discount rate reflects the yield required by the market. If the bond were priced differently, arbitrage would occur until the price returned to PV.

• Future cash flows = coupons + face value
• Discount rate = yield required by investors
• No‑arbitrage forces price = present value

Step-by-Step: Why Bond Price = Present Value

Step 1: A Bond Is a Stream of Future Payments

A coupon bond pays:

• Periodic coupon payments
• Face value at maturity

Step 2: Money in the Future Is Worth Less Today

The time value of money states:

\[ \text{PV} = \frac{\text{Future Cash Flow}}{(1+r)^t} \]

Step 3: Investors Require a Yield

The yield reflects:

• interest rates
• inflation expectations
• credit risk
• opportunity cost

Step 4: No-Arbitrage Forces the Price

If the bond were priced above PV, investors would sell it. If priced below PV, investors would buy it. Arbitrage pushes the price to:

\[ P = \sum_{t=1}^{T} \frac{C}{(1+y)^t} + \frac{F}{(1+y)^T} \]

Step 5: Yield and Price Move Opposite

Higher yields → higher discounting → lower present value → lower bond price. Lower yields → higher bond price.

Numerical Example

Consider a bond with:

• Face value = 1000
• Coupon = 5% annually → 50 per year
• Maturity = 3 years
• Yield = 6%

Step 1: Discount Coupons

\[ PV_{\text{coupons}} = \frac{50}{1.06} + \frac{50}{1.06^2} + \frac{50}{1.06^3} \]

Step 2: Discount Face Value

\[ PV_{\text{face}} = \frac{1000}{1.06^3} \]

Step 3: Add Them

Total price ≈ 973. Because yield (6%) > coupon rate (5%), the bond sells at a discount.