Bond Pricing

A bond is the present value of its cash flows

Finance FundamentalsBonds & Fixed IncomeFree preview

⏱️ About 16 min

A $1,000 bond pays a 5% coupon for three years, but the market demands a 6% return. Why does it trade for only $973.27, less than its face value?

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The big idea: A bond price is the present value of every cash flow it promises - the coupons as an annuity plus the face value as a lump sum - all discounted at the yield the market requires.

🎯 By the end, you'll be able to

Write the bond price as the present value of coupons plus face value.
Compute the price of a coupon bond at a given required yield.
Explain why price falls when the required yield rises above the coupon rate.

📎 Helpful to know first

The Module 1 lessons on present value and annuities.

A bond is a package of cash flows

A coupon bond promises two things: a stream of equal coupon payments and a single face-value repayment at maturity. Its fair price today is simply the present value of that whole package, discounted at the yield $y$ investors require.

\[ P = C \times \dfrac{1 - (1 + y)^{-n}}{y} + \dfrac{F}{(1 + y)^{n}} \]

Reading the formula

$C$ is the coupon paid each period, $y$ is the required yield per period, $n$ is the number of periods, and $F$ is the face value repaid at maturity. The first term is the present value of the coupons (an annuity); the second is the present value of the face value (a lump sum).

⚠️ Price and yield move in opposite directions

Because the yield sits in the denominator, a higher required yield lowers every discounted cash flow, so the price falls. This inverse relationship is the single most important fact about bonds.

🎮 Bond Price Explorer LIVE

Predict first: Predict first: if the required yield rises above the coupon rate, will the bond trade above or below its face value?

Bond price as the required yield changes; the face value is marked for reference.

📝 Worked example: A $1,000 face-value bond pays a 5% annual coupon ($50) for 3 years. The market requires a 6% yield. What is the price?

1. Coupon $C = 50$, yield $y = 0.06$, periods $n = 3$, face $F = 1000$.
2. Annuity factor: $(1 - 1.06^{-3})/0.06 = 0.160381/0.06 = 2.673017$.
3. PV of coupons: $50 \times 2.673017 = 133.65$.
4. PV of face: $1000 \times 1.06^{-3} = 1000 \times 0.839619 = 839.62$.
5. Price: $133.65 + 839.62 = 973.27$.

✓ Price = $973.27 (a discount bond, because the 5% coupon is below the 6% yield)

✏️ Practice: The same 5% coupon, 3-year, $1,000 bond, but now the market requires only 4%. Is the price above or below $1,000, and roughly what is it?

💡 Hint

With $y = 0.04$: price $= 50 \times (1 - 1.04^{-3})/0.04 + 1000 \times 1.04^{-3}$.

Answer

Above par: price = $1,027.75

Check your understanding

1. When the required yield rises above the coupon rate, the bond trades:

A yield above the coupon rate discounts the cash flows more heavily than the coupons compensate, so the price falls below face value.

2. In the bond price formula, the term $F/(1+y)^{n}$ represents:

It discounts the single face-value repayment back to today; the coupons are the annuity term.

✅ Key takeaways

A bond price is the present value of its coupons plus its face value.
Price $= C(1-(1+y)^{-n})/y + F/(1+y)^{n}$.
Price and required yield move in opposite directions.

➡️ If you know the price the market sets, you can invert the formula to find the yield it implies - that yield is the yield to maturity, next.

Want to test yourself on this? Try the Finance Fundamentals Aptitude test →