Finance-Theory · Unit 5 · Video 3 · Interactive Practice

Pricing a Coupon Bond: Why It Sells at Par When the Market Rate Equals the Coupon Rate

IKey Formulas

Formula	Name	What it says
$C = c \times F$	Coupon	The coupon rate $c$ is written into the bond and fixes the payment: $C = 0.05 \times 1{,}000 = 50$ a year, whatever the market does
$P_0 = \dfrac{C}{1+r} + \dfrac{C}{(1+r)^2} + \dfrac{C+F}{(1+r)^3}$	Price of a three-year coupon bond	Each dated cash flow discounted to today at the market rate $r$ . At $r = 5\%$ the terms are 47.62, 45.35 and 907.03 to the cent, and the price is exactly $1,000
$r = c \;\Longrightarrow\; P_0 = F$	Par	$rF = C$ : a year's interest at the market rate on the face value is exactly the coupon. With $r \lt c$ every discount factor is larger and $P_0 \gt F$ , above par; with $r \gt c$ , $P_0 \lt F$ , below par
$d_t = \dfrac{1}{(1+r)^t}$	Discount factor	Today's price of a dollar paid at date $t$ : the exchange rate between dated dollars that an auction of the bond works out

Key Insight: The coupon rate fixes the cash flows; the market rate prices them. The bond sells at par exactly when the two rates are equal, because then one year's interest on $1,000 is the $50 coupon.

IIThe Market Rate Sets the Price

The 5% coupon fixes the cash flows; only the market rate $r$ that discounts them moves the price.

Coupon rate 5% fixed by the bond Market rate r 5.00%

IIIWhy Exactly Par: Roll Back One Year at a Time

At $r = 5\%$ a year's interest on $1,000 is the $50 coupon, so each step back lands on $1,000.

Market rate r Step 5

💡 Nothing in the argument depends on the bond having three years: at $r = c$ , rolling back from any maturity lands on the face value at every coupon date, so the bond sells at par.

IVA Second Answer: Auction It Off

An auction clears at the present value: each dated payment converted into today's dollars at the market's discount factors.

Your bid $1,000

💡 The market need not discount every date at the same rate: it prices a dollar at each date separately, and zero-coupon bonds make those prices directly visible.

VQuiz Questions

Problem 1 · Price a Three-Year Bond

Given: a three-year bond with face value $1,000 and a 6% annual coupon, and a market rate of $r = 4\%$ . Find its price today.

✅ Correct! The coupon is

C = 0.06 \times 1{,}000 = 60

, and each payment is discounted at the market rate:

P_0 \approx 57.69 + 55.47 + 942.34 = 1{,}055.50

. The market rate is below the coupon rate, so the bond sells above par.

❌ That discounts at the coupon rate. The 6% coupon fixes the $60 payments; the market rate of 4% is what discounts them. Discounting at the coupon rate returns par by construction.

❌ The last payment is coupon plus principal. At date 3 the bond pays $60 + $1,000 = $1,060, not $1,000.

❌ Each payment has its own date. Discounting all $1,180 over three years treats the coupons as if they arrived at date 3. The date-1 coupon is discounted one year and the date-2 coupon two.

Show solution

The coupon rate fixes the payments: $C = 0.06 \times 1{,}000 = 60$ at dates 1 and 2, and $C + F = 1{,}060$ at date 3. The market rate discounts them:

P_0 = \frac{60}{1.04} + \frac{60}{(1.04)^2} + \frac{1{,}060}{(1.04)^3} \approx 57.69 + 55.47 + 942.34 = 1{,}055.50

The price is $1,055.50. The market rate (4%) is below the coupon rate (6%), so the bond sells above par: each year it pays $60 when a year's market interest on $1,000 is only $40.

Problem 2 · The Rate Falls, the Coupon Does Not

Given: you hold the video's bond (face value $1,000, 5% coupon rate, three years to maturity), bought at par when the market rate was 5%. The same day, the market rate falls to 4%.

What does the bond pay at date 1?

And what happens to its price?

✅ Correct! The coupon is written into the bond, so it stays $50. The lower market rate shrinks every denominator:

P_0 = \frac{50}{1.04} + \frac{50}{(1.04)^2} + \frac{1{,}050}{(1.04)^3} \approx 1{,}027.75

, above par.

❌ The coupon does not follow the market. $50 is 5% of the face value, printed on the bond; the new 4% rate only changes how the market discounts that $50.

❌ That is a present value, not a payment. The date-1 coupon is worth $48.08 today at 4% (and $47.62 at 5%), but the bond still hands over $50 at date 1.

❌ The face value is fixed, but the price is not. The price is the present value of $50, $50 and $1,050, and every one of those terms grows when the discount rate falls.

❌ The direction is reversed. A lower rate makes each denominator

(1+r)^t

smaller, so each discount factor, and each present value, is larger.

❌ Close, but the extra $10 a year must be discounted too. The bond pays $50 when market interest on $1,000 is $40, and those three extra $10 payments are worth

\frac{10}{1.04} + \frac{10}{(1.04)^2} + \frac{10}{(1.04)^3} \approx 27.75

today, not 30.

Show solution

The payment. The coupon rate is part of the contract: $C = 0.05 \times 1{,}000 = 50$ at dates 1 and 2, and $1,050 at date 3, wherever market rates go.

The price. The market now discounts those same cash flows at $r = 4\%$ :

P_0 = \frac{50}{1.04} + \frac{50}{(1.04)^2} + \frac{1{,}050}{(1.04)^3} \approx 1{,}027.75

Every term is larger than at 5%: to the cent, 48.08 against 47.62, 46.23 against 45.35, and 933.45 against 907.03 (rounded separately they add to 1,027.76; unrounded, to 1,027.75). The bond now sells above par.

Another route to the premium. Each year the bond pays $50 when a year's market interest on $1,000 is only $40. The three extra $10 payments are worth

P_0 - F = \frac{10}{1.04} + \frac{10}{(1.04)^2} + \frac{10}{(1.04)^3} \approx 27.75

so the price is about $1,000 + $27.75 = $1,027.75, the same answer.

Problem 3 · Price It From the Market's Discount Factors

Given: the market prices dated dollars at $0.96 today for a dollar paid at date 1, $0.92 for a dollar at date 2 and $0.88 for a dollar at date 3. Find the price of the three-year bond with face value $1,000 and a 5% annual coupon.

✅ Correct! Each payment is converted at its own date's price:

P_0 = 50(0.96) + 50(0.92) + 1{,}050(0.88) = 48 + 46 + 924 = 1{,}018

. That sum is exactly what an auction of the bond computes.

❌ A 5% coupon does not guarantee par. The price is set by the market's discount factors, and these are all larger than the 5% ones (

\frac{1}{1.05} \approx 0.952

\frac{1}{(1.05)^2} \approx 0.907

\frac{1}{(1.05)^3} \approx 0.864

), so every payment is worth more today and the bond sells above par.

❌ The date-3 payment is $1,050, not $1,000. The last coupon arrives together with the principal.

❌ Each date has its own exchange rate. Applying the date-3 factor to all $1,150 undervalues the coupons that arrive sooner.

Show solution

A discount factor is today's price of a dollar paid at a later date, so each dated payment converts to today's dollars by multiplication:

P_0 = 50 \times 0.96 + 50 \times 0.92 + 1{,}050 \times 0.88 = 48 + 46 + 924 = 1{,}018

The bond sells for $1,018, above par. Each of the market's factors is larger than the corresponding 5% factor (0.952, 0.907 and 0.864 to three places), which is the same as saying the market discounts at less than the 5% coupon rate: a date-1 dollar priced at $0.96 corresponds to $r_1 = \frac{1}{0.96} - 1 \approx 4.17\%$ .

No single interest rate is needed. The auction prices each date's dollars on their own, and the present value uses exactly those prices.

Problem 4 · A Ten-Year Bond at Its Coupon Rate

Given: a ten-year bond with face value $1,000 and a 6% annual coupon, when the market rate is also 6%. Find its price without writing out ten terms.

✅ Correct! With

r = c

, a year's interest on $1,000 at 6% is the $60 coupon, so rolling back from year 10 lands on $1,000 at every date. The annuity formula agrees: the coupons are worth 441.61 and the principal 558.39, which sum to 1,000.00.

❌ That adds the cash flows without discounting. Ten coupons of $60 plus the $1,000 principal make $1,600, but a dollar at date 10 is worth much less than a dollar today.

❌ That is only the principal,

\frac{1{,}000}{(1.06)^{10}} \approx 558.39

. The ten $60 coupons are worth another 441.61 today.

❌ That is only the coupons, an annuity worth

\frac{60}{0.06}\left(1 - \frac{1}{(1.06)^{10}}\right) \approx 441.61

. The $1,000 principal at date 10 is worth another 558.39.

Show solution

Split the bond into an annuity of ten $60 coupons and the principal at date 10. Because the coupon equals a year's market interest on the face value, $C = rF$ , the annuity's $\frac{C}{r}$ is exactly $F$ :

P_0 = \frac{C}{r}\left(1 - \frac{1}{(1+r)^{10}}\right) + \frac{F}{(1+r)^{10}} = F\left(1 - \frac{1}{(1+r)^{10}}\right) + \frac{F}{(1+r)^{10}} = F

Numerically, $\frac{60}{0.06} = 1{,}000$ and $\frac{1}{(1.06)^{10}} \approx 0.558395$ , so the coupons are worth about 441.61 and the principal about 558.39: $1,000 in total. Nothing here used the number of years, so the same holds at any maturity.

It is the roll-back argument again: with $r = c$ the bond is worth $1,000 just after every coupon, all the way back to today.

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