Lesson 1 of 6 · All lessons in this module

Free lesson: the first lesson of every module is open to everyone.

Lesson 1 of 6 · 12 min

Pricing a bond with a market discount rate

A bond's price is its promised coupons and principal discounted at the market discount rate, and comparing that rate with the coupon rate tells you at once whether the bond trades at par, a discount or a premium.

Video lesson · 7 min. Watch it first, then use the text below to go deeper.

In short

Bond pricing is discounted cash flow analysis: price = PV of every coupon + PV of the face value.
The market discount rate is the return investors require for the bond's risk; it is also called the required yield or required rate of return.
Coupon rate = market discount rate → par; coupon rate < rate → discount; coupon rate > rate → premium.
A discount compensates the buyer for a deficient coupon; a premium charges the buyer for an excess coupon.
Work in periods: semiannual coupons mean coupon ÷ 2, rate ÷ 2 and years × 2. Annual rates are the periodic rate × periods per year.

Price = present value of the promised cash flows

A fixed-rate bond promises a known list of cash flows: a coupon (PMT) each period and the face value (FV, also called par or principal) at maturity. Its price is the present value of that list. Throughout this module we assume the issuer pays everything in full and on time.

Each cash flow is discounted at the market discount rate r: the rate of return investors require given the bond's risk. One coupon paid in t periods is worth $PMT/(1+r)^t$ today. Add up all of them, plus the face value discounted over the full N periods, and you have the price.

Prices are normally stated per 100 of par value. A price of 97.25 means the bond costs 97.25% of its face value. On the BA II Plus enter N, I/Y, PMT and FV = 100, then compute PV; it shows as negative because it is cash you pay out.

The 4-year 5% bond discounted at 6%. Each cash flow shrinks more the later it arrives, and the four present values add up to the price, 96.535. Most of the value sits in the final coupon plus principal.

The coupon is fixed when the bond is issued. The market discount rate changes every day. The price is the variable that moves to reconcile the two.

Par, discount and premium

If the coupon rate equals the market discount rate, the coupons give investors exactly the return they require. On a coupon date the bond is then worth its face value: it trades at par. This is why most bonds are issued with a coupon close to the market rate at the time.

Suppose rates rise after issue. A new bond would now pay a higher coupon, so the old bond's coupon is deficient. Nobody will pay par for it. Its price falls below 100 until the lower price makes up for the smaller coupons. The bond trades at a discount. (This 'discount' has nothing to do with the word in 'market discount rate'.)

If rates fall instead, the old coupon is excessive compared with what new bonds pay. Buyers compete for it and bid the price above 100: the bond trades at a premium. The premium is the extra price paid for coupons that are larger than the market requires.

You can measure the gap directly. The discount equals the PV of the coupon shortfall (market-rate coupon minus actual coupon) over the bond's life. The premium equals the PV of the coupon excess.

Fixed-rate bond on a coupon date
Price vs face value	Coupon rate vs market discount rate	Label
PV = FV	Coupon rate = market discount rate	Par
PV < FV	Coupon rate < market discount rate	Discount
PV > FV	Coupon rate > market discount rate	Premium

One bond, many market rates. Left of the coupon rate the coupon is excessive and the bond trades at a premium; right of it the coupon is deficient and the bond trades at a discount. The curve bows slightly: it is not a straight line.

Periods, periodic rates and annual quotes

The pricing formula works period by period, so the coupon, the rate and N must all refer to the same period. With semiannual coupons, divide the annual coupon by 2, divide the annual market discount rate by 2 and double the number of years.

Rates are quoted as annual figures unless stated otherwise. A periodic rate is annualized by multiplying it by the number of periods per year: a 2.2% semiannual rate is quoted as 4.4%. So you can make the par/discount/premium judgement either with periodic rates or with annual ones; the answer is the same.

If a question shows you the pricing equation itself, read the periodic coupon and the periodic rate straight from it. A coupon of 2 per 100 discounted at 2.5% per period is a deficient coupon, so the bond is below par without any arithmetic.

Formulas

Present value of one coupon: $PV = \dfrac{PMT_t}{(1+r)^t}$
PMT_t = coupon paid in t periods, r = market discount rate per period.
Bond price on a coupon date: $PV = \dfrac{PMT}{(1+r)^1} + \dfrac{PMT}{(1+r)^2} + \dots + \dfrac{PMT + FV}{(1+r)^N}$
FV = face value, N = periods to maturity. Use per-period coupon and rate.
Discount (or premium) as PV of the coupon gap: $FV - PV = \sum_{t=1}^{N} \dfrac{(r \times FV) - PMT}{(1+r)^t}$
Positive → discount (deficient coupon); negative → premium (excess coupon).

Worked examples

Example 1 One bond, three market rates

A 4-year bond pays a 5% annual coupon on a face value of 100. What is its price if the market discount rate is (a) 5%, (b) 6% and (c) 4%?

Example 2 Semiannual bond and the deficient coupon

A 6-year corporate bond pays a 3.6% coupon semiannually. The market discount rate is 4.4% (annual, semiannual compounding). What is the price, and how big is the discount explained by the coupon shortfall?

Example 3 Reading the pricing equation

An analyst writes: $PV = \frac{1.5}{1.0125} + \frac{1.5}{1.0125^{2}} + \dots + \frac{101.5}{1.0125^{12}}$ . Is the bond at par, a discount or a premium, and what are its annual coupon rate and market discount rate if coupons are semiannual?

On the exam

Expect a straight price calculation, often with semiannual coupons and an annual-pay distractor, or a stem asking whether a bond trades at par, a discount or a premium given its coupon and the market rate.

Converting only some inputs for a semiannual bond. Coupon, rate and N must all be per half-year.
Thinking 'discount bond' relates to the 'market discount rate'. A discount bond simply has a price below par.
Comparing a periodic rate with an annual coupon rate. Put both on the same basis first.
Assuming a bond issued at par stays at par. Its price moves whenever the market discount rate moves.

Check yourself

Q1An 8-year bond pays a 4.8% coupon semiannually. The market discount rate is 5.6%. The bond's price per 100 of par is closest to:
Q2A bond has a coupon rate of 3.25%. The market discount rate for its risk is 2.90%. On a coupon date, the bond most likely trades:
Q3A bond is priced as $PV = \frac{2}{1.025} + \frac{2}{1.025^{2}} + \dots + \frac{102}{1.025^{20}}$ . Relative to par, its price is most likely:
Q4A 3-year bond pays a 6% annual coupon. If the market discount rate is 4.5%, its price per 100 of par is closest to:
Q5Which statement about bond pricing is least accurate?

0/5Check yourself

Answer the 5 questions: +10 XP for each one you get right first time.

Liked this lesson? Get the rest free for 7 days

Create a free account to open the other 5 lessons in this module, with videos, flashcards that come back before you forget them, the question bank and full mocks. No card needed.

Start your 7-day free trial Log in