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Lesson 1 of 8 · 13 min

Present value and discount (zero-coupon) bonds

A cash flow received later is worth less today, and a zero-coupon bond's price is simply its principal discounted back at the market rate.

Video lesson · 6 min. Watch it first, then use the text below to go deeper.

In short

Compounding and discounting are one equation: $FV_t = PV(1+r)^t$ , so $PV = FV_t/(1+r)^t$ .
Fixed-income cash flows follow three patterns: discount, periodic interest (coupon) and level payment.
A discount (zero-coupon) bond pays only its principal at maturity. Its whole return is the gap FV − PV.
Price and yield move in opposite directions. If the yield does not change, the price drifts toward par as maturity approaches.
With a negative yield the price is above par and falls (amortizes) to par over time.
More frequent compounding at the same stated rate lowers the PV. Continuous compounding gives the lowest PV: $PV = FV e^{-rt}$ .
The effective annual rate $(1 + R_s/m)^m - 1$ rises with compounding frequency, by ever smaller steps, toward $e^{R_s} - 1$ .

Why a later cash flow is worth less

Money you hold today can be invested to earn a return. So a promise to receive the same amount later is worth less than having it now. The time value of money converts cash flows at different dates into one comparable figure, usually today's present value (PV).

The link between dates is the discount rate $r$ per period. Compounding moves money forward: $FV_t = PV(1+r)^t$ . Discounting runs the same equation backwards: $PV = FV_t/(1+r)^t$ , which you can also write as $PV = FV_t(1+r)^{-t}$ .

The right $r$ depends on the instrument, on when its cash flows arrive and on how risky they are. A riskier or more distant cash flow normally gets a higher rate, and so a lower PV.

If compounding happens continuously, the factor $(1+r)^t$ becomes $e^{rt}$ . Then $FV_t = PV e^{rt}$ and $PV = FV_t e^{-rt}$ . You will need this form for forward exchange rates later in this module.

Discounting asks one question: how much would I have to invest today, at rate r, to end up with exactly this future amount? That amount is what a rational buyer will pay.

Three cash flow patterns in fixed income

A fixed-income instrument, such as a bond or a loan, is a contract. The investor lends money now and the issuer promises specified payments later. Its discount rate is an interest rate, usually quoted as the yield-to-maturity (YTM).

Almost every bond or loan fits one of three patterns. Identify the pattern first, because it tells you which formula to use.

Pattern	Investor pays	Investor receives	Example
Discount	PV today	One principal payment (FV) at maturity	Zero-coupon bond, stripped principal
Periodic interest	PV today	Coupons (PMT) each period, plus FV with the last coupon	Ordinary coupon bond
Level payment	PV today	Equal payments (A), each mixing interest and principal	Mortgage, amortizing loan, annuity

Pricing a discount bond

A discount bond pays no interest along the way. The investor buys it below its principal and receives the full principal at maturity. With no coupons, it is also called a zero-coupon bond.

Its price is the principal discounted over the full term: $PV = FV/(1+r)^t$ . The difference FV − PV is all the interest the investor earns. There is no other source of return.

Many governments let individual coupon or principal payments be separated from a coupon bond and traded on their own. Each of these stripped cash flows is a zero-coupon bond and is priced the same way.

The 8-year zero at 4.5%: one payment out today, one payment back at maturity, nothing in between. The whole 29.68 gap is the investor's interest.

How the price behaves: time, rates and negative yields

Passage of time. If the yield stays the same, a zero's price rises each year toward par, because fewer periods of discounting remain. This upward drift (accretion) is the interest the investor earns.

Change in yield. A higher yield lowers the price at once, and a lower yield raises it. Price and yield always move in opposite directions. The longer the time to maturity, the bigger the price change.

Negative yields. When $r < 0$ , the factor $(1+r)^t$ is below 1, so the price is above par. The investor knowingly pays more than they will get back. If the yield does not change, this premium price falls (amortizes) toward par as maturity approaches.

Compounding frequency. Take the same stated annual rate but discount semiannually: half the rate, twice the periods. The price is slightly lower than with annual discounting, because $(1+r/2)^{2t} > (1+r)^t$ for any positive $r$ . Continuous discounting gives the lowest price of all.

Read right to left as time passes: every curve is pulled to 100 at maturity. A higher yield sits lower and the gap is widest for long maturities; a negative yield keeps the price above par until it slides down to 100.

Pull to par: whatever the yield, a zero is worth exactly its principal at maturity. A discount price climbs up to par, and a premium price (negative yield) slides down to par.

Stated rate versus effective annual rate

Banks and bond markets quote a stated (quoted) annual rate $R_s$ together with a compounding frequency m. The rate actually applied each period is the periodic rate $R_s/m$ , and it is applied m times a year.

The effective annual rate (EAR) is what one year of that compounding really earns: $EAR = (1 + R_s/m)^m - 1$ . It is the same calculation as annualizing a periodic return (Rates and Returns module). With annual compounding (m = 1) the EAR equals the stated rate; with any more frequent compounding it is higher.

Holding $R_s$ fixed, the EAR rises as m rises, but each step adds less than the one before. The limit is continuous compounding, where the EAR is $e^{R_s} - 1$ . So compare deposits or loans by their EAR, never by their stated rates alone.

Stated annual rate of 8% at different compounding frequencies
Compounding	m	EAR
Annual	1	8.000%
Semiannual	2	8.160%
Quarterly	4	8.243%
Monthly	12	8.300%
Daily	365	8.328%
Continuous	∞	$e^{0.08} - 1$ = 8.329%

The first extra compounding step adds 16 basis points; going from monthly to continuous adds less than 3. The EAR climbs ever more slowly toward the

e^{0.08} - 1

ceiling.

Formulas

Future value (discrete compounding): $FV_t = PV(1+r)^t$
r = rate per period, t = number of periods.
Discount bond price: $PV = \dfrac{FV_t}{(1+r)^t} = FV_t(1+r)^{-t}$
FV = principal paid at time t. Use per-period r and t (halve r and double t for semiannual).
Continuous compounding: $FV_t = PV e^{rt} \qquad PV = FV_t e^{-rt}$
r = continuously compounded annual rate, t in years.
Effective annual rate: $EAR = \left(1 + \dfrac{R_s}{m}\right)^{m} - 1 \qquad EAR_{\text{continuous}} = e^{R_s} - 1$
R_s = stated annual rate, m = compounding periods per year. EAR = stated rate only when m = 1.

Worked examples

Example 1 Pricing a zero-coupon bond

An 8-year zero-coupon government bond has a principal of EUR 100. The market discount rate is 4.5% a year (annual compounding). What should an investor pay?

Example 2 Time passes, then the yield changes

Use the bond from Example 1. (a) If the yield is still 4.5% three years from now, what is the price? (b) If instead the yield jumps to 5.5% right after purchase, what is the price?

Example 3 Compounding frequency and a negative yield

(a) A 10-year zero pays GBP 100. The stated annual rate is 6%. Find the price with annual, semiannual and continuous discounting. (b) A 6-year CHF zero yields −0.30%. What is its price, and is it above or below par?

On the exam

Expect a direct zero-coupon price, or a question on how the price behaves as time passes, as yields change or when yields are negative. Semiannual versus annual discounting is a favourite source of wrong options.

Using simple interest, 100 / (1 + r × t), to price a multi-year zero quoted on a compound yield. It overstates the price. (Short money-market bills quoted on a simple annual yield are the exception: they are priced as face ÷ (1 + y × days/365).)
Assuming a zero always trades below par. With a negative yield it trades above par.
Mixing per-period and annual figures. With semiannual discounting, halve the rate and double the periods.
Confusing the two price effects: time passing (pull to par) and a change in yield (inverse move).
Treating a stated rate as the effective rate. With monthly compounding, 4% stated earns more than 4% a year.

Check yourself

Q1A 12-year zero-coupon bond has a principal of GBP 100 and a yield of 3.8% (annual compounding). Its price is closest to:
Q2An investor holds a zero-coupon bond with a positive yield. If the yield does not change, the bond's price over its remaining life will most likely:
Q3A 5-year EUR zero-coupon bond is issued at a yield of −0.40%. Its price per EUR 100 of principal is closest to:
Q4Which statement about a zero-coupon bond issued at a negative yield is least accurate?
Q5A cash flow of NOK 1,000 is due in 4 years. With a continuously compounded rate of 3%, its present value is closest to:
Q6Compared with annual discounting at the same stated annual rate, semiannual discounting of a zero-coupon bond's principal gives a price that is:

0/6Check yourself

Answer the 6 questions: +10 XP for each one you get right first time.

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