Lecture 4: Bonds, yield to maturity and duration

A bond is the simplest security to value, because its payments are fixed in advance. This lecture shows how to compute a bond's price from its yield, what happens between coupon dates, and how to measure a bond's sensitivity to interest rates with duration. It ends with the risks of bonds and credit ratings.

Start here if this is new to you

Bond. A loan cut into tradable pieces. The issuer (a government or company) borrows from many investors at once. Each bond promises fixed payments on fixed dates.
Nominal value (face value, par value). The amount repaid at maturity, usually €1,000 per bond. Coupons are computed on this amount.
Coupon rate. The yearly interest as a percentage of the nominal value. An 8% coupon on €1,000 pays €80 a year. The coupon is the payment itself (€80).
Time to maturity. The number of years until the nominal value is repaid.
Frequency of coupon payments. Annual (once a year) or semiannual (twice a year, each coupon being half the yearly amount). US bonds mostly pay semiannually; the examples here are annual unless stated.
Yield to maturity (YTM). The interest rate that makes the present value of the bond's payments equal to its price: the return you earn if you buy at that price and hold to maturity. It is the discount rate for bonds, and it is set by the market according to the bond's risk.
Price and yield move in opposite directions. The payments are fixed, so if investors demand a higher yield, they will only pay a lower price, and vice versa. This single fact explains most of this lecture.

The cash flows of a bond

To determine a bond's cash flows you need four things: the nominal value, the coupon rate, the time to maturity and the frequency of coupon payments.

Worked example (from the slides)

A bond has a nominal value of €1,000, a coupon rate of 8%, matures in 3 years and pays coupons annually. Its cash flows are:

€80 in 1 year
€80 in 2 years
€1,080 in 3 years (the last coupon plus the nominal value)

The price of a bond

The price is the present value of the cash flows, discounted at the yield to maturity. For a bond with coupon C, nominal value F, n years to maturity and yield y:

P = \frac{C}{1 + y} + \frac{C}{( 1 + y ) ^{2}} + \dots + \frac{C + F}{( 1 + y ) ^{n}} = \frac{C}{y} (1 - \frac{1}{( 1 + y ) ^{n}}) + \frac{F}{( 1 + y ) ^{n}}

The coupons are an annuity and the nominal value a single amount, so the second form is just lecture 2's formulas combined. For an exam with three or four cash flows it is easiest to discount each cash flow separately.

Worked example (from the slides)

Suppose the bond above has a YTM of 10%. Its price is

P = \frac{80}{1.10} + \frac{80}{1.1 0 ^{2}} + \frac{1 , 080}{1.1 0 ^{3}} = 72.73 + 66.12 + 811.42 = C = 950.26

The bond pays 8% coupons but investors demand 10%, so it sells below par (at a discount). The buyer gets 8% in coupons plus a capital gain from 950.26 to 1,000 at maturity, which together give 10%. If the YTM were 8%, the price would be exactly €1,000 (at par); if it were 6%, the price would be above 1,000 (at a premium).

The YTM is determined by the bond's risk: a safe government bond has a low yield and therefore a high price; a risky corporate bond a high yield and a low price.

Buying between coupon dates: accrued interest, clean and dirty price

Worked example (from the slides)

Suppose you buy the same bond six months from now, and it still has a 10% YTM. The payments are now 0.5, 1.5 and 2.5 years away, so the discounting uses those exponents:

P = \frac{80}{1.1 0 ^{0.5}} + \frac{80}{1.1 0 ^{1.5}} + \frac{1 , 080}{1.1 0 ^{2.5}} = C = 996.64

This is the price you would actually pay to buy the bond (the dirty price or full price). It is higher than 950.26 because the first coupon is now only six months away, and half a year of interest has already been "earned".

Bond prices in the press are clean prices: they do not include accrued interest. Accrued interest is the interest due between the last coupon payment and now. Here, six months of the €80 annual coupon have accrued:

accrued interest = 80 \times \frac{180}{360} = C = 40

The bond's quoted (clean) price would be 996.64 − 40 = €956.64.

In plain words

When you buy a bond halfway between coupon dates, the seller has earned half the coupon but you will receive all of it. So you pay the seller for the half he earned: the accrued interest. Quoted prices leave it out so that they do not jump every coupon date; what you pay is the quoted price plus accrued interest.

Duration: how sensitive is a bond to interest rates?

A bond's price falls when yields rise. But by how much? That depends on how far in the future its payments lie: money due in 10 years is discounted much more heavily than money due next year, so a change in the discount rate moves its present value more. Duration captures this: it is the average time until the bond's payments, where each payment is weighted by its share in the bond's price.

A bond's duration is calculated in three steps:

Calculate the present value of every cash flow.
Divide each present value by the bond price (these ratios are the weights; they add up to 1).
Multiply each ratio by the time until that cash flow, and add everything up.

Duration = t \sum T_{t} \times \frac{P V ( C _{t} )}{P}

where PV(C_t) is the present value of each cash flow and T_t the time to it.

Worked example (from the slides)

The 3-year 8% bond with YTM 10% and price 950.26:

T	C	PV(C)	PV(C) / P	(PV(C) / P) × T
1	80	72.73	0.076	0.076
2	80	66.12	0.070	0.139
3	1,080	811.42	0.854	2.562
		950.26	1.000	2.777

Duration = 2.777 years. It is a bit less than the maturity of 3 years because some money (the coupons) arrives earlier. For a zero-coupon bond, which has only one payment at maturity, duration equals maturity.

What duration tells you. The duration of a bond shows its sensitivity to interest rate changes. For the bond above, a half-point change in yield moves the price by about 1.25% in each direction:

Change in YTM	Price	Change (%)
−0.5% (to 9.5%)	962.37	+1.274%
+0.5% (to 10.5%)	938.37	−1.251%
Sum of the two		2.525%

The sum of the two changes, 2.525% per percentage point of yield, is no coincidence: it equals the modified duration.

Modified duration and the price-change approximation

M D = \frac{Duration}{1 + Y T M} = \frac{2.777}{1.10} = 2.525

Modified duration (MD) is widely used in the bond market as the main indicator of a bond's risk. It gives the approximate percentage change in the price for a change in the yield:

\frac{Δ P}{P} \approx - M D \times (Y T M_{t} - Y T M_{t - 1})

The minus sign is the inverse relation between price and yield. A bond with MD = 2.525 loses about 2.525% of its price for every percentage point rise in its yield.

Worked example (from the slides)

The YTM moves from 10% to 10.5%. The bond's price will drop by approximately

\frac{Δ P}{P} = - 2.525 \times (0.105 - 0.10) = - 1.26%

and will become about 950.26 × (1 − 0.0126) = €938.27. The exact price at 10.5% is €938.37, so the approximation is off by 10 cents.

How good is the approximation? For the annual-coupon bond:

YTM	Exact price	Approximation	Difference
9%	974.69	974.26	0.43
9.5%	962.37	962.26	0.11
10%	950.26	950.26	0
10.5%	938.37	938.27	0.10
11%	926.69	926.27	0.42

The error grows with the size of the yield change, and it is always in the same direction: the true price is above the approximation. The reason is convexity: the true price-yield relation is a curve that bends upward, while the MD approximation is a straight line touching the curve at the current yield. The more convex the curve, the larger the error. For small yield changes the straight line is good enough; for large changes you should compute the exact price.

Worked example (from the slides): a zero-coupon bond

A zero-coupon bond matures in 5 years, has a nominal value of €1,000 and a YTM of 10%. It pays nothing until maturity, so its price is a single discounted amount:

P = \frac{1 , 000}{1.1 0 ^{5}} = C = 620.92

Its duration is 5 (one payment, in 5 years), so MD = 5 / 1.10 = 4.545. If the YTM becomes 12.2%, the estimated price change is

\frac{Δ P}{P} \approx - 4.545 \times (0.122 - 0.10) = - 4.545 \times 0.022 = - 10%

The exact price at 12.2% is 1,000 / 1.122⁵ = €562.39, a real change of 562.39 / 620.92 − 1 = −9.4%. The approximation overstates the loss by 0.6 percentage points, again because of convexity, and the error is larger than before because the yield change (2.2 points) is larger.

Bond risks

Interest rate risk: if market yields rise, the price of your bond falls. Measured by duration. Long bonds are riskier than short bonds.
Credit risk (default risk): the issuer may not pay. Assessed by credit rating agencies such as S&P and Moody's (table below).
Reinvestment risk: the YTM assumes that you can reinvest all coupons at the YTM. If rates fall, your coupons earn less and your realised return falls short of the YTM. Note that this works against interest rate risk: falling rates raise the bond's price but hurt the reinvested coupons.
Call risk: if the bond's terms allow the issuer to repay early (a call provision), it will do so exactly when rates have fallen and you would rather keep the bond.
Inflation risk: the payments are fixed in euros; unexpected inflation reduces what those euros buy.

Credit ratings

Moody's	S&P	Meaning
Aaa	AAA	Very good credit quality, minimum risk of default
Aa	AA
A	A
Baa	BBB	Average quality debt
Ba	BB	Speculative debt, high default risk
B	B
Caa	CCC
Ca	CC
C	C
D	D	In default

Bonds rated Baa/BBB and above are called investment grade; below that they are speculative grade or "junk" bonds, which must pay a much higher yield to find buyers. Many pension funds are only allowed to hold investment grade bonds.

Exercises from the slides, fully worked out

Exercise 1: a 5-year zero-coupon bond

YTM 6%, nominal value €1,000. Price and duration?

P = \frac{1 , 000}{1.0 6 ^{5}} = \frac{1 , 000}{1.3382} = C = 747.26

A zero-coupon bond has a single cash flow at maturity, so its duration is 5 years (the weight of the only payment is 1, times T = 5). Modified duration: 5 / 1.06 = 4.717.

Exercise 2a: a 2-year bond with semiannual coupons

Nominal €1,000, coupon 4% paid semiannually, annualized return 8%. Price and duration?

With semiannual payments, work in half-years: the coupon is 4% / 2 = 2% of 1,000 = €20 per half-year, the yield is 8% / 2 = 4% per half-year, and there are 2 × 2 = 4 periods. Cash flows: 20, 20, 20 and 1,020.

P = \frac{20}{1.04} + \frac{20}{1.0 4 ^{2}} + \frac{20}{1.0 4 ^{3}} + \frac{1 , 020}{1.0 4 ^{4}} = 19.23 + 18.49 + 17.78 + 871.90 = C = 927.40

Duration (in half-years), with the weights PV/P:

D = 1 \times \frac{19.23}{927.40} + 2 \times \frac{18.49}{927.40} + 3 \times \frac{17.78}{927.40} + 4 \times \frac{871.90}{927.40} = 3.879 half-years

In years: 3.879 / 2 = 1.94 years. Modified duration: 1.94 / 1.04 = 1.865 (divide by 1 plus the yield per period).

Exercise 2b: the price if the YTM drops to 7.5%

Using modified duration: the yield falls by 0.5 percentage points, so

\frac{Δ P}{P} \approx - 1.865 \times (0.075 - 0.08) = + 0.93%, P \approx 927.40 \times 1.0093 = C = 936.05

Actual price: the yield per half-year is now 7.5% / 2 = 3.75%:

P = \frac{20}{1.0375} + \frac{20}{1.037 5 ^{2}} + \frac{20}{1.037 5 ^{3}} + \frac{1 , 020}{1.037 5 ^{4}} = C = 936.10

The approximation is within 5 cents, as expected for a small yield change on a short bond.

Exercise 3: which bond to buy if you expect a rate cut?

Choice: 2-year bonds with a 3% annual coupon and YTM 5%, or 3-year bonds with a 6% coupon and YTM 6%.

If interest rates fall, bond prices rise, and the bond with the higher duration rises the most. So compute both durations.

2-year, 3%, YTM 5%: price = 30/1.05 + 1,030/1.05² = 28.57 + 934.24 = 962.81; duration = (1 × 28.57 + 2 × 934.24) / 962.81 = 1.97 years; MD = 1.97 / 1.05 = 1.88.
3-year, 6%, YTM 6%: price = 1,000 (coupon equals yield, so it trades at par); duration = (1 × 56.60 + 2 × 53.40 + 3 × 890.00) / 1,000 = 2.83 years; MD = 2.83 / 1.06 = 2.67.

Buy the 3-year bonds. If yields fall by 1 percentage point, the 2-year bond gains about 1.9% and the 3-year bond about 2.7%. The longer bond is the bigger bet on falling rates. (If you expected rates to rise, the answer would reverse: the short bond loses less.)

In plain words

Lecture 4 in one breath: price = present value of coupons and nominal value at the YTM; yields up means prices down. Duration is the weighted average time to the payments; modified duration = duration / (1 + YTM) tells you the percentage price change per percentage point of yield. The approximation is a straight line; convexity makes the true price a bit higher than the line predicts.

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