1

Callable bonds

Bonds that may be repurchased by the issuer at a specified call price during the call period

A call usually occurs after a fall in market interest rates that allows issuers to refinance outstanding debt with new bonds.

Generally, the call price is above the bond’s face value. The difference between the call price and the face value is the call premium

Bonds are not usually callable during the first few years of a bond’s life. During this period the bond is said to be call-protected.

2

Investors are typically interested in knowing what the yield will be if the bond is called by the issuer at the first possible date. This is called yield to call (YTC).

Suppose that we have a 3-year, $1,000 par value, 6% semiannual coupon bond. We observe that the value of the bond is $852.48. The first call price is $1,060 in 2 years. Find YTC.

N= 4, FV = 1060 PMT=30 PV = -852.48 I/YR = 8.85*2= 17.707%

3

More on Bond Prices

2M2M

2YTM1

FV

2YTM1

11

YTM

CpriceBond

$1,107.41

2.081

1000

2.081

11

.08

90priceBond

5050

Now assume a bond has 25 years to maturity, a 9% coupon,and the YTM is 8%. What is the price? Is the bond selling atpremium or discount?

Now assume the same bond has a YTM of 10%. (9% coupon &25 years to maturity) What is the price? Is the bond selling at premium or discount?

$908.72

2.101

1000

2.101

11

.10

90priceBond

5050

4

More on Bond Prices (cont’d)

$1,040.55

2.081

1000

2.081

11

.08

90priceBond

1010

Now assume the same bond has a YTM of 10%. (9% coupon &5 years to maturity) What is the price? Is the bond selling at premium or discount?

$961.39

2.101

1000

2.101

11

.10

90priceBond

1010

Now assume the same bond has 5 years to maturity (9% coupon& YTM of 8%) What is the price? Is the bond selling at premium or discount?

5

More on Bond Prices (cont’d)

Where does this leave us? We found:Coupon Years YTM Price 9% 25 8% $1,107 9% 25 10% $ 908 9% 5 8% $1,040 9% 5 10% $ 961

$900

$950

$1,000

$1,050

$1,100

$1,150

8% 9% 10% 11%

25 years

5 years

6

Decreasing yields cause bond prices to rise, but long-term bonds increase more than short-term. Similarly, increasing yields cause long-term bonds to decrease in price more than short-term bonds.

7

Malkiel’s Theorems

Summarizes the relationship between bond prices, yields, coupons, and maturity:

all theorems are ceteris paribus:

1) Bond prices move inversely with interest rates.

2) The longer the maturity of a bond, the more sensitive is it’s price to a change in interest rates.

8

3) The price sensitivity of any bond increases with it’s maturity, but the increase occurs at a decreasing rate. A 10-year bond is much more sensitive to changes in yield than a 1-year bond. However, a 30-year bond is only slightly more sensitive than a 20-year bond .

9

Bond Prices and Yields (8% bond)

Time to Maturity

Yields 5 years 10 years 20 years

7 percent $1,041.58 $1,071.06 $1,106.78

9 percent 960.44 934.96 907.99

Price Difference $81.14 $136.10

67.7%

$198.79

46.1%

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4) The lower the coupon rate on a bond, the more sensitive is it’s price to a change in interest rates.

If two bonds with different coupon rates have the same maturity, then the value of the one with the lower coupon is proportionately more dependent on the face amount to be received at maturity.

As a result, all other things being equal, the value of lower coupon bonds will fluctuate more as interest rates change.

Put another way, the bond with the higher coupon has a larger cash flow early in its life, so its value is less sensitive to changes in the discount rate

11

20-Year Bond Prices and Yields

Coupon Rates

Yields 6 percent 8 percent 10 percent

6 percent $1,000.00 $1,231.15 $1,462.30

8 percent 802.07 1,000.00 1,197.93

10 percent 656.82 828.41 1,000.00

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5) For a given absolute change in a bond’s yield to maturity, the magnitude of the price increase caused by a decrease in yield is greater than the price decrease caused by an increase in yield

13

Malkiel’s Theorems (#5)8% coupon, 20 year bond

Yield Price Falls 2% Rises 2% Increase Decrease6% $1,231 $1,547 $1,000 25.70% 18.80%8% $1,000 $1,231 $828 23.10% 17.20%10% $828 $1,000 $699 20.80% 15.60%

Price when yieldPercentage price

change

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Duration– Price sensitivity tends to increase with time to

maturity– Need to deal with the ambiguity of the

“maturity” of a bond making many payments.– Duration measures a bond’s sensitivity to

interest rate changes. – More specifically, duration is a weighted

average of individual maturities of all the bond’s separate cash flows.

– The weight is the present value of the payment divided by the bond price.

15

Calculate a duration for a bond with three years until maturity. 8% of Coupon rate and yield.

16

Figure 10.3: Calculating bond duration

Discount Present Years x Present valueYears Cash flow factor value / Bond price

0.5 40 0.96154 38.4615 0.01921 40 0.92456 36.9822 0.0370

1.5 40 0.88900 35.5599 0.05332 40 0.85480 34.1922 0.0684

2.5 40 0.82193 32.8771 0.08223 1040 0.79031 821.9271 2.4658

$1,000.00 2.7259Bond price Bond duration

17

Calculating Par Value Bond Duration

2M

2YTM

1

11

YTM2

YTM1

durationbondvaluePar

Calculating Macaulay’s Duration for a par value bond is aspecial case, as follows:

years8.51

2.091

11

.092

.091durationMac.

30

18

To calculating Macaulay’s Duration for any other bond:

C = annual coupon rateM = maturity (years)

12

YTM1CYTM

YTMCM2

YTM1

YTM2

YTM1

MD2M

19

Assume you have a bond with 9% coupon, 8% YTM, and 15 years to maturity. Calculate Macaulay’s

Duration.

years8.78

12.081.09.08

.08.09152.081

.082

.081Dur.Mac.

30

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Price Change & Duration

2YTM

1

YTMinChangeMDpricebondinΔ%

To compute the percentage change in a bond’s priceusing Macaulay Duration:

To compute the Modified Duration:

2YTM

1

durationMacaulaydurationModified

To compute the percentage change in a bond’s priceusing Modified Duration:

YTMinChangeDurationModifiedpricebondinΔ%

21

Calculating Price Change

16.27%

2.09

1

.11.098.5pricebondinΔ%

Assume a bond with Macaulay’s duration of 8.5 years,with the YTM at 9%, but estimated the YTM will go to 11%,calculate the percentage change in bond price and thenew bond price.

Change in bond price, assuming bond was originally at par:

Approx. new price = $1,000 + (-16.27% x $1,000) = $837.30

22

Price Change & Duration

Assume you have a bond with Macaulay’s duration of8.5 years and YTM of 9%, calculate the modified duration.

years8.134

2.09

1

8.5durationModified

Using the bond above with modified duration of 8.134years and a change in yields from 9% to 11%, calculatethe percentage change in bond price.

16.27%.11.098.134pricebondinΔ%

Note this is the same percentage change as computed previously.

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Figure 10.3: Bond duration and maturity

0

2

4

6

8

10

12

0 5 10 15 20 25 30

Bond maturity (years)

Bo

nd

du

rati

on

(ye

ars

) 0% Coupon

5% Coupon 10% Coupon

15% Coupon

24

Zero coupon bond: duration = maturity Duration Properties

– Longer maturity, longer duration

– Duration increases at a decreasing rate as maturity lengthens

– Lower coupon, longer duration

– Higher yield, shorter duration

25

YTM = 10.498%

66

2YTM1

1000

2YTM1

11

YTM

80937$

What is the Macaulay duration of an 8% coupon bond with 3 years to maturity and a current price of $937.10? What is the modified duration?Solution:First calculate the yield:

26

Now calculate the Macaulay’s duration.

Solution:

12

10498.108.10498.

10498.08.32

10498.1

10498.2

10498.1

.Dur.Mac6

Mac. Duration = 2.715 yearsModified duration

= 2.715 / (1 + .10498/2) = 2.58 years