Citibank: basic of corporate finance
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9-30 FIXED INCOME SECURITIES
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ANSWER KEY
8. A new Eurobond issue is available to investors. The bonds carry a 7.375% coupon rate,
have a maturity of 10 years, and are priced to yield 8.0%. What price percentage would
investors expect to pay for the new bonds? Give your answer in percentage terms.
95.81
Use the bond features on your calculator to solve this problem. Enter
7.375% as the coupon rate and 8.0% as the yield to maturity. Enter the
appropriate dates for a 10 year maturity. Press the price key to arrive at the
answer 95.8062.
9. A Eurobond with five years to maturity has a percentage price of 97.85%. If the
coupon rate of the security is 6.325%, what is the yield to maturity of the bond?
6.85%
Use the bond features on your calculator to solve this problem. Enter
6.325% as the coupon rate, 97.85 as the percentage price of the security,
and enter the appropriate dates for a five year maturity. Press the yield to
maturity key and you should get 6.8472.
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PRACTICE EXERCISE
(Continued)
10. What is the Treasury security equivalent yield for the Eurobond in Problem 9?
_____________%
11. An investor is considering the purchase of a Treasury note from another investor. The
annual coupon of the note is $90, it has been 36 days since the last coupon payment,
and the coupon payment period is 183 days. How much would the new investor have
to add to the purchase price of the security to compensate the old investor for
accrued interest?
$______________
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ANSWER KEY
10. What is the Treasury security equivalent yield for the Eurobond in Problem 9?
4.79%
Use the yield equivalence formula with Y
E
= 0.0485 and solve for Y
T
.
(1 + Y
E
) = [1 + (Y
T
/ 2)]
2
(1 + 0.0485) = [1 + (Y
T
/ 2)]
2
(1.0485)
1/2
= [1 + (Y
T
/ 2)]
(1.0240) - 1 = Y
T
/ 2
0.0240 x 2 = Y
T
0.0479 = Y
T
or Y
T
= 4.79%
An annual Eurobond yield of 4.85% is equivalent to an annual Treasury yield of
4.79%.
11. An investor is considering the purchase of a Treasury note from another investor. The
annual coupon of the note is $90, it has been 36 days since the last coupon payment,
and the coupon payment period is 183 days. How much would the new investor have
to add to the purchase price of the security to compensate the old investor for
accrued interest?
$8.85
Use the accrued interest formula with C = $90 and D
H
= 36.
AI = (C / 2) x (D
H
/ 183)
AI = ($90 / 2) x (36 / 183)
AI = 45 x 0.1967
AI = $8.85
The new investor would have to pay the old investor $8.85 to compensate for
the accrued interest.
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FIXED INCOME SECURITIES 9-33
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DURATION
Factors That Affect Bond Prices
In a competitive market, a company's bonds must offer fair expected
rates of return to entice investors. The return to the investor will be in
the form of price gains during the life of the bond plus the coupon
payments made by the issuer of the bond.
Sensitivity to
interest rates
In Unit Four, we discussed the relationship between market interest
rates and bond prices. We said that as interest rates fall, bond prices
rise and vice versa. For example, a bond with an 8% coupon will sell
at par (face value) when other competitive yields are also 8%. If the
market rate rises to 9%, the bond price will fall until its expected
return matches the 9% market rate.
Effect of
maturity on
sensitivity to
interest rates
So, we know that bond prices are sensitive to changes in market
interest rates. In addition, the prices of long-term bonds are more
sensitive to changes in interest rates than the prices of short-term
bonds. Given this relationship between the sensitivity of bond prices
to changes in market interest rates and the life of a bond, it is
particularly important to have an appropriate definition of the life or
term of a bond.
Effective life
of the bond
Just as the quoted coupon rate doesn't convey the true yield of a bond,
the effective life of a bond is more complicated than just looking at
the maturity date. Bond price sensitivity is more appropriately related
to the weighted average life of a bond than to the maturity date.
If you were to receive $1,000 after 20 years and no coupon
payments during that time, the effective life of the bond would be 20
years. However, if, in addition to the $1,000, you receive $100 per
year for each of the 20 years, the weighted average term of the
payout is less than 20 years. The higher the coupon payments
relative to the par value, the shorter the average life of the payout.
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9-34 FIXED INCOME SECURITIES
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Duration of a Bond
Average
maturity of
bond's cash
flows
Duration is the measure we use to estimate the average maturity of
a bond's cash flows. It represents the weighted average life of the
bond, where the weights are based on the present value of the
individual cash flows relative to the present value of the total cash
flows (current price of the bond). The formula for calculating the
duration of a bond is:
T
[ CF
t
/ (1 + y)
t
]
D = å t
_______________________________
t = 1
P
Where:
D = Duration of the security
t = Time in years until each payment is made
T = Time in years until the security matures
CF
t
= Cash flow (coupon or principal) payments made by
the security
y = Yield to maturity of the security. If the coupons are
semi-annual, then this rate needs to be a semi-annual
rate (y / 2).
P = Price of the security
Example
To illustrate the duration calculation, we will calculate the duration
of a three-year note that pays semi-annual coupons and has the
following characteristics:
• Percentage selling price of 95.65
• Annual coupon of 7%
• Annual yield to maturity (bond equivalent yield) of 8.678%
Adjust annual
coupon rate
for semi-
annual
payments
Because the coupons are paid semi-annually, it is necessary to
convert the discount rate of 8.678% to a fully-compounded annual
discount rate. To do this, we divide the annual discount rate by two,
add one, square the result, and subtract one.
0.08678 / 2 = 0.04339
(1 + 0.04339)
2 -
1 = 0.887 = 8.87%
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FIXED INCOME SECURITIES 9-35
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The fully-compounded annual discount rate is 8.87%. We will use
this rate in our calculations. You can see the results arranged in the
table in Figure 9.3.
(1)
Time
(2)
Cash Flow
(3)
Present Value
at Semi-annual
Rate
(4)
Present
Value /
Bond Price
(5)
(1) x (4)
Time-
weighted
Value
0.5 3.50 3.3545 0.0351 0.0175
1.0 3.50 3.2150 0.0336 0.0336
1.5 3.50 3.0813 0.0322 0.0483
2.0 3.50 2.9531 0.0309 0.0617
2.5 3.50 2.8303 0.0296 0.0740
3.0 103.50 80.2159 0.8386 2.5159
Total 95.6501 1.0000 2.7511
Figure 9.3: Duration for a Three-year Coupon Bond
Explanation
of the table
An explanation of each column will help you understand the
calculations.
Time: The time in years from the present time until each
coupon payment. The first coupon is paid in 0.5 years,
and so on.
Cash Flow: The cash flows in percentage terms that are
received as coupon payments.
Present Value Each cash flow discounted at the appropriate rate (in
at Semi-annual this case, it is the fully-compounded annual rate of
Rate: 0.0887). The present value of the first semi-annual
coupon is [3.5 / (1.0887)
0.5
] = 3.3545. The sum of
the values in Column 3 should be equal to the market
price of the bond. (There may be a slight rounding
error.)
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Present Value / Weights for each cash flow are determined by
Bond Price: dividing the present value of the cash flow by the market
price of the bond (95.65). The calculation forthe first cash flow is 3.3545 /
95.65 = 0.0351. The weights in Column 4 should add up to 1.0.
Time-weighted The time until the cash flow is received (Column 1)
Value: is multiplied by the weight for the cash flow (Column 4).
The first calculation is 0.0351 x .05 = 0.0175. The sum
of the time-weighted values in Column 5 is the weighted
average life (duration) of the bond. In this example, the
duration of the 3-year coupon bond is 2.7511 years.
Two points
There are two important points to consider when calculating the
duration of a bond:
Appropriate
discount rate
1. The discount rate used when discounting cash flows must be
appropriate for the type of bond. For example, the bond
equivalent yield is the correct rate to use for an annual bond.
For a semi-annual bond, the fully-compounded annual
discount rate is appropriate.
Consistent use
of percentages
or dollars
2. It is important to be consistent in the use of percentages or
dollar values for cash flows and prices. Either one is
appropriate, but do not mix them.
Example
Another example of a duration calculation follows.
What is the duration of a 3-year Eurobond, with a coupon of 7.5% and a
percentage price of 96.50? First we calculate the BEY (yield to
maturity) of the security. Using your calculator, enter the appropriate
dates for maturity, the coupon rate of 7.5%, and the percentage price of
96.50. The yield to maturity should be 8.8797%. This is the appropriate
rate to use to discount annual cash flows of the Eurobond. We can now
build the worksheet and fill in the information as we make the
calculations.
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FIXED INCOME SECURITIES 9-37
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(1)
Time
(2)
Cash Flow
(3)
Present
Value at
Annual Rate
(4)
Present
Value /
Bond Price
(5)
Time-
weighted
Value
1.0
2.0
3.0
Total
7.50
7.50
107.50
6.8883
6.3266
83.2851
96.5000
0.0714
0.0656
0.8631
1.0001
0.0714
0.1311
2.5892
2.7917
Figure 9.4: Duration for a Three-year Eurobond
The duration of this Eurobond is 2.7917 years. Use the totals to
ensure that the calculation is correct. The sum of the values in
Column 3 equals the quoted percentage price of the Eurobond. The
sum of the weights equals 1.0 in Column 4.
Duration of a
perpetuity
The duration of a perpetuity with equal payments is calculated as (1
+ r) / r where r equals the yield of the security. For example,
consider a perpetuity that is priced to yield 10% annually. The
duration would be:
(1 + r) / r
(1 + 0.10) / 0.10 = 11 years
Even though the instrument makes an annual payment forever, the
duration is only 11 years.
Duration of a Portfolio
Weighted
average
The duration of the portfolio is a weighted-average of the durations
of each of the bonds held in the portfolio. For example, suppose an
investor has a portfolio with 86.6% of the funds invested in 4-year
notes with a duration of 3.6554 years and 13.4% of the funds
invested in 10-year bonds with a duration of 8.9652 years. The
duration of the portfolio is:
D
p
= (3.6554)(0.866) + (8.9652)(0.134) = 4.3669 years
The weights are the percentage of the total portfolio invested in each
security.
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9-38 FIXED INCOME SECURITIES
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Duration Relationships
Influences on
duration
calculation
As you can see from the calculations, duration is influenced by
three important relationships:
• Maturity date
• Coupon rate
• Yield to maturity
Duration is positively correlated with maturity, but it moves in the
opposite direction of coupon rates and yield to maturity.
Maturity
• A bond's duration generally increases with the time to maturity,
holding the coupon rate constant.
Duration is a measure of the average maturity of the bond's
payments, and the largest payment is generally made at maturity.
Therefore, an increase in the time to maturity will increase the
duration of the instrument.
The duration of a zero-coupon bond equals the time to maturity.
One cash flow is received when the bond matures in the amount of
the face value. Compared to coupon bonds of the same maturity,
zero-coupon bonds have the greatest duration and, therefore, have
the greatest sensitivity to changes in market interest rates.
Coupon rate
• A bond's duration is lower when the coupon rate is higher.
High coupon rate bonds produce higher cash flows during the life
of the bond and, therefore, duration is weighted toward the early or
middle years. For low coupon bonds, duration is weighted more
heavily toward the final payment at maturity.
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FIXED INCOME SECURITIES 9-39
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Yield to
maturity
• The duration of a coupon bond is higher when the bond's yield to
maturity is lower, holding all other factors constant.
Recall that our calculation of duration requires that the cash flows
be discounted by the yield to maturity rate (bond equivalent rate).
A smaller discount rate places less value on earlier cash flows in
the discounting process.
Analysts use duration to track the sensitivity of bond prices to
changes in interest rates.
Example
Suppose that you have to decide whether to invest in an 8% coupon
rate with a 20-year maturity or a 12% coupon rate bond with a 25-
year maturity. Which bond will have the larger increase in price if
interest rates decline?
The bond with the shorter maturity (8% coupon rate for 20 years)
has a longer duration than the bond with the greater maturity (12%
for 25 years) and, thus, is the most price sensitive.
Summary
Duration represents the weighted average life of the bond. The
weights are based on the present value of the individual cash flows
relative to the present value of the total cash flows (current price of
the bond).
The longer the duration, the greater the impact of a change in
interest rates on price.
The duration of a coupon bond is higher when the bond's yield to
maturity is lower.
The higher the coupon rate, the lower the duration.
Duration is primarily used as a measure to judge bond price
sensitivity to interest rate changes.
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