Problems and Solutions 1 1.1 1.1
CHAP CHAPTE TER R 1 — Prob Proble lems ms Probl Problem ems s on Bonds Bonds
Exercise Exercise 1.1
On 12/04/01, consider a fixed-coupon bond whose features are the following: • • • •
face value: $1,000 coupon rate: 8% coupon frequency: semiannual maturity: 05/06/04
What are the future cash flows delivered by this bond? Solution 1.1
1. The coupon cash flow is equal to $40 8% × $1,000 = $40 2 It is delive delivered red on the follow following ing future future dates dates:: 05/06/ 05/06/02, 02, 11/06/ 11/06/02, 02, 05/06/ 05/06/03, 03, 11/06/03 and 05/06/04. The redemption value is equal to the face value $1,000 and is delivered on maturity date 05/06/04. Coupon =
Exercise Exercise 1.2
Consider the same bond as in the previous exercise. We are still on 12/04/01. 1. Compute the accrued interest taking into account the Actual/Actual day-count basis. 2. Same question if we are now on 09/06/02.
Solution 1.2
1. The last coupon has been delivered on 11/06/01. There are 28 days between 11/06/01 and 12/04/01, and 181 days between the last coupon date (11/06/01) and the next coupon date (05/06/02). Hence, the accrued interest is equal to $6.188 28 × $40 = $6.188 Accrued Accrued Interest Interest = 181 2. The last coupon has been delivered on 05/06/02. There are 123 days between 05/06/02 and 09/06/02, and 184 days between the last coupon date (05/06/02) and the next coupon date (11/06/02). Hence, the accrued interest is equal to $26.739 123 × $40 = $26.739 Accrued Accrued Interest Interest = 184
2 Problems and Solutions
Exercise 1.3
An investor has a cash of $10,000,000 at disposal. He wants to invest in a bond with $1,000 nominal value and whose dirty price is equal to 107.457%. 1. What is the number of bonds he will buy? 2. Same question if the nominal value and the dirty price of the bond are respectively $100 and 98.453%.
Solution 1.3
1. The number of bonds he will buy is given by the following formula Number of bonds bought =
Cash Nominal Value of the bond × dirty price
Here, the number of bonds is equal to 9,306 n
2.
n
=
=
9,306.048
is equal to 101,562 n
Exercise 1.4
10,000,000 1,000 × 107.457%
=
10,000,000 100 × 98.453%
=
101,571.31
On 10/25/99, consider a fixed-coupon bond whose features are the following: • • • •
face value: Eur 100 coupon rate: 10% coupon frequency: annual maturity: 04/15/08
Compute the accrued interest taking into account the four different day-count bases: Actual/Actual, Actual/365, Actual/360 and 30/360. Solution 1.4
The last coupon has been delivered on 04/15/99. There are 193 days between 04/15/99 and 10/25/99, and 366 days between the last coupon date (04/15/99) and the next coupon date (04/15/00). •
The accrued interest with the Actual/Actual day-count basis is equal to Eur 5.273 193 × 10% × Eur 100 = Eur 5.273 366
•
The accrued interest with the Actual/365 day-count basis is equal to Eur 5.288 193 365
•
×
10% × Eur 100 = Eur 5.288
The accrued interest with the Actual/360 day-count basis is equal to Eur 5.361 193 360
×
10% × Eur 100 = Eur 5.361
There are 15 days between 04/15/99 and 04/30/99, 5 months between May and September, and 25 days between 09/30/99 and 10/25/99, so that there
3 Problems and Solutions
are 190 days between 04/15/99 and 10/25/99 on the 30/360 day-count basis 15 + (5 × 30) + 25 = 190 •
Finally, the accrued interest with the 30/360 day-count basis is equal to Eur 5.278 190 × 10% × Eur 100 = Eur 5.278 360
Exercise 1.5
Some bonds have irregular first coupons. • A long first coupon is paid on the second anniversary date of the bond and starts accruing on the issue date. So, the first coupon value is greater than the normal coupon rate. • A long first coupon with regular value is paid on the second anniversary date of the bond and starts accruing on the first anniversary date. So, the first coupon value is equal to the normal coupon rate. • A short first coupon is paid on the first anniversary date of the bond and starts accruing on the issue date. The first coupon value is smaller than the normal coupon rate. • A short first coupon with regular value is paid on the first anniversary date of the bond and has a value equal to the normal coupon rate. Consider the four following bonds with nominal value equal to 1 million euros and annual coupon frequency: • Bond 1: issue date 05/21/96, coupon 5%, maturity date 05/21/02, long first coupon, redemption value 100%; • Bond 2: issue date 02/21/96, coupon 5%, maturity date 02/21/02, long first coupon with regular value, redemption value 99%; • Bond 3: issue date 11/21/95, coupon 3%, maturity date 3 years and 2 months, short first coupon, redemption value 100%; • Bond 4: issue date 08/21/95, coupon 4.5%, maturity date 08/21/00, short first coupon with regular value, redemption value 100%. Compute the future cash flows of each of these bonds.
Solution 1.5
Bond 1 pays 100,000 euros on 05/21/98, 50,000 euros on 05/21/99, 05/21/00, 05/21/01 and 1,050,000 euros on 05/21/02. Bond 2 pays 50,000 euros on 02/21/98, 02/21/99, 02/21/00, 02/21/01 and 1,040,000 euros on 05/21/02. Bond 3 pays 5,000 euros on 01/21/96, 30,000 euros on 01/21/97, 01/21/98 and 1,030,000 euros on 01/21/99. Bond 4 pays 45,000 euros on 08/21/96, 08/21/97, 08/21/98, 08/21/99 and 1,045,000 euros on 08/21/00.
Exercise 1.8
An investor wants to buy a bullet bond of the automotive sector. He has two choices: either invest in a US corporate bond denominated in euros or in a French corporate bond with same maturity and coupon. Are the two bonds comparable?
Solution 1.8
The answer is no. First, the coupon and yield frequency of the US corporate bond is semiannual, while it is annual for the French corporate bond. To compare the yields
4 Problems and Solutions
on the two instruments, you have to convert either the semiannual yield of the US bond into an equivalently annual yield or the annual yield of the French bond into an equivalently semiannual yield. Second, the two bonds do not necessarily have the same rating, that is, the same credit risk. Third, they do not necessarily have the same liquidity. Exercise 1.13
Treasury bills are quoted using the yield on a discount basis or on a moneymarket basis. 1. The yield on a discount basis denoted by yd is computed as yd
=
F − P
×
B
F n where F is the face value, P the price, B the year-basis (365 or 360) and n is
the number of calendar days remaining to maturity. Prove in this case that the price of the T-bill is obtained using the equation
P = F 1 −
n × yd B
2. The yield on a money-market basis denoted by ym is computed as ym
=
B
B
×
yd
−
n × yd
Prove in this case that the price of the T-bill is obtained using the equation P =
3. Show that
Solution 1.13
yd
=
yd
=
F
n×ym B
1+
B
×
+
n × ym
B
ym
1. From the equation F − P F
×
B n
we find n × yd
−
B
1=−
P F
and finally, we obtain
P = F 1 −
2. From the equation ym
=
B
n × yd B
B
×
yd
−
n × yd
5 Problems and Solutions
we find ym
=
B
B
×
−
n
F −P B × F n = F −P B × × F n
B F −P × n F F −P 1 − F
Then, we have n × ym B
F −P F = = P F
F − P
F
=
P
P
−
1
Finally, we obtain F
P =
1+
3. From the equation ym
=
B
n×ym B
B
×
yd
−
n × yd
we find ym (B
−
n × yd ) − B
×
yd
=
0
Then, we have yd (−n × ym − B)
= −B ×
ym
Finally, we obtain yd
=
B
B
×
ym
+
n × ym
Exercise 1.15
What is the price P of the certificate of deposit issued by bank X on 06/06/00, with maturity 08/25/00, face value $10,000,000, an interest rate at issuance of 5% falling at maturity and a yield of 4.5% as of 07/31/00?
Solution 1.15
Recall that the price P of such a product is given by P = F ×
nc B n + ym × m B
1+c×
1
where F is the face value, c the interest rate at issuance, nc is the number of days between issue and maturity, B is the year-basis (360 or 365), ym is the yield on a money-market basis and nm is the number of days between settlement and maturity. Then, the price P of the certificate of deposit issued by bank X is equal to P = $10,000,000 ×
80 1 + 5% × 360
25 1 + 4.5% × 360
=
$10,079,612.3
Indeed, there are 80 calendar days between 06/06/00 and 08/25/00, and 25 calendar days between 07/31/00 and 08/25/00
6 Problems and Solutions
Exercise 1.16
On 01/03/2002, an investor buys $1 million US T-Bill with maturity date 06/27/2002 and discount yield 1.76% on the settlement date. 1. What is the price of the T-Bill? 2. What is the equivalent money-market yield?
Solution 1.16
1. The settlement date of the transaction is 01/04/2002 (trading date plus 1 working day). There are 174 calendar days between the settlement date and the maturity date. The price P of the T-Bill is equal to
100 × 1 − 1.76% ×
174 360
=
99.1493
2. The equivalent money-market yield is equal to 1.775% 1.76% × 360 360 − 174 × 1.76%
2
=
1.775%
CHAPTER 2—Problems
Exercise 2.1
Suppose the 1-year continuously compounded interest rate is 12%. What is the effective annual interest rate?
Solution 2.1
The effective annual interest rate is R
=
e0.12 − 1 = 0.1275
=
12.75%.
Exercise 2.2
If you deposit $2,500 in a bank account that earns 8% annually on a continuously compounded basis, what will be the account balance in 7.14 years?
Solution 2.2
The account balance in 7.14 years will be $2,500.e8%×7.14
=
$4,425.98
Exercise 2.3
If an investment has a cumulative 63.45% rate of return over 3.78 years, what is the annual continuously compounded rate of return?
Solution 2.3
The annual continuously compounded rate of return R is such that 1.6345 = e3.78R We find R c
Exercise 2.7
=
c
ln(1.6345)/3.78 = 13%.
1. What is the price of a 5-year bond with a nominal value of $100, a yield to maturity of 7% (with annual compounding frequency), a 10% coupon rate and an annual coupon frequency? 2. Same question for a yield to maturity of 8%, 9% and 10%. Conclude.
7 Problems and Solutions
Solution 2.7
1. The price P of a bond is given by the formula n
P =
i =1
which simplifies into P =
N × c y
N × c (1 + y) i
1−
+
N (1 + y) n
1 (1 + y) n
+
N (1 + y) n
where N , c, y and n are respectively the nominal value, the coupon rate, the yield to maturity and the number of years to maturity of the bond. Here, we obtain for P P =
10 1 1− 7% (1 + 7%)5
+
100 (1 + 7%)5
P is then equal to 112.301% of the nominal value or $112.301. Note that we can also use the Excel function “Price” to obtain P .
2. Prices of the bond for different yields to maturity (YTM) are given in the following table YTM (%) 8 9 10
Price ($) 107.985 103.890 100
Bond prices decrease as rates increase. Exercise 2.10
1. What is the yield to maturity of a 5-year bond with a nominal value of $100, a 10% coupon rate, an annual coupon frequency and a price of 97.856? 2. Same question for a price of 100 and 105.451.
Solution 2.10
1. The yield to maturity y of this bond is the solution to the following equation P =
N × c y
1−
1 (1 + y) n
+
N (1 + y) n
where N , c, P and n are respectively the nominal value, the coupon rate, the price and the number of years to maturity of the bond. Here, y is solution to 97.856 =
10 y
1−
1 (1 + y) 5
+
100 (1 + y) 5
Using, for example, Newton’s three points method (or the “Solver” function in Excel), we obtain 10.574%. Note that we can also use the Excel function “Yield” to obtain y . 2. Yields to maturity (YTM) of the bond for different prices are given in the following table
8 Problems and Solutions
Price 100 105.451 Exercise 2.13
YTM (%) 10 8.613
Consider the following bond: annual coupon 5%, maturity 5 years, annual compounding frequency. 1. What is its relative price change if its required yield increases from 10% to 11%? 2. What is its relative price change if its required yield increases from 5% to 6%? 3. What conclusion can you draw from these examples? Explain why.
Solution 2.13
1. The initial price P is equal to 5 (1 + 10%)
P =
+
+
5 (1 + 10%)2
5 (1 + 10%)4
+
5 (1 + 10%)3
+
105 (1 + 10%)5
=
81.046
After the yield change, the price becomes P
5 (1 + 11%)
=
+
+
5 (1 + 11%)4
5 (1 + 11%)2 +
+
105 (1 + 11%)5
5 (1 + 11%)3 =
77.825
Hence, the bond price has decreased by P
−
P
P
=
3.97%
2. The initial price P is equal to P = =
5 (1 + 5%)
+
5 (1 + 5%)2
+
5 (1 + 5%)3
+
5 (1 + 5%)4
+
105 (1 + 5%)5
+
5 (1 + 6%)4
+
105 (1 + 6%)5
100
After the yield change, the price becomes P
=
5 (1 + 6%)
=
95.788
+
5 (1 + 6%)2
+
5 (1 + 6%)3
Hence, the bond price has decreased by P
−
P
P
=
4.21%
3. In low interest-rate environments, the relative price volatility of a bond is higher than in high interest-rate environments for the same yield change (here, in our example +1%). This is due to the convexity relationship between the price of a bond and its yield.
9 Problems and Solutions
Exercise 2.14
We consider the following zero-coupon curve: Maturity (years) 1 2 3 4 5
Zero-Coupon Rate (%) 4.00 4.50 4.75 4.90 5.00
1. What is the price of a 5-year bond with a $100 face value, which delivers a 5% annual coupon rate? 2. What is the yield to maturity of this bond? 3. We suppose that the zero-coupon curve increases instantaneously and uniformly by 0.5%. What is the new price and the new yield to maturity of the bond? What is the impact of this rate increase for the bondholder? 4. We suppose now that the zero-coupon curve remains stable over time. You hold the bond until maturity. What is the annual return rate of your investment? Why is this rate different from the yield to maturity? Solution 2.14
1. The price P of the bond is equal to the sum of its discounted cash flows and given by the following formula P = =
5 1 + 4%
+
5 (1 + 4.5%)2
+
5 (1 + 4.75%)3
+
5 (1 + 4.9%)4
+
105 (1 + 5%)5
$100.136
2. The yield to maturity R of this bond verifies the following equation 4
100.136 =
i =1
5 (1 + R) i
+
105 (1 + R)5
Using the Excel function “Yield”, we obtain 4.9686% for R . 3. The new price P of the bond is given by the following formula: P = =
5 1 + 4.5%
+
5 (1 + 5%)2
+
5 (1 + 5.25%)3
+
5 (1 + 5.4%)4
+
105 (1 + 5.5%)5
$97.999 The new yield to maturity R of this bond verifies the following equation 4
97.999 =
i =1
5 (1
+ R) i
+
105 (1 + R) 5
Using the Excel function “yield”, we obtain 5.4682% for R . The impact of this rate increase is an absolute capital loss of $2.137 for the bondholder. Absolute Loss
=
97.999 − 100.136 = −$2.137
10 Problems and Solutions
and a relative capital loss of 2.134% Relative Loss
=
−2.137
100.136
= −2.134%
4. Before maturity, the bondholder receives intermediate coupons that he reinvests in the market: •
after one year, he receives $5 that he reinvests for 4 years at the 4-year zerocoupon rate to obtain on the maturity date of the bond 5 × (1 + 4.9%)4
•
=
after two years, he receives $5 that he reinvests for 3 years at the 3-year zerocoupon rate to obtain on the maturity date of the bond 5 × (1 + 4.75%)3
•
=
$5.7469
after three years, he receives $5 that he reinvests for 2 years at the 2-year zero-coupon rate to obtain on the maturity date of the bond 5 × (1 + 4.5%)2
•
$6.0544
=
$5.4601
after four years, he receives $5 that he reinvests for 1 year at the 1-year zerocoupon rate to obtain on the maturity date of the bond 5 × (1 + 4%) = $5.2
•
after five years, he receives the final cash flow equal to $105. The bondholder finally obtains $127.4614 five years later 6.0544 + 5.7469 + 5.4601 + 5.2 + 105 = $127.4614 which corresponds to a 4.944% annual return rate
127.4614 100.136
1/5
−
1 = 4.944%
This return rate is different from the yield to maturity of this bond (4.9686%) because the curve is not flat at a 4.9686% level. With a flat curve at a 4.9686% level, we obtain $127.6108 five years later 6.0703 + 5.7829 + 5.5092 + 5.2484 + 105 = $127.6108 which corresponds exactly to a 4.9686% annual return rate.
Exercise 2.15
127.6108 100.136
1/5 −
1 = 4.9686%
Let us consider the two following French Treasury bonds whose characteristics are the following:
11 Problems and Solutions
Name
Maturity (years) 6 20
Bond 1 Bond 2
Coupon Rate (%) 5 0
Price 100 31.18
Your investment horizon is 6 years. Which of the two bonds will you select? Solution 2.15
It depends on the level of the reinvestment rate, at which you can reinvest the coupons of Bond 1, as well as on the yield to maturity of Bond 2 at horizon. If you suppose, for example, that the reinvestment rate is equal to the yield to maturity of Bond 2 at horizon, then the total return of Bond 2 will decrease as the reinvestment rate increases, as opposed to Bond 1. Indeed, while the unique source of return for Bond 1 is its reinvested coupons, it lies for Bond 2 in its price appreciation. Bond 1 and Bond 2 will yield nearly the same annualized return (5.15%) for a reinvestment rate of 6.365%.
Exercise 2.18
We consider three bonds with the following features Bond Bond 1 Bond 2 Bond 3
Maturity (years) 1 2 3
Annual Coupon 10 8 8
Price 106.56 106.20 106.45
1. Find the 1-year, 2-year and 3-year zero-coupon rates from the table above. 2. We consider another bond with the following features Bond Bond 4
Maturity 3 years
Annual Coupon 9
Price 109.01
Use the zero-coupon curve to price this bond. 3. Find an arbitrage strategy. Solution 2.18
1. The 1-year zero-coupon rate denoted by R( 0, 1), verifies 110 1 + R(0, 1)
=
106.56
We find the expression 110 − 1 = 3.228% 106.56 The 2-year zero-coupon rate denoted by R( 0, 2), verifies R(0, 1)
=
8 1 + 3.228%
+
108 (1 + R(0, 2))2
=
106.20
We find the expression R(0, 2) =
108 106.2 −
8 1+3.228%
1/2 −
1 = 4.738%
12 Problems and Solutions
The 3-year zero-coupon rate denoted by R(0, 3), verifies 8 1 + 3.228%
8 (1 + 4.738%)2
+
+
108 (1 + R(0, 3))3
=
106.45
We find the expression R( 0, 3) =
108 106.45 −
8 8 1+3.228% − (1+4.738%)2
1/3 −
1 = 5.718%
2. The price P of Bond 4 using the zero-coupon curve is given by the following formula: 9 9 109 + + P = 1 + 3.228% (1 + 4.738%)2 (1 + 5.718%)3 =
109.177
3. This bond is underpriced by the market compared to its theoretical value. There is an arbitrage if the market price of this bond reverts to the theoretical value. We have to simply buy the bond at a $109.01 price and hope that it is mispriced by the market and will soon revert to around $109.177. Exercise 2.20
We consider two bonds with the following features Bond Bond 1 Bond 2
Maturity (years) 10 10
Coupon Rate (%) 10 5
Price 1,352.2 964.3
YTM (%) 5.359 5.473
YTM stands for yield to maturity. These two bonds have a $1,000 face value, and an annual coupon frequency. 1. An investor buys these two bonds and holds them until maturity. Compute the annual return rate over the period, supposing that the yield curve becomes instantaneously flat at a 5.4% level and remains stable at this level during 10 years. 2. What is the rate level such that these two bonds provide the same annual return rate? In this case, what is the annual return rate of the two bonds? Solution 2.20
1. We consider that the investor reinvests its intermediate cash flows at a unique 5.4% rate. For Bond 1, the investor obtains the following sum at the maturity of the bond 9
100 ×
(1 + 5.4%)i
+
1,100 = 2,281.52
i =1
which corresponds exactly to a 5.3703% annual return rate.
2,281.52 1,352.2
1/10
−
1 = 5.3703%
13 Problems and Solutions
For Bond 2, the investor obtains the following sum at the maturity of the bond 9
50 ×
(1 + 5.4%)i
1,050 = 1,640.76
+
i =1
which corresponds exactly to a 5.4589% annual return rate.
1,640.76 964.3
1/10
−
1 = 5.4589%
2. We have to find the value R , such that 100 ×
9
i i =1 (1 + R)
+
1,100
1,352.2
50 ×
=
9
i =1 (1 +
R) i
+
1,050
964.3
Using the Excel solver, we finally obtain 6.4447% for R . The annual return rate of the two bonds is equal to 5.6641%
9
100 ×
Exercise 2.24
i =1 (1 + 6.4447)
i
+
1,100
1,352.2
1/10 −
1 = 5.6641%
Assume that the following bond yields, compounded semiannually: 6-month Treasury Strip: 5.00%; 1-year Treasury Strip: 5.25%; 18-month Treasury Strip: 5.75%. 1. What is the 6-month forward rate in six months? 2. What is the 1-year forward rate in six months? 3. What is the price of a semiannual 10% coupon Treasury bond that matures in exactly 18 months?
Solution 2.24
1.
1+
R2 (0, 1)
2
1.026252 ⇒
=
2
=
1+
1.025 1 +
R2 (0, 0.5)
2
1+
F 2 (0, 0.5, 0.5)
2
F 2 (0, 0.5, 0.5) = 5.5003%
F 2 (0, 0.5, 0.5)
2
2.
1+
R2 (0, 1.5)
2
3
=
3
1.02875
=
1+
R2 (0, 0.5)
2
1.025 1 + ⇒
1+
F 2 (0,0.5,1)
2
F 2 (0, 0.5, 1)
2 2
2
F 2 (0, 0.5, 1) = 6.1260%
3. The cash flows are coupons of 5% in six months and a year, and coupon plus principal payment of 105% in 18 months. We can discount using the spot rates
14 Problems and Solutions
that we are given: 5
P =
Exercise 2.26
5
+
1 + 0.205
1 + 0.0525 2
2
105
+
1
0.0575 + 2
3
=
106.0661
Consider a coupon bond with n = 20 semesters (i.e., 10 years) to maturity, an annual coupon rate c = 6.5% (coupons are paid semiannually), and nominal value N = $1,000. Suppose that the semiannually compounded yield to maturity (YTM) of this bond is y2 = 5.5%. 1. Compute the current price of the bond using the annuity formula. 2. Compute the annually compounded YTM and the current yield of the bond. Compare them with y2 . 3. If the yield to maturity on the bond does not change over the next semester, what is the Holding Period Return (HPR) obtained from buying the bond now and selling it one semester from now, just after coupon payment? At what price will the bond sell one semester from now just after coupon payment?
Solution 2.26
1. For the current price of the bond, we use the formula P 0
=
c
N ×
y2
×
so that P 0
=
1,000 ×
6.5% 5.5%
×
1−
1 (1 + y2
/2)n
1 (1 + 0.0275)20
1−
+
N
+
(1 + y2 /2)n
1,000 (1 + 0.0275)20
=
1,076.14
2. The annually compounded yield to maturity (YTM) denoted by y and the current yield denoted by yc are obtained using the following formulas: y
=
y2 2
1+
yc
−
2
=
cN P 0
1=
=
1+
0.055 2
0.065 × 1,000 1,076.14
2
=
−
1 = 0.055756
0.060401
Therefore, they are both larger than y2 . 3. First, we compute P 1 , the price of the bond one semester from now: P 1
=
N ×
c y2
=
1,000 ×
=
1,073.2
×
1
1−
6.5% 5.5%
(1 + y2 /2)n−1
×
1−
+
N (1 + y2 /2)n−1
1 (1 + 0.0275)19
+
1,000 (1 + 0.0275)19
The Holding Period Return from buying the bond now and selling it one semester from now is then: HPR =
P 1 − P 0 + cN 2 P 0
=
1,073.20 − 1,076.14 + 32.5 1,076.14
=
2.75%
