Tutorial Work Sheet: Module 5 Part A Solutions

Tutorial Work Sheet: Module 5 Part A Solutions

Question 1 What is the duration of a par value bond with a coupon rate of 8% and a remaining time to maturity of 5 years?

Solution:

CF $80 $80 $80 $80 $1,080

Time 1 2 3 4 5

PV of CF@08%

$80/1.08 = $74.07 $80/(1.08)2 = $68.59 $80/(1.08)3 = $63.51 $80/(1.08)4 = $58.80 $1,080/(1.08)5 = $735.03

Price=$1000.00

Time * PV of CF 74.07 * 1 = 74.07 68.59 * 2 = 137.18 63.51 * 3 = 190.53 58.80 * 4 = 235.20 735.03 * 5 = 3675.15

4312.13

Duration = 4312.12 / 1000 = 4.312

Question 2

Find the duration of a 6% coupon bond making annual coupon payments if it has three years until maturity and a yield to maturity of 6%. What is the duration if the yield to maturity is 10%?

Solution: a. YTM = 6%

(1)

(2)

(3)

PV of CF

Cash Flow

Time

(Discount

rate = 6%)

$60.00

1

$56.60

$60.00

2

$53.40

$1,060.00

3

$890.00

Price=$1,000.

Duration = 2833.40/1000 = 2.833 yea0r0s

(4)

Time * PV of CF

56.60 106.80 2670.00 2833.40

b. YTM = 10%

(1) Time until Payment

(years)

1 2 3

(2)

Cash Flow

$60.00 $60.00 $1,060.00

(3) PV of CF (Discount rate = 10%)

$54.55 $49.40 $796.39

1

(4)

Weight

0.0606 0.0551 0.8844

(5)

Column (1) Column (4)

0.0606 0.1102 2.6532

Column Sums $900.53

1.0000

2.8240

Duration = 2.824 years, which is less than the duration at the YTM of 6%.

Question 3

You own a fixed-income asset with a duration of five years. If the level of interest rates, which is currently 8%, goes up by 10 basis points, how much do you expect the price of the asset to go down (in percentage terms)?

Solution:

DM = 5/(1.08) = 4.63

When interest rate goes up by 100 basis points, price goes down by 4.63%. When interest rates go up by 10 basis points, price goes down by 0.463%.

Question 4

Rank the following bonds in order of descending duration.

Bond A B C D E

Coupon 15% 15 0 8 15

Time to Maturity 20 years 15 20 20 15

Yield to Maturity 10% 10 10 10 15

Solution:

The ranking of bonds in descending order of Duration: C Higher the time to maturity higher the duration D Lower the coupon higher the duration A B Lower the YTM higher the Duration E

Question 5

You are managing a portfolio of $1 million. Your target duration is 10 years. You can choose from 2 bonds: a zero coupon bond with 5 years to maturity and a perpetuity each currently yielding 5%.

a) How much of each bond will you hold in the portfolio? b) How will these fractions change next year if the target duration is now 9 years?

2

Solution: a.The duration of the perpetuity is: 1.05/0.05 = 21 years

Call w the weight of the zero-coupon bond. Then: (w 5) + [(1 ? w) 21] = 10 w = 11/16 = 0.6875

Therefore, the portfolio weights would be as follows: 11/16 invested in the zero and 5/16 in the perpetuity.

b.

Next year, the zero-coupon bond will have a duration of 4 years and the

perpetuity will still have a 21-year duration. To obtain the target duration of nine years,

which is now the duration of the obligation, we again solve for w:

(w 4) + [(1 ? w) 21] = 9 w = 12/17 = 0.7059

So, the proportion of the portfolio invested in the zero increases to 12/17 and the proportion invested in the perpetuity falls to 5/17.

3

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