Question

FINC 3380 Homework 4

Seungho Baek
Due: Wednesday, April 1 2020, 6:29 P.M.

For this homework assignment, you need to use Excel. Once you done, upload your excel file to Blackboard. Do not submit HW4 via email.

Part 1 Reading Assignments Read LN6.

Part 2 Practical Assignments using Excel Consider two annual coupon bonds. Now current market rate of interest (i.e. discount rate) is 7 percent. Bond A has just been issued. Its face value is $1,000, it bears a 7 percent coupon rate and will mature in 10 years. Bond B was issued five years ago, when the interests were higher. This bond has $1,000 face value and bears a 13 percent coupon rate. When issued, this bond had a 15-year maturity, so its remaining maturity is 10 years.

1) Compute the respective bond prices using the above equation.

P=?N Ct

(1+r)t

1

Note that this equation is to price a bond with payments Ct, where t=1,2,...,N. Ordinarily,thefirstN−1paymentswillbeinterest rates, and CN will be the sum of the repayment of principal and the last interest payment. If the term structure is flat and the discount rate for all of the payment is r.

2. 2) Compute the respective Macaulay durations. The Macaulay duration is de- fined as

   whereD=−1 dP. P dy

3. 3) Which bond is more exposure to interest risk? Explain it thoroughly. 1

D=
1

1?N t·Ct P (1+r)t

   

4) Compute the respective modified durations for two bonds using the below equation and interpret two values.

1 ?N t·Ct P 1 (1+r)t

Dmod =
5) Here is another equation to price a bond with a continuously compounding

  

interest rate of rc.

1+r

N
Pc =?Cte−rct

1

1. (1) Using the above equation, calculate the price of the bond.

   Note: You need to convert an annual compounding rate to a continuously compounded rate.

   (1+r)t = erct
   rc = ln(1+r)

2. (2) Calculate the Macaulay duration.

Dc = 1 ?t·Cte−rct

P

1

(3) Compare two bond prices from problem 1) and two bond prices from problem 5)and provide your findings. What about the durations (from problem 1) and problem 5))?

6) Let’s evaluate price sensitivity to interest rate.
(1) Takethederivativeofthebond’spricewithrespecttothecurrentinterest

rate, i.e., dP , and then multiply − 1 and dP . dr Pdr

N

   

(2) Takethederivativeofthebond’spricewithrespecttothecurrentinterest rate, i.e., dPc , and then multiply − 1 and dPc .

  

drc Pc drc (3) Discuss with two results.

4. (4) When the market rate falls to 5% per year, what are the prices for those bonds?

5. (5) Are there any big differences between them (from problem 1) and prob- lem 5))? Discuss it.

7) Let’s find the respective new bond prices using the durations.

(1) What are the new bond prices when the interest falls to 5% per year? Using the below equation, calculate the new bond prices approximately.

P1 ≈ P0+∆P0 =P0+dP0∆r=P0+(−D·P0)∆r dr

2

(2) What are the new bond prices (with continuously compounding interest rate) when the interest falls to 5% per year? Using the below equation, calculate the new bond prices approximately.

dPc
P1c ≈ P0c +∆P0c = P0c + 0 ∆rc = P0c +(−Dc ·P0c)∆rc drc

8) Compare bond prices from problem 6-4) with the results from problem 7-1) and 7-2).

3

s Window Help . 100% FINC3380 HW4.pdf (page 1 of 3) Edited FINC 3380 Homework 4 Seungho Baek Due: Wednesday, April 1 2020, 6:29 P.M. For this homework assignment, you need to use Excel. Once you done, upload your excel file to Blackboard. Do not submit HW4 via email. Part 1 Reading Assignments Read LN6. Part 2 Practical Assignments using Ercel Consider two annual coupon bonds. Now current market rate of interest (i.e. discount rate) is 7 percent. Bond A has just been issued. Its face value is $1,000, it bears a 7 percent coupon rate and will mature in 10 years. Bond B was issued five years ago, when the interests were higher. This bond has $1,000 face value and bears a 13 percent coupon rate. When issued, this bond had a 15-year maturity, so its remaining maturity is 10 years. 1) Compute the respective bond prices using the above equation. PC

Answer 1

As per HOMEWORKLIB RULES, I'm only supposed to answer the first 4 sub-parts of any question. I have answered.

Details given:

	Bond A	Bond B
Interest rate	7%	7%
Face Value	$1000	$1000
Coupon rate	7%	13%
Time to maturity	10	10

The given formulae have been used. The results are presented in a tabular format.

1. Bond A Price:

Time	Bond A Cash Flows	Discounting Factor = 1/(1+r)^t	Present Value = Cash Flows * 1/(1+r)^t
0	70	1	70
1	70	0.934579439	65.42056075
2	70	0.873438728	61.14071098
3	70	0.816297877	57.14085138
4	70	0.762895212	53.40266484
5	70	0.712986179	49.90903256
6	70	0.666342224	46.64395567
7	70	0.622749742	43.59248193
8	70	0.582009105	40.74063732
9	70	0.543933743	38.07536198
10	1070	0.508349292	543.9337426
		Bond price = Sum of Present values of cash flows	1070

Bond B Price:

Time	Bond A Cash Flows	Discounting Factor = 1/(1+r)^t	Present Value = Cash Flows * 1/(1+r)^t
0	130	1	130
1	130	0.934579439	121.4953271
2	130	0.873438728	113.5470347
3	130	0.816297877	106.118724
4	130	0.762895212	99.17637757
5	130	0.712986179	92.68820333
6	130	0.666342224	86.6244891
7	130	0.622749742	80.95746644
8	130	0.582009105	75.66118359
9	130	0.543933743	70.71138654
10	1130	0.508349292	574.4347001
		Bond price = Sum of Present values of cash flows	1551.414892

2. Bond A Macaulay Duration:

	Present Value = Cash Flows * 1/(1+r)^t	Fraction of Price	Fraction of price * Time
0	70	0.065420561	0
1	65.42056075	0.061140711	0.061140711
2	61.14071098	0.057140851	0.114281703
3	57.14085138	0.053402665	0.160207995
4	53.40266484	0.049909033	0.19963613
5	49.90903256	0.046643956	0.233219778
6	46.64395567	0.043592482	0.261554892
7	43.59248193	0.040740637	0.285184461
8	40.74063732	0.038075362	0.304602896
9	38.07536198	0.03558445	0.320260054
10	543.9337426	0.508349292	5.083492921
	1070	Macaulay Duration	7.023581541

Bond B Macaulay duration:

Time	Present Value = Cash Flows * 1/(1+r)^t	Fraction of Price	Fraction of price * Time
0	130	0.083794477	0
1	121.4953271	0.078312596	0.078312596
2	113.5470347	0.073189342	0.146378683
3	106.118724	0.068401254	0.205203762
4	99.17637757	0.063926406	0.255705622
5	92.68820333	0.059744304	0.298721521
6	86.6244891	0.055835798	0.33501479
7	80.95746644	0.052182989	0.365280924
8	75.66118359	0.048769149	0.39015319
9	70.71138654	0.045578644	0.410207793
10	574.4347001	0.370265042	3.702650419
	1551.414892	Macaulay Duration	6.187629299

3. Since the Macaulay duration of Bond A is greater than that of Bond B, the interest rate risk of Bond A is more. This is because the price of a bond with greater maturity is more sensitive to changes in interest rates.

4. Bond A Modified Duration:

Modified Duration = Macaulay Duration / (1+r)

6.564094898

Interpretation: When the interest rate increases by 1 percentage point, the price of the bond shall fall by 6.564094898

Bond B Modified Duration:

Modified Duration = Macaulay Duration / (1+r)

5.782831121

Interpretation: When the interest rates increase by 1 percentage point, the price of the bond B will fall by 5.782831121