Assignment 10 Answers Managing FIPortfolios PDF

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FNCE 30001 – Investments Semester 1, 2021 Module 10: Managing Fixed Income Portfolios Tutorial and Assignment QuestionsPart B, the marked Quiz part of this assignment is graded pass/fail purelyfor the attempt at answering. You must fill in an answer for full credit. PartB Assignment Answers are due ...

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FNCE 30001 – Investments Semester 1, 2021 Module 10: Managing Fixed Income Portfolios Tutorial and Assignment Questions

Part B, the marked Quiz part of this assignment is graded pass/fail purely for the attempt at answering. You must fill in an answer for full credit. Part B Assignment Answers are due 9 am Monday 17 May via the LMS. You will need to use a spreadsheet program like Excel or Google Sheets to solve this assignment.

Part A: This part is unmarked A1. A junior portfolio manager has been asked to establish a fund that will be worth $175 million in four years’ time. Her supervisor has suggested to her that an appropriate investment would be 5-year 15% coupon bonds at a yield of 7.6% pa. Although the junior manager has some knowledge of bonds, she does not understand the reason for this suggestion. (a)

Explain the reason to the junior manager in simple terms.

“Duration” is a measure of the weighted average time period it takes for cash flows to arrive. The weights are the present value of each cash flow. In turn, duration is closely related to a measure of the sensitivity of a bond’s price to changes in required yield. As shown in the table below, the duration of a 5-year 15% bond priced to yield 7.6% pa is almost exactly 4 years. yield =

0.076

Time 1 2 3 4 5 Totals D=

Cash flows 15 15 15 15 115

t × cash 15 30 45 60 575

PV(cash) 13.9405204 12.955874 12.0407751 11.1903114 79.7327023 129.860183

PV(t × cash) 13.9405204 25.911748 36.1223253 44.7612458 398.663512 519.399351

3.999681333

It can be shown that if duration matches the investment horizon, then the future value of the investment is “immunised” against parallel shifts in the yield curve. Hence, as suggested by the supervisor, an investment in this 5-year bond is appropriate for an investment horizon of 4 years. (b)

How much should she invest to establish the fund? What annual coupon interest will this investment produce? If the par value of one bond is $10 million, how many bonds should be bought?

Given that the target sum in 4 years’ time is $175 million, the amount to be invested today is $175m / (1.076)4 = $130,553,634. To find the annual coupon interest (C):

 0.15  Par  1 Par 1 − + 5 0.076  ( 1.076 )  ( 1.076 )5   = 0.605274  Par + 0.693328  Par

$130,553,634 =

$130,553,634 1.298602 = $100,533,985 C = 0.15  $100,533,985 = $15,080,098

Par =

If “one bond” has a par value of $10 million, then the annual coupon interest is $1.5 million per bond. Using the standard bond pricing formula, the price of one such bond is:

 $1.5m  1 $10 m 1 −  + 5 0.076  ( 1.076)  ( 1.076) 5   = $12,986,018

P=

The number of these bonds purchased at the start of Year 1 is therefore $130,553,634 / $12,986,018 = 10.0534001. (c)

Immediately after the fund is established, yields increase by 100 basis points. Show that, if no further yield shifts occur, the fund will achieve the target in four years’ time.

Yields increase by 100 basis points – that is, by 1% – so yields are 8.6% pa. Therefore, the price of one bond becomes:

 $1.5m  1 $10 m 1 − + 5 0.086  ( 1.086)  ( 1.086) 5   = $12,515,430

P=

Therefore, the bond holding after the increase in yield is worth 10.0534001 × $12,515,430 = $125,822,625. The investor should rebalance the portfolio now but we will ignore this. We will also ignore the need to rebalance on future coupon dates.

At the end of Year 1 The coupon interest received = 10.0534001 × $1,500,000 = $15,080,100. The ex-interest price of one bond is:

 $1.5m  1 $10 m 1 − + 4 0.086  ( 1.086)  ( 1.086) 4   = $12,091,757

P=

Therefore, the number of new bonds purchased is $15,080,100 / $12,091,757 = 1.2471389 bonds. The total bond holding therefore increases to 10.0534001 + 1.2471389 = 11.3005390 bonds. The value of the bond holding is 11.3005390 × $12,091,757 = $136,643,372. At the end of Year 2 The coupon interest received = 11.3005390 × $1,500,000 = $16,950,809. The ex-interest price of one bond is:

 $1.5m  1 $10 m 1 − + 3 0.086  ( 1.086)  ( 1.086) 3   = $11,631, 648

P=

Therefore, the number of new bonds purchased is $16,950,809 / $11,631,648 = 1.4573007 bonds. The total bond holding therefore increases to 11.3005390 + 1.4573007 = 12.7578397 bonds. The value of the bond holding is 12.7578397 × $11,631,648 = $148,394,701. At the end of Year 3 The coupon interest received = 12.7578397 × $1,500,000 = $19,136,760. The ex-interest price of one bond is:

 $1.5m  1 $10 m 1 − + 2 0.086  ( 1.086)  ( 1.086) 2   = $11,131,969

P=

Therefore, the number of new bonds purchased is $19,136,760 / $11,131,969 = 1.7190813 bonds. The total bond holding therefore increases to 12.7578397 + 1.7190813 = 14.476921 bonds. The value of the bond holding is 14.476921 × $11,131,969 = $161,156,636.

At the end of Year 4 The coupon interest received = 14.476921 × $1,500,000 = $21,715,382. The ex-interest price of one bond is:

$11.5m 1.086 = $10,589, 319

P=

At this point the portfolio is liquidated, as follows: Sale of bonds = 14.476921 × $10,589,319 = $153,300,735 Coupon interest received = $21,715,382 Hence, cash held = $153,300,735 + $21,715,382 = $175,016,117. Therefore, the objective of having at least $175 million has been achieved.

The above is summarised in the following table:

Progress of the investment if yields increase from 7.6% pa to 8.6% pa. Date = investment period expired (years) 0 1 2 3 Bond term left (years) 5 4 3 2 Coupon interest received ($) $0 $15,080,100 $16,950,809 $19,136,760 Price of 1 bond ($); Par = $10m $12,515,430 $12,091,757 $11,631,648 $11,131,969 No. of extra bonds bought 0 1.2471389 1.4573007 1.7190813 No. of bonds held 10.0534001 11.3005390 12.7578397 14.476921 Value of bonds held ($) $125,822,625 $136,643,372 $148,394,701 $161,156,636 Bond price is the present value of the remaining cash flows; par value for one bond is $10m. At the end of Year 4, the cash holding is $21,715,382 + $153,300,735 = $175,016,117.

4 1 $21,715,382 $10,589,319 0 14.476921 $153,300,735

A2. Consider a 7-year 12% coupon bond, with a par value of $100, and which has just paid a coupon. The yield curve is flat at 9.25% pa. Coupons are paid annually. (a)

Calculate the duration. Use the duration to make a first approximation of the percentage capital gain or loss if the yield increases by 25 basis points.

See spreadsheet on the next page. Duration is 5.2349 years. First approximation is a capital loss of 1.19792%. (b)

Calculate the convexity adjustment. Use this adjustment to make a second approximation of the percentage capital gain or loss if the yield increases by 25 basis points.

See spreadsheet. Second approximation is a capital loss of 1.18810%. (c)

Calculate the exact percentage capital gain or loss if the yield increases by 25 basis points.

See spreadsheet. Exact change is a capital loss of 1.18817%. (d)

Assuming yields do not change, what will be the duration of the bond three months later?

If there is no coupon payment and no change in yield, then duration falls 1-for-1 with term to maturity. In the next three months there is no coupon payment and the question tells us that there has been no change in yield. Hence, in three months’ time, the duration will be 0.25 years less than it is now. That is, duration will be 5.2349 – 0.25 = 4.9849 years. yield =

0.0925

multiples Time 1 2 3 4 5 6 7 Totals D= X= Delta i ADJ

Cash flows

0.095

PV(cash) 12 12 12 12 12 12 112

5.23492484 15.711715 0.0025 9.8198E-05

1st approx % 2nd approx %

-1.19792%

Exact %

-1.18817%

-1.18810%

10.98398 10.05399 9.202735 8.423556 7.710349 7.057527 60.29314 113.7253

tx PV(cash) 10.98398 20.10798 27.6082 33.69422 38.55174 42.34516 422.052 595.3433

t x (t+1) 2 6 12 20 30 42 56

PV(cash x multiples) 21.96796 60.32393 110.4328 168.4711 231.3105 296.4162 3376.416 4265.338

New price 10.9589 10.00813 9.139846 8.346892 7.622732 6.961399 59.33613 112.374

(e)

What will be the duration of the bond (i) immediately before the next coupon payment? (ii) immediately after the next coupon payment?

IMMEDIATELY BEFORE Time

Cash flows 0 1 2 3 4 5 6

t x cash 12 12 12 12 12 12 112

0 12 24 36 48 60 672

PV(cash) 12 10.9839817 10.0539878 9.20273485 8.42355592 7.71034867 65.8702556 124.244865

PV(t x cash) 0 10.9839817 20.1079756 27.6082045 33.6942237 38.5517434 395.221534 526.167663

Totals D= 4.234924835 IMMEDIATELY AFTER Time Cash flows t x cash PV(cash) PV(t x cash) 0 0 0 0 1 12 12 10.9839817 10.9839817 2 12 24 10.0539878 20.1079756 3 12 36 9.20273485 27.6082045 4 12 48 8.42355592 33.6942237 5 12 60 7.71034867 38.5517434 6 112 672 65.8702556 395.221534 Totals 112.244865 526.167663 D= 4.687676934

A3. What is the “convexity” of a coupon bond? Why do investors have a positive view of convexity? The bond price is negatively related to the yield (first derivative). The curve is convex to the origin (second derivative). The more convex the curve, the greater is the gain if yields fall and the smaller is the loss if yields rise. If the current yield-to-maturity increases or decreases, the high-convexity bond is the better one to have (and hence, contrary to the diagram, the bond prices today would not be equal – the higher convexity bond would have a higher price).

7

Bond Price

High-convexity bond

Low-convexity bond Current price

Current yield to maturity

Yield

Part B: Assignment, worth 1.25 marks on a pass/fail basis. You will receive full marks if you make an honest attempt at completing the assignment. This is due 9 am Monday 17 May 2021 via the LMS Quizzes Page. Given the following bond portfolio, comprised of bonds paying annual coupons: Bond

Yield

A B

0.5% 0.6%

FV per Bond Number of Coupon Rate Bonds Held $100 10 2% $100 20 2.5%

Years Maturity 3 5

to

Question 1: What is the price per $100 FV of Bond A? Question 2: What is the price per $100 FV of Bond B? Question 3: What is the total value of the portfolio? (For each bond, multiply the Number of Bonds Held by the price.) To solve Questions 4 and 5, you should fill out a table that looks something like this: Year 0 1 …

Bond 1 Bond 2 ? …

? …

Portfolio Cash Flows -[Answer to Question 3] ? ….

8

Question 4: What are cash flows of the portfolio in Year 3? Question 5: What is the yield of the portfolio? (Hint: Use the IRR function in Excel on the “Portfolio Cash Flows” column.) Bond

Yield

A B

FV per Bond

0.50% 0.60%

Year

Bond 1 0 1 2 3 4 5

100 100

Coupon Rate 2% 2.50%

Years to Bonds Maturity Held 3 5

Price

10 $104.46 20 $109.33

Value of Bond Holding $1,044.55 $2,186.63 $3,231.18

Bond 2

Portfolio Cash Flows -$3,231.18 50 70 50 70 50 1070 50 50 2050 2050

20 20 1020

Yield

0.5773%

9...