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  • Name a benchmark index that is used to track the performance of a stock/securities exchange in each of Australia, USA, Hong Kong, Japan, England and China.
  • Identify the major asset classes that are likely to feature in each of the types of managed funds listed below. In addition, propose and briefly justify  a “typical” asset allocation for each type of fund.
  • Stable fund
  • Balanced fund
  • Growth fund.

How is duration related to interest elasticity of a fixed income security? What is the relationship between duration and the price of a fixed income security?

Consider a $1,000 Treasury bond paying a semi-annual coupon of 10 per cent p.a. and currently selling at par in the secondary market, and with a maturity date of 11 years.

  • What is the duration, modified duration and dollar duration of this bond?
  • What will be the estimated price change on the bond if interest rates increase by 0.10 per cent (10 basis points)? If interest rates decrease by 0.20 per cent ( 20 basis points)?.
  • What would be the actual price of the bond be under each interest rate change in part (b) using the traditional present value bond pricing techniques? What is the amount of error in each case? Why does this error occur?

Blue Sky Limited has just paid a dividend of 20 cents per share. Investors require a 16 per cent return from investments such as this. If the dividend is expected to grow at a steady 8 per cent per year, what is the current value of each share? What will the shares be worth in 5 years?

Now assume that the dividend is expected to grow at 20 per cent for the next 3 years and then settle down to 8 per cent per year. What price would the share sell for today?

Question 1
  1. a) Benchmark index is a group of stocks/securities which are used as benchmarks to compare the performance of stocks. These are also called the market index. The benchmark index for certain countries is given below:

Country

Benchmark Index

Australia

ASX 200

USA

Dow Jones

Hong Kong

Hang Seng

Japan

Nikkei Index

England

FTSE 100

China

Shanghai

(Economics, 2017)

  1. b) There are various types of mutual funds options available to an investor. Each mutual fund has its own differential features and advantages. The asset classes and the allocation of different assets for three mutual funds i.e. stable fund, balance fund and growth fund are discussed below:
  2. i) Stable – A stable fund is a fund for retirement plans. Under this fund, the main investment asset is wrapped bonds which is the guaranteed investment certificate (GICs), these GICs are paired with insurance policies to give a minimum yield on the principal amount invested and also to preserve the principal amount. The bonds of the fund can be short term and intermediate term.
  3. ii) Balance fund – A balance fund comprises of three asset classes which are stock, bond and money market. The differential feature of balance fund is that the mix of stocks and bonds does not change and remains constant as the minimum and maximum amount invested in each asset class is fixed. The fund can have higher equity component making the fund moderate or a higher bond component making the fund conservative. Balance fund is preferred by investors who are looking for safety, income and a modest capital appreciation.

iii) Growth fund – A growth fund comprises of only stocks. The fund consists of a portfolio of diversified stocks. The primary goal of the fund is capital appreciation. The fund comprises of stocks of companies which have above average growth and these companies mostly reinvest their earning for expansion and acquisitions. Growth funds can be large cap, mid cap and small cap.

Interest elasticity is the sensitivity of the price of the fixed income securities to the interest rates. Higher interest elasticity means that with a small change in the interest rates, the prices of securities fluctuate by a higher amount. Duration of a fixed income security measures the average life of a bond. Duration provides an estimate of the sensitivity of the bond prices to the interest rates. Higher the duration, higher is the sensitivity of the interest rates to the bond prices and vice versa (Bodie, Kane, & Marcus, 2016). Like if the interest rate increases by 1%, and the duration of a bond is 5 years, the prices of bond will decline by 5%.

Prices and duration of a bond or fixed income security are related from the point of view of interest rates. Price and interest rates have an inverse relationship. The sensitivity of price and interest rate is measured by duration. Duration determines the change in price as a result of a change in interest rate.

Face value (F) = $1000

Market price (P) = $1000

Coupon rate (C) = 10% paid semi annually

Maturity (n) = 11 years

  1. a) Duration of the Treasury bond = present value of cash flows weighted by the length of time to receiving the cash and divided by the current selling price of the bond.

For discounting the cash flows, the yield to maturity needs to be calculated to be used as discount rate.

Yield to maturity = [(C+ (F-P))/n] / [F+P]/2

    = [50 + (0/22)] / [2000/2]

    = 5%

The Duration of bond is calculated as follows:

Period

Cash flow

Period*cash flow

PV @ 5%

Present value of cash flow

1

 $       50

 $       50

 $   0.952

 $         47.62

2

 $       50

 $      100

 $   0.907

 $         90.70

3

 $       50

 $      150

 $   0.864

 $       129.58

4

 $       50

 $      200

 $   0.823

 $       164.54

5

 $       50

 $      250

 $   0.784

 $       195.88

6

 $       50

 $      300

 $   0.746

 $       223.86

7

 $       50

 $      350

 $   0.711

 $       248.74

8

 $       50

 $      400

 $   0.677

 $       270.74

9

 $       50

 $      450

 $   0.645

 $       290.07

10

 $       50

 $      500

 $   0.614

 $       306.96

11

 $       50

 $      550

 $   0.585

 $       321.57

12

 $       50

 $      600

 $   0.557

 $       334.10

13

 $       50

 $      650

 $   0.530

 $       344.71

14

 $       50

 $      700

 $   0.505

 $       353.55

15

 $       50

 $      750

 $   0.481

 $       360.76

16

 $       50

 $      800

 $   0.458

 $       366.49

17

 $       50

 $      850

 $   0.436

 $       370.85

18

 $       50

 $      900

 $   0.416

 $       373.97

19

 $       50

 $      950

 $   0.396

 $       375.95

20

 $       50

 $   1,000

 $   0.377

 $       376.89

21

 $       50

 $   1,050

 $   0.359

 $       376.89

22

 $   1,050

 $ 23,100

 $   0.342

 $     7,896.73

Total

 $ 13,821.15

Macauley Duration of bond = 13821.15 / 1000

= 13.82 years

Modified Duration = Macauley Duration / (1+yield to maturity)

= 13.82 / 6%

= 13.16

Dollar duration = -modified duration * (bond price / 100)

= -13.16*(1000/100)

= $131.6

  1. b) Estimated price change on the bond if the interest rate increases by 0.1% = 13.82*0.1%

= 1.38%

New bond price = 1000-(1.38%*1000)

  = $986.18

Estimated price change on the bond if the interest rate decreases by 0.2% = 13.82*0.2%

= 2.76%

New bond price = 1000 + (2.76%*1000)

  = $1,027.64

  1. c) Actual bond price using traditional present value bond pricing technique

When interest rate increases by 0.1%, the new yield is 5.1%.

Period

Cash flow

PV @ 5.1%

Present value of cash flow

1

$50

$0.951

$47.6

2

$50

$0.905

$45.3

3

$50

$0.861

$43.1

4

$50

$0.820

$41.0

5

$50

$0.780

$39.0

6

$50

$0.742

$37.1

7

$50

$0.706

$35.3

8

$50

$0.672

$33.6

9

$50

$0.639

$32.0

10

$50

$0.608

$30.4

11

$50

$0.579

$28.9

12

$50

$0.551

$27.5

13

$50

$0.524

$26.2

14

$50

$0.498

$24.9

15

$50

$0.474

$23.7

16

$50

$0.451

$22.6

17

$50

$0.429

$21.5

18

$50

$0.408

$20.4

19

$50

$0.389

$19.4

20

$50

$0.370

$18.5

21

$50

$0.352

$17.6

22

$1,050

$0.335

$351.5

Total

$987.0

Bond price when the interest rate decreases by 0.2%, the new interest rate is 4.8%

Period

Cash flow

PV @ 4.8%

Present value of cash flow

1

$50

$0.954

$47.7

2

$50

$0.910

$45.5

3

$50

$0.869

$43.4

4

$50

$0.829

$41.5

5

$50

$0.791

$39.6

6

$50

$0.755

$37.7

7

$50

$0.720

$36.0

8

$50

$0.687

$34.4

9

$50

$0.656

$32.8

10

$50

$0.626

$31.3

11

$50

$0.597

$29.9

12

$50

$0.570

$28.5

13

$50

$0.544

$27.2

14

$50

$0.519

$25.9

15

$50

$0.495

$24.7

16

$50

$0.472

$23.6

17

$50

$0.451

$22.5

18

$50

$0.430

$21.5

19

$50

$0.410

$20.5

20

$50

$0.392

$19.6

21

$50

$0.374

$18.7

22

$1,050

$0.356

$374.3

Total

$1,026.8

Change in interest rate

Estimated bond price

Actual bond price

Error

Increases by 0.1%

$986.18

$987

$0.78

Decreases by 0.2%

$1,027.64

$1026.8

$0.83

From the above we see that the change in the estimated bond price is higher than the actual bond price. This error occurs because the estimated price due to change in interest rate is due to the interest rate elasticity of the bond.

Dividend paid = $0.2, rate of return = 16%, growth rate = 8%

Value of share = expected dividend / (rate of return – growth rate)

= (0.2*1.08) / (0.16-0.08)

= $2.7

Price of the stock in five years = dividend in year 5 / (rate of return – growth rate)

Dividend in year 5

Year

Expected dividend

1

$0.216

2

$0.233

3

$0.251

4

$0.272

5

$0.293

Value of share = 0.293 / 0.08

= $3.67

Price of share with growth rate being 20% for next three years and 8% thereafter.

Expected dividend for the high growth phase:

Year

Growth rate

Expected dividend

0

$0.2

1

20%

$0.24

2

20%

$0.28

3

20%

$0.34

4

8%

$0.37

Using the Gordon’s growth model formula, we arrive at the PV of perpetual dividend from 4th year onwards

= 0.37 / (0.16-0.08)

= $4.66

Value of share today = PV of dividends of high growth phase + PV of terminal value

Year

Cash Flows

PV of $1 @16%

PV of cash flows

1

$0.24

$0.862

$0.21

2

$0.29

$0.743

$0.21

3

$0.35

$0.641

$0.22

3

$4.67

$0.641

$2.99

Total

$3.63

Hence, the share would sell for a price of $3.63 today.

References

Bodie, Z., Kane, A., & Marcus, A. (2016). Essentials of Investments. Australia: McGraw Hill Education.

Economics, T. (2017, October 6). Stocks. Retrieved October 6, 2017, from Trading Economics: https://tradingeconomics.com/stock

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