Evolving of Finance Principles - Math Problem Example

The paper  “Evolving of Finance Principles”  is a detailed example of a finance & accounting math problem. The historical price return for a stock is given by

 Ln (Pt/Pt-1)100

Where

Ln is the natural logarithm

Pt -the price of the stock for the month

 Pt-1-price of stock for the previous month

1. EXPECTED RETURNS FROM HISTORICAL RETURNS

BHP= -2.521%

QBE= -2.246%

WES= 3.136%

WOW= 3.278%

1. Monthly volatility

Risk is the Standard deviation of the monthly returns

R=

Volatility

BHP= 4.693%

QBE= 8.145%

WES= 4.571%

WOW= 1.964%

1. Risk-return diagram
2. WOW is the best investment based on past performance. It has the expected return (3.278%) and the lowest risk (964%) for all available investment options.

SEE EXCEL FILE ATTACHED

QUESTION TWO

1. Completely independent

Stock AAA

Expected return, = 10%

Volatility, σ = 25%

Stock BBB

Expected return, = 5%

Volatility, σ = 3%

Weight in AAA, W1=0.20

Weight in BBB, W2=0.80

E [Rp] = E [Ri]

E [Rp] = E [RAAA] +  E [RBBB]

            = 0.225% + 0.8 5%

            =9%

σp2=∑∑WAWB σA, B

Where:

σp2= variance of returns for the portfolio

σA, B= covariance of the stock returns for AAA and BBB

σA2 or σB2 = variance of AAA and BBB stocks returns

WA = proportion of funds invested in AAA stocks

WB = proportion of funds invested in BBB stocks

Expanding the formula,

σp2 = WA σA2 + WB σB2 + 2 WAWB σA, B

= WA σA2+ (1- WA) σB2 + 2WA (1- WA)ρA,BσB σA

If AAA and BBB are independent, ρA,B=0

Hence the formula collapses to

σp2= WA σA2+ (1- WA) σB2

            = 0.2  0.252+ 0.8  0.032

            = 0.0125 + 0.00072

            σp =0.01322

σp= 11.5%

1. Perfectly correlated

The portfolio return is the same as in A above.

There are two scenarios for perfect correlation,

• Perfect negative correlation, where ρ= -1

σp2 = WA σA2+ (1- WA) σB2 + 2WA (1- WA)ρA,BσB σA

The correlation component is given as

2 0.2  0.8   -1  0.25  0.03

=0.0024

σp2= 0.01322-0.0024

             =0.01082

σp= 10.4%

• Perfect positive correlation, where ρ=1

Therefore the correlation component is added

σp2= 0.01322+0.0024

            =0.01562

σp= 12.5%

Question three
Cost of capital to the Bank of its ordinary shares
Outstanding shares            2.2731 billion
Closing price            25.06
Dividend       1.80 (0.9 already paid)The dividend is not expected to grow but remains constant, therefore, g=0
25.06=, but since g is 0,
r= 7%

1. Cost of capital to the Bank of NABHA floating rate notes

NABHA, a perpetual floating rate note, interest paid quarterly,

Pre tax cost of debt is given by

 Coupon rate is 5.04/4=1.126

PV =  =

68=

YMT3months= 0.0185

Effective rate (1+0.0185)4-1

            = 7.6%

Post tax rate for NABHA floating rate

Effective tax rate= 22.4%

= 0.076203 (1-0.224) = 0.05913

=5.913%

1. WACC

Market value of equity = $25.06 per share ´ 2.2731 billion shares outstanding = $56.963886 billion

Market value of floating rate notes (debt) = 100M ´ 68 = 6.8 billion

Market value of assets = $56.963886 + 6.8 = 63.763886

WACC = (0.076203 (1-0.224) + (0.07183)

            =0.06417+ 0.006306

            = 0.0705

            =7.05%

Question four
expected return on investment
Jennifer, investment amount = $100,000
Debt capital, 50,000 at I = 5%
Investment rate= 12%

The interest from portfolio investment before interest expenses is given as
P (1+r)n- P

Where P is the initial investment
n- Period in years
r= rate of interest
(150,000  1.12) – 150,000 = 18,000

The interest gained from the loan is given by
(50,000  1.05) – 50,000 = 2500

 Interest after expenses,
18,000 – 25000 = 15,500

Expected return= 15,500/150,000
=10.33%

1. Using CAPM equation

E [ Rp ]= Rf + βp*( E[Rm ] - Rf)

Where:
E [ Rp ]- expected portfolio return
Rf – the market free rate, given as 5%
E[Rm ] – the expected market return
βp- the portfolio risk
10.33% = 5% + βp* (12% - 5%)
10.33% = 5% + 7%* βp
βp=0.0533/0.07
= 0.7614

1. The volatility of Jennifer’s portfolio

βp = (σ Portfolio returns*ρ portfolio and Market returns) / σm

Therefore, σ Portfolio returns= (βp σm)/ ρ portfolio and Market returns

Assuming the ρ portfolio and Market returns and βp are equal, the standard deviation of market returns equals the portfolio returns, therefore,

σ Portfolio returns=

 =0.4472

1. Increasing expected to return to 18%

Jennifer’s returns are contingent on the initial investment into the portfolio. She has to increase the amount either by additional savings or by borrowing (most likely considering she had borrowed earlier).

Her net interest from the investment is 7% if she borrows the extra amount,

(X) (1.07) + 100000(1.05) – X =interest
Expected return, 18% = Interest/X
By substitution, interest= 0.18X
1.07X –105000 – X = 0.18X
0.11X= 105000
X = 954545

This is the amount that gives an expected return of 18% at the given market rate. 

Question five

XYZ Corporation
Profit  $1M (constant value)
Total equity capital  $10M

New strategy

20%    profits as dividend
80%    paid out as a dividend
Required rate of return = 10%
g= 0%

The Gordon growth model,

Year

1

2

3

4

Profit Amount

1,000,000

1,000,000

1,000,000

1,000,000

Share price before the buyback

10.00

8.20

8.30

8.50

Share Price

8.20

8.30

8.50

8.70

Shares bought back

20000

24390.24

24096.39

23529.41

Dividend payment per share

0.816

0.837162

0.858818

0.881073

Outstanding shares

980,000

955,610

931,513

907,984