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Q1. Financial Mathematics
(A)
Using equation: S=P (1+rt)
According to the question:
Therefore: $10,250=$10,000[1+r (60/365)]
r= 0.15208≈15.21%
That is, the annual simple interest rate r=0.15208or approximately15.21 per cent per annum.
(B)According to this question, the series of deposits consists of the initial deposit plus an ordinary annuity with a term of 10 years.
i)
The future value of the first deposit is:
=$2000*1.7908
=$3581.69
The future value of the 10 later deposits is:
So, the value of the investment at the date of the last deposit is $3581.69+$26359.74=$29941.43
ii)
The amount to be deposited now is the present value of $29941.43 in 10 years’ time.
If Judy wished to accumulate the same sum by making a single deposit now,
So, by Simple Interest:
By Compound Interest:
So Judy needs to invest $18713.39, if she pays a simple interest.
Otherwise, she needs to invest $16719.58, if she pays a compound interest.
(C) Calculate the monthly repayment.
Where, months
From the formula above, we can change it to:
Therefore, the monthly repayment is $282.24.
(D)
Calculate the current share price of a Bank has the required rate of return of 15 percent.
i) Constant Dividends
Where, P0= The current share price
D0= The current earning per share=$2.50
Ke= Required rate of return on the shares=15%
The current share price of a Bank is $16.67.
ii) Growth in Dividends
The dividend per share will grow at a constant rate.
From the question, Ke=15%, the required rate of return is greater than g=5%, which is the expected rate of return.
So the current share price is $7.35.
iii) Growth in Dividends
The dividend per share will grow at changeable rate.
Where, D0=0.6 g’=0.12 g=0.05 Ke=0.15
The current share price is $7.5272.
Q2. Option Pricing Model
Using equation：
Call option:
In this problem, Current share price S=$21
Strike price X=$20
Time until option expiration T=5 months=0.42 year
Volatility σ2=0.2 per annum, σ=0.45
Risk-free rate r’=0.12
Step 1:
Step 2:
Using Normal Distribution table:
N(0.48) is given as 0.5+0.1844=0.6844
N(0.49) is given as 0.5+0.1879=0.6879
An estimate of N(0.4842) that is sufficiently accurate for the present purpose is
N (0.4842) =0.6859
Similarly: N(0.19)=0.5+0.0753=0.5753
N(0.20)=0.5+0.0793=0.5793
N(0.1926)=0.5763
e-rT=e-0.12*0.42=e-0.0504=0.95
So, the value of the call option is approximately $3.4542.
Q3. Volatility Forecasting
a)
The Exponentially Weighted Moving Average (EWMA) Model approach is designed to track changes in the volatility.
Using equation:
In this question, λ=0.94 σ2n-1=0.013
Therefore, the volatility is 1.22% per day.
b)
The GARCH (1, 1) model : ω=0.000002, = 0.04 and = 0.94.
The long-run average variance:
Q4. Calculations
a)
●calculate the average return
Using equation:
The return for XYZ
The average return for XYZ,
The return for ABC
The average return for ABC,
●calculate the risk&CV
Risk is the SD of the stocks.
For XYZ:
For ABC:
The XYZ stocks has about 14% of return more than ABC stock’s 10.9%from 2001 to 2005 on average return. The average return for XYZ is better than ABC’s, this is, the inventors who hold the XYZ stock can gain more return.
The data of XYZ stock on the Standard Deviation (risk) and coefficient of variation (CV) are bigger than the ABC stock’s, which means the stock of XYZ have the higher variability than the stock of ABC. In other words, the stock of XYZ is more risky.
b) Calculate the geometric mean
When calculating the geometric mean the negative return cannot be used. So, we use RR(Relative return) to solve this problem.
Relative return = 1 + daily return (which have already got in a)
YEAR RR for XYZ RR for ABC
2002 1.25 1.012
2003 1.27 1.204
2004 1.053 1.09
2005 0.986 1.132
The geometric mean measures the change in wealth overtime. It shows how the money grows over a specific period. The geometric mean of XYZ is 0.133>0.1073, the ABC’s, so the money in XYZ may grow a little more than the money in ABC.
c)
Using equation:
For XYZ,
$10,000 invested at the end of 2001would has been worth $16,480 by the end of 2005.
For ABC,
$10,000 invested at the end of 2001would has been worth $15,034 by the end of 2005.
d)
i)Value at risk (VaR) at 5% level
For XYZ
For ABC
For the equally weighted portfolio
Where, WXYZ=WABC=50%
Correlation between ABC and XYZ is ρ =+0.65
σxyz=0.1418
σABC=0.08
The VaR of XYZ is bigger than the VaR of ABC (0.233975>0.132), which means that chance of loss investment in XYZ is more than in ABC. The VaR of equally weighted portfolio is between the VaR of XYZ and VaR of ABC. That means the portfolio can distribute the chance of loss investment and make it at middle level.
ii)Value at risk (VaR) at 1% level
For XYZ
For ABC
For the equally weighted portfolio
Where, WXYZ=WABC=50%
Correlation between ABC and XYZ is ρ =+0.65
σxyz=0.1418
σABC=0.08
Finally, we get the similar result at 1%level as the one at 5%level.The VaR of XYZ is bigger than the VaR of ABC (0.3304>0.1864), , which means that it will get more chance of loss investment in XYZ. The VaR of equally weighted portfolio is between the VaR of XYZ and VaR of ABC. That means the portfolio can distribute the chance of loss investment and make it at middle level.
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