Risk Management Models
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Risk Management Models Semester 2/2011
Case Study No:3
Q1 Financial Mathematics
Mr. Abbey borrowed $12,000 and repaid the loan 90 days latter with a single payment of $15,250. What is the implied annual simple interest rate?
90days=3month
((15,250-12,000)/3×12)/12,000×100%=108.33%
(B) Brown invests $5000 today in a bank account that pays interest annually at a rate of 6 percent. He then makes ten more deposits of $5000 each at annual intervals.
What is the value of the investment at the date of the last deposit?
FV10=5000×〖(1+i)〗^10+5000/i[(1+i)n-1]
=5000(1+0.06)10+ 5000/0.06[(1+0.06)10-1]
=74,858
If Brown wished to accumulate the same sum by making a single deposit now, how much would he need to invest?
FV10=P(1+i)n
74,858=P(1.06)10
P=74,858/(1.06)10
=41,800.32
(C) Mr. Steve borrowed $27,000 from a bank to buy a car. He agreed to pay a fixed interest rate of 9 percent per year (calculated quarterly) and to repay by equal quarterly installments over 30 years. Calculate the quarterly repayment.
Quarterly interest rate=9%/4=2.25%
PMT=P×i/[1-(1+i)-n]
=27,000×0.0225/(1-1.0225-30×4)
=653.226
(D) Calculate the current share price of a Bank has the required rate of return 16 percent.
The current earnings per share of bank are $1.50. The bank does not reinvest any of its earning, which are expected to remain constant.
K=RRR=16%
EPS0=D0=1.50
P=D0/k=EPS0/K=1.50/0.16=9.38
The current dividend per share is 80 cents. This dividend is expected to grow at 6 percent per year.
K=16% EPS0=D0=0.80 G=6%
D1=D0(1+g)
=0.80×(1+0.06)
=0.848
P0=D1/k-g=0.848/(0.16-0.06)=8.48
Current dividend per share is 70 cents. The dividend of the company has been growing at 12 percent per year in recent years, a rate expected to be maintained for a further 3 years. It is then envisaged that the growth rate will decline to 6 per cent per year and remain at the level indefinitely.
EPS0=D0=70=$0.7
g=12%
g2=6%
D1=D0(1+g)=0.70(1+0.12)=0.78
D2=0.70(1.12)2=0.88
D3=0.70(1.12)3=0.98
=8.61
Q2. Option Pricing Model
Calculate the value of a five-month call option when the price is $65, the strike price is $57, the risk-free rate is 12% per annum, and the volatility of future price is 20 % per annum.
=0.12
T=5/12=0.4167
=20%=0.2
=0.447
= =0.77
d2= d1- T
=65(0.7794)-57×0.95(0.6808)
=13.79
Q3. Volatility Forecasting
Suppose that the current price of gold at close of trading yesterday was $350 and its volatility was estimated as 1.25% per day. The price at the close of trading today is $330. Update the volatility estimate using
(a). The Exponentially weighted moving average (EWMA) model with
= 0.94
σn2=σn-12+(1-)Un-12
= 0.94
Si-1=350
Si=330
σn-12=1.25%
σn2=0.94(1.25)+(1-0.94) Un-12
Un-12=[(330-350)/350]2=0.003
σn2=0.94*1.25%+(1-0.94)*0.003=0.01193
(b). The GARCH(1,1) model with = 0.000002, = 0.04 and = 0.94. What is the long-run average volatility.
σ2=ω/(1-α-β)
=0.000002/(1-0.04-0.94)
=0.000002/0.002
=0.0001
σ=0.01
Q4. USE ONLY the CALCULTOR to answer the Following Questions.
XYZ Stock Price
ABC Stock
Price