Risk Management Models

Risk Management Models Semester 2/2011

Case Study No:3

Q1 Financial Mathematics

(A) 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

iii. 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

P3=D4/k-g2=D3(1+g2)/k-g2=0.98×1.06/(0.16-0.06)=1.03/0.1=10.4

P0=D1/(1+k)+D2/(1+k)2+D3/(1+k)3+P3/(1+k)3

=0.78/1.16+0.88/1.162+0.98/1.163+10.4/1.163

=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.

X=57

S=65

=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

1. 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.

Year XYZ Stock Price ABC Stock

Price

2003

2004