邪恶漫画肉机器人:Financial Markets
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Using time series data, let the b of stock Z beestimated at -1.5.
Q1. How does the return on stock Z change ifthe whole market goes up by 10 percent?
Answer:
According toformula: Rj = Rf + bj ( Rm - Rf ) (Watson and Head, 2004)
Where: Rj = the rate of return of security j predicted by the model
Rf = the risk-free rate of return
bj = the beta coefficient of security j
Rm= the return of the market
Now, the b of stock Z isestimated at -1.5, then the whole market goes up by 10 percent, so:
Rj =Rf -1.5(Rm - Rf ) ①
Rj =Rf -1.5(1.1Rm - Rf ) ②
②-① : Rj = -0.15 Rm
So, if stock Zhas a beta of -1.5, and the market return increases by 10 per cent, the returnof stock Z will decrease by 0.15 percent.
Q2. How does the return on stock Z changeif the whole market goes down by 10 percent?
Answer:
According toformula: Rj = Rf + bj ( Rm - Rf ) (Watson and Head,2004)
Where: Rj = the rate of return of security j predicted by the model
Rf = the risk-free rate of return
bj = the beta coefficient of security j
Rm= the return of the market
Now, the b of stock Z isestimated at -1.5, then the whole market goes down by 10 percent, so:
Rj =Rf -1.5(Rm - Rf ) ①
Rj =Rf -1.5(0.9Rm - Rf ) ②
②-① : Rj = 0.15 Rm
So, if stock Zhas a beta of -1.5, and the market return decreases by 10 percent, the returnof stock Z will increase by 0.15 percent.
Q3. Suppose that you have $1000 invested inthe market portfolio. You just learn that you inherited $100. Which one of thefollowing investment opportunities will yield the overall safest portfolioreturn? Explain your answer.
a) Investthe extra $
b) Investthe extra $
c) Investthe extra $
Answer:
a) According toformula:
Portfolio variance = x12σ12 +x22σ22+2(x1x2ρ12σ1σ2) (Brealeyand Myers, 2003)
Then, let x1=proportions invested in market portfolio
x2=proportions invested in Treasury bills
σ1=variance of market portfolio return
σ2=variance of Treasury bills
However, normally, Treasury bills are no risk. So, σ2= 0
And, x1=1000/1000+100≈0.91
x2=100/1000+100≈0.09
Then, Portfolio variance = 0.912σ12 =0.8281σ12
Namely, the portfolio risk including investing the extra $
b) According toformula:
Portfolio variance = x12σ12 +x22σ22+2(x1x2ρ12σ1σ2) (Brealeyand Myers, 2003)
Then, let x1=proportions invested in the market portfolio
x2=proportions invested extra $
σ1=variance of the market portfolio return
σ2=variance of extra $
However, σ1=σ2 because of investing the samemarket portfolio. By definition, the beta of the market is always 1 and acts asa benchmark. (Watson and Head,2004)
Then, according toformula: ρ= σ1b /σ2
Then, ρ=1
And, x1=1000/1000+100≈0.91
x2=100/1000+100≈0.09
So, Portfolio variance = 0.912σ12+0.092σ12+2(0.91×0.09σ12)
=σ12
Namely, the portfolio risk including investing the extra $
c) According toformula:
Portfolio variance = x12σ12 +x22σ22+2(x1x2ρ12σ1σ2) (Brealeyand Myers, 2003)
Then, let x1= proportions invested inmarket portfolio
x2=proportions invested in stocks Z
σ1=variance of market portfolio return
σ2=variance of stock Z return
ρ12=correlation between returns on market portfolio and stock Z
So, x1=1000/1000+100≈0.91
x2=100/1000+100≈0.09
Portfolio variance = 0.912σ12+0.092σ22+2(0.91×0.09ρ12σ1σ2)
According to formula:
ρ(Ri, Rm)= cov(Ri, Rm)/ σiσm
So, ρ(Ri, Rm) = σmbi/σi then, ρ12=σ1 b/σ2 , and the beta of stock Z is -1.5
So, ρ12= -1.5σ1/σ2
Then, Portfolio variance = 0.912σ12+0.092σ22+2(0.91×0.09ρ12σ1σ2)
= 0.6σ12
Namely, the portfolio risk including investing the extra $
Compared with threeportfolio risks, they are 0.6σ12﹤0.8281σ12﹤σ12.Therefore, the safest portfolio return is to invest the extra $
Q4. You considerinvesting in an option on an individual stock as apposed to an option on themarket portfolio. Which one of the two investment opportunities is worth moreand why?
Answer:
Suppose individual stock and market portfolio are the same S0, K,r, q and T separately.
Then, let σ of individual stock and σ of market portfolio are the same,
According to formula:
C= S0e-qTN(d1)-Ke-rTN(d2)
P= Ke-rTN(-d2)- S0e-qTN(-d1)
d1=[ln(S0/K)+(r-q+σ2/2)T]/ σT1/2
d2= d1-σT1/2
So, the price C of a European call is the same as the price P of aEuropean put on an individual stock and market portfolio.
Then, according to formula:
Rj = Rf + bj ( Rm - Rf )
So, bj = Rj - Rf / Rm - Rf ①
According to formula ρ(Rj, Rm) = σmbj/σj
So, bj=σjρ(Rj, Rm)/ σm ②
Because of ① and ②, then Rj - Rf / Rm - Rf =σjρ(Rj, Rm)/σm and -1≦ρ(Rj, Rm)≦1,
Then,
If 0﹤bj﹤1, Rm ﹥ Rj, namely,investment on the market portfolio is worth more because of the return ofmarket portfolio is larger than the return of individual stock.
If bj=1, namely, ρ(Rj, Rm)=1, then Rm= Rj, namely,investment on the market portfolio or an individual stock has the same return, andreturn of both individual stock and market portfolio always move in the samedirection, but not the same percentage amount.
If -1﹤bj﹤0, return of both individualstock and market portfolio always not move in the same direction so that it isdifficult to decide which the return is more worth.
If bj= -1, namely, ρ(Rj, Rm)= -1, it doesn'treally occur.
However, let σof individual stock and σ of market portfolio are not the same. Investing incall or put option on an individual stock or the market portfolio will affectthe return of individual stock and the return of the market portfolio. Thereason is that the price of call or put option depends on different variancesof both individual stock and the market portfolio if individual stock andmarket portfolio are the same S0, K, r, q and T. For example, theprice of European call option will increase if σ of individual stock increasesand other variables are fixed.
(Word count: 1078)