Financial Markets

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: R_j = R_f + b_j ( R_m - R_f ) (Watson and Head, 2004)

 

Where: R_j = the rate of return of security j predicted by the model

R_f = the risk-free rate of return

b_j = the beta coefficient of security j

R_m= 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:

 

R_j =R_f -1.5(R_m - R_f )  ①

 

R_j =R_f -1.5(1.1R_m - R_f ) ②

 

②-① : R_j = -0.15 R_m

 

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: R_j = R_f + b_j ( R_m - R_f ) (Watson and Head,2004)

 

Where: R_j = the rate of return of security j predicted by the model

R_f = the risk-free rate of return

b_j = the beta coefficient of security j

R_m= 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:

 

R_j =R_f -1.5(R_m - R_f )  ①

 

R_j =R_f -1.5(0.9R_m - R_f ) ②

 

②-① : R_j = 0.15 R_m

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 $100 inTreasury bills.

b)     Investthe extra $100 in themarket portfolio.

c)     Investthe extra $100 in stockZ.

 

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 $100 in Treasury bills is measured by 0.8281σ₁².

 

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 $100 in the market portfolio

σ₁=variance of the market portfolio return

σ₂=variance of extra $100 in the market portfolio

 

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 $100 in the market portfolio is measured by σ₁².

 

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(R_i, R_m)/ σ_iσ_m

 

So, ρ_{(Ri, Rm)} = σ_mb_i/σ_i  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 $100 in stock Z is measured by 0.6σ₁².

 

Compared with threeportfolio risks, they are 0.6σ₁²﹤0.8281σ₁²﹤σ₁².Therefore, the safest portfolio return is to invest the extra $100 in stock Z.

 

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 S_0, 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]/ σT^1/2

 

d₂= d₁-σT^1/2

 

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:

 

R_j = R_f + b_j ( R_m - R_f )

 

So, b_{j =} R_j - R_f / R_m - R_f  ①

 

According to formula ρ_{(Rj, Rm)} = σ_mb_j/σ_j

 

So, b_j=σ_jρ_{(Rj, Rm)}/ σ_m   ②

 

Because of ① and ②, then R_j - R_f / R_m - R_f =σ_jρ_{(Rj, Rm)}/σ_m and -1≦ρ_{(Rj, Rm)}≦1,

 

Then,

If 0﹤b_j﹤1,  R_m ﹥ R_j, namely,investment on the market portfolio is worth more because of the return ofmarket portfolio is larger than the return of individual stock.

 

If b_j=1, namely, ρ_{(Rj, Rm)}=1, then R_m= R_j, 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﹤b_j﹤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 b_j= -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 S_0, K, r, q and T. For example, theprice of European call option will increase if σ of individual stock increasesand other variables are fixed.

 

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