# 辅导代写接单-Solutions to EC333 20-21

Solutions to EC333 20-21

a) True- According to CAPM for an asset that has the same expected return as the risk-free rate it must have a zero beta. Zero beta implies that there is zero correlation.

b) True- We consider the possible cases of the stock price in 1-year. If it is lower than 200 or higher than 300 than the two alternatives yield the same payoff. However, when the stock price is strictly between 200 and 300 then the combination of a put option with a strike of 200 combined with an option with a strike of 300 yields a higher payoff.

c) True- Since the put price is increasing in the strike price while the call price is decreasing we have that P(300)>P(200) while C(300)<C(200).

d) False- A’s Sharpe ratio is (10-2)/30=0.2666 While B’s Sharpe ratio is (8-2)/20=0.3. If we must pick one of the two it must be the one with the higher Sharpe ration.

e) False- While A has a lower Sharpe ratio it has a return that exceeds the risk free rate. If the correlation between B and A is low or zero than it would be best to invest in both securities.

2.

a) There is an arbitrage opportunity: borrow \$100 in the US and convert to British pounds:

100 × 0.7 = 70. Additionally, lock in a future exchange at the current forward price of 72.1 British pounds for 72.1 = 103 US dollars. Invest the 70 pounds and get 1.03 × 70 = 72.1

0.7

pounds. Convert back to 103 US dollars. The loan + interest equals 101. This leaves a profit of

\$2. b)

Since the put-call parity doesn't hold there is an opportunity for arbitrage:

��� ��� = ���(���) − ���(���)

1+��� 1+��� ������ ������

in this case the asset is 1 USD and ��� = ��� = 0.7 and ���(���) = ���(���) and therefore: 0.7 − 0.7 = −0.0135 ≠ 0

This means that

0 <

��� ��� + ���(���) − ���(���)

1.03 1.01

1+��� 1+��� ������ ������

Therefore, in order to make a profit we do the following:

Borrow 1 dollar, convert it to 0.7 GBP and invest it in Britain. Additionally, we buy 1.01 call options and sell 1.01 put option.s The total cost of this plan is 0 in year 1. In 1 year we will have 0.721 GBP, owe 1.01 USD and either owe or not owe 1.01 put options, depending on the exchange rate ���. The following table summarizes the strategy and the realizations:

1

Today

1

-1 USD=-0.7 GBP

-1.01P 1.01P

Value at Expiration date

��� ≤ 0.7

-1.01 USD 0.721 GBP

0

1.01(1USD -0.7GBP)= 1.01USD-0.707GBP

Borrow 1 USD Invest in Britain

Buy 1.01 call options Sell 1.01 Put options

��� > 0.7

-1.01 USD

0.721 GBP

1.01(USD -0.7GBP)= 1.01USD-0.707GBP

0

Total in USD

0

1.01 + 0.721 − 0.707 − 1.01 = ���

0.014 ≥ 0.014 = 0.02 > 0 ��� 0.7

0.721 − 0.707

1.01 + ��� − 1.01 =

0.014 ���>0

3.

a) We are looking for a portfolio ��� = ��� ��������������� + (1 − ���) ⋅ ������������������ such that ������ =��������������������� +(1−���)��������������������� =11%

Insert the values given and get

��� ⋅ 10 + (1 − ���)12.5 = 11 ⇒ 12.5 − 2.5��� = 11 ⇒ 1.5 = 2.5��� ��� = 0.6

The risk is the standard deviation of the portfolio. Since the assets are independent we get that:

��� = √���2���2 + (1 − ���)2���2 = √0.620.12 + 0.420.122 ��� ��������������� ������������������

= √0.36 ⋅ 0.01 + 0.16 ⋅ 0.0144 = √0.0036 + 0.002304 = 0.0768

b)

Denote by Q the tangency portfolio. We know that the tangency portfolio is efficient. Additionally, we now that the weights on the risky assets in any efficient portfolio are given by ���Σ−1������ for some ���. Since the assets are uncorrelated we know that

Σ=(0.12 0 )⇒Σ−1 = 0 0.122

1 (0.122 0 )=62500(0.0144

0 ) 0.01

We also know that Therefore:

������ = (��� ���������������

��� ,��� ��� )′ = (0.07,0.095)′ ��� ������������������ ���

0.12 ⋅ 0.122 − 0 ⋅ 0 0 0.12 9

0

0 )( 0.07 ) = 62500(0.07 ⋅ 0.0144) = (

The tangency portfolio has 0 weight on the risk-free asset and therefore (���Σ−1������)′��� = 1. This

means that ��� = 1 = 0.0735 and the tangency portfolio is given by 7+6.5972

Σ−1������ = 62500(0.0144

9 0 0.01 0.095 9 0.095 ⋅ 0.01 6.5972

7 )

2

��� = 0.5148 ⋅ ��������������� + 0.4852 ⋅ ������������������

Or simply ��� = (0.5148). Therefore, the return of the tangency portfolio is 0.4852

And the excess return is

������ = 0.5148 ⋅ 10 + 0.4852 ⋅ 12.5 = 11.2131

������ = 8.2131 ���

Since the assets are uncorrelated the standard deviation is

��� = √0.51482 ⋅ ���2 + 0.48522���2 = √0.51482 ⋅ 0.12 + 0.485220.122 = 0.0777 ��� ��������������� ������������������

We know that any efficient portfolio is a combination of the risk-free asset and the tangency portfolio. We are looking for an excess return of 11 − 3 = 8, therefore the weight we will put on the tangency portfolio, ���, needs to solve

��������� =8⇒���= 8 =0.9741 ��� 8.2131

Meaning invest 0.9741 in the tangency portfolio Q and 0.0259 in the risk-free asset.

c) From part b) we obtain the tangency portfolio that we denote by Q. Consider a CAPM like equation with respect to Q of an investment in Facebook. We will invest a positive amount in Facebook when the return on Facebook (FB) exceeds what can be expected from this CAPM relation. Using the linearity of the covariance we get that

���������(���, ������) = ���������(0.5148 ⋅ ��������������� + 0.4852 ⋅ ������������������, ������) = 0.5148 ⋅ ���������(���������������, ������) + 0.4852 ⋅ ���������(������������������, ������)

= 0.5148 ⋅ ��� ��� ��� + 0.4852 ⋅ ��� ��� ���

��������������� ������ ���������������,������ ������������������ ������ ������������������,������

= 0.5148 ⋅ 0.1 ⋅ 0.1 ⋅ 0.1 + 0.4852 ⋅ 0.12 ⋅ 0.1 ⋅ 0 = 0.0005148

And therefore the ��� of FB with respect to ��� is

��������� = ���������(���, ������) = 0.0005148 = 0.0853

���������(���) 0.07772

We now check whether the excess return on FB exceeds the predicted return: The excess return of FB is equal to: 13 − 3 = 10

The predicted excess return according to the CAPM is:

��� ⋅(��� ���)=��� ⋅(������)=0.0853⋅8.2131=0.7003 ������ ��� ��� ������ ���

3

The return on FB exceeds the return predicted by the CAPM-like relationship and hence we would invest in FB.

4

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