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 portfoli