problem 1: Assume that a risk neutral agent has $100,000 today that he wants to save for one year. Compare the given two savings plans. 
 
Bank A offers a standard savings account with 4% p.a.
 
Bank B offers the following alternative:
 
There is a basis interest rate of 1% p.a. and 50% participation on the performance of the S&P500. The maximum interest rate is capped at 7% p.a. (ex; If the S&P increases by 6%, there is a bonus of 3% so that the total return is 4% p.a. If the return of the S&P is 20%, the plan has a return of 7%. Note, if the S&P has a negative return, the interest rate remains at 1% p.a.)
 
Assume that the S&P has 1000 points at t = 0. At t = 1 it can have {900, 990, 1000, 1020, 1040, 1100, 1120, 1200, 1260, 1300} points with equal probability.
 
a) Draw the payoff of alternative B as a function of S&P (with S&P performance on the X-axis, and the return of the plan on the Y-axis.)
 
b) The agent maximizes the expected amount at t = 1. Which plan is more better? How much more can the agent spend in expectation at t = 1, if he chooses the better one?

problem 2: Assume that Real Option Inc. has a product that produces the following cash flow.
 
At t = 1, the demand can be high or low with equal probability. If demand is high (low) the cash flow is CFH = 400 (CFL = 200).
 
At t = 2, the demand can as well be high or low. If demand was high at t = 1, then a high demand at t = 2 occurs with probability 0.8. If demand was low at t = 1, then a high demand at t = 2 arises with probability 0.2. If demand is high (low) at t = 2 then CFH = 400 (CFL = 200). The (risk adjusted) interest rate for this project is 10%.
 
a) Draw the event and decision tree.
 
b) Determine the market price (expected value) of Real Option Inc. at t = 0?
 
Now assume that Real Option Inc. can rent a platform to run a marketing campaign. For this  purpose Real Option Inc. should sign a two year contract with the platform provider. The costs for using the platform are 200 per period. Marketing itself doesn't cost anything and has the following effect. In the high demand state, marketing doubles the demand. In the low demand state it has no effect.
 
c) Should Real Option Inc. rent the platform at t = 0 and run the marketing campaign?
 
Contracts can only be signed at t = 0. However suppose Real Option Inc. can terminate the contract  after the first year by paying the fine of 10. 
 
d) Determine the optimal strategy of Real Option Inc. and the maximum market value of the firm?
 
problem 3: Consider the expected return and standard deviation of the given two assets:
 
Asset 1:   E[r1] = 0.1   and  σ₁= 0.3 
 
Asset 2:  E[r2] = 0.2   and  σ₂= 0.4

a) Draw (ex with Excel) the set of achievable portfolios for the cases: (i) ρ₁₂= -1, (ii) ρ₁₂= 0.

b) Suppsoe ρ₁₂ = 0. Which portfolio has the minimal variance? Find out the minimal variance?

problem 4: Consider an economy with three states which take place with probability (0.2, 0.4, 0.4). Assuma that a firm has a project which generates the state dependent cash flows (100, 200, 200) at t = 1. The investment costs are 170 at t = 0. The firm owns this money. The market portfolio produces the payoff (200, 250, 300) and has an expected return of 8%. The risk free rate is 3%. Assuma that the CAPM holds. 
 
a) What is the beta of this project?
 
b) Describe whether the firm should conduct the project.