This problem set consists of four problems.

problem 1) Assume a risk neutral agent has $100,000 today that he wants to save for one year. Compare the following 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. (E.g. 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.)
Suppose the S&P has 1000 points at t=0. At t=1 it can have {900, 990, 1000, 1020, 1040, 1100, 1120, 1230, 1300, 1400} points with equal probability.

(a) Draw the payoff of alternative B as a function of the S&P (with the 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 better? How much more can the agent spend in expectation at t=1, if he chooses the better one?

problem 2) Assume Real Option Inc. has a product which generates the following cash flow. At t=1, the demand could 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 could also be high or low. If demand was high at t=1, then a high demand at t=2 arises 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) What is the market price (expected value) of Real Option Inc. at t=0?

Now assume Real Option Inc. could 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 does not 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. But suppose Real Option Inc. can terminate the contract after the first year by paying a 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 following two assets:
Asset 1: E[r1]=0.1 und σ1=0.3
Asset 2: E[r2]=0.2 und σ12=0.4

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

(b) Assume ρ12=0. Which portfolio has the minimal variance? What is the minimal variance?

problem 4

Consider an economy with three states which occur with probability (0.2, 0.4, 0.4). Assume a firm has a project that generates the state dependent cash flows (100, 200, 200) at t=1. The investment costs are 165 at t=0. The firm owns this money. The market portfolio generates the payoff (200, 250, 300) and has an expected return of 10%. The risk free rate is 3%. Suppose the CAPM holds.

(a) Determine the beta of this project?

(b) Describe whether the firm should conduct the project.