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1 Understanding Discounting, Risk Premia, the SD, and the EMM (Easy, 40 points) This problem is quite easy, but it will help you understand what is going on with the SDF and EMM. Consider the simple one-period binomial model from class with two assets: a risk-free money- market (or bond) and a risky stock. There are two time periods: t = 0,1. Let S(t) denote the price of the stock, and let B(t) denote the time t value of $1 invested at time 0. Obviously B(0) = 1, and let's make life easier by supposing the risk-free rate is so that B(1) = 1.25. For the stock, suppose it starts at S(0) = So, and goes up to S(1) = us, and down to S(1) = ds, with equal probabilities (p = 0.5). Let's make it even easier by setting d = 1. Here, u > 1.25 > d=1, so that there is no arbitrage in this model. In summary, we are plugging into our standard binomial model: r = log 1.25 e' = 1.25 e" = 0.8 p=0.5 d=1 u >1.25 = (a) (2 points) With these nice numbers, it's nice to express interest rates and expected returns uncompounded, instead of continuous compounding. The risk-free rate is 25% per period. That is, if we solve r in 1 B(0) -B(1) 1+TB we get rb = 0.25 = 25%. Find the expected return on stock. That is, solve forrs in 1 S(0) = E(S(1)] 1+rs Your answer should be a simple expression in terms of u. (b) (5 points) For what values of u is rs > rb = 0.25? Your answer should be in the form of an inequality. (c) (3 points) Find the stochastic discount factor, 8(t). That is, set &(0) = 1, and find Eu and Ed. Your answer should still be in terms of u. = (d) (5 points) Under what condition is Eu re, then it means that the stock has a risk premium. This means that the stock has a high expected return to help compensate for the risk. Put differently, you are more worried about the low return of the stock (1) than the upside (u), and you need a price below the expected value discounted at the risk-free rate. How is this worry about the "down" state reflected in the SDF? (f) (5 points) Find the equivalent martingale measure Q. That is, find the probability of the up state q under the EMM Q. Again, you should have expression in terms of q. (g) (5 points) As before, under what condition on u is q

1.25 > d=1, so that there is no arbitrage in this model. In summary, we are plugging into our standard binomial model: r = log 1.25 e' = 1.25 e" = 0.8 p=0.5 d=1 u >1.25 = (a) (2 points) With these nice numbers, it's nice to express interest rates and expected returns uncompounded, instead of continuous compounding. The risk-free rate is 25% per period. That is, if we solve r in 1 B(0) -B(1) 1+TB we get rb = 0.25 = 25%. Find the expected return on stock. That is, solve forrs in 1 S(0) = E(S(1)] 1+rs Your answer should be a simple expression in terms of u. (b) (5 points) For what values of u is rs > rb = 0.25? Your answer should be in the form of an inequality. (c) (3 points) Find the stochastic discount factor, 8(t). That is, set &(0) = 1, and find Eu and Ed. Your answer should still be in terms of u. = (d) (5 points) Under what condition is Eu re, then it means that the stock has a risk premium. This means that the stock has a high expected return to help compensate for the risk. Put differently, you are more worried about the low return of the stock (1) than the upside (u), and you need a price below the expected value discounted at the risk-free rate. How is this worry about the "down" state reflected in the SDF? (f) (5 points) Find the equivalent martingale measure Q. That is, find the probability of the up state q under the EMM Q. Again, you should have expression in terms of q. (g) (5 points) As before, under what condition on u is q

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