Consider a model where the annual return on the S&P500 is positive 70% of the ti

Question

Consider a model where the annual return on the S&P500 is positive 70% of the time and negative 30% of the time and whether the S&P500 goes up or down is independent from one year to the next. Hence if X is the number of times that the market goes up in the next 10 years then X is a Binomial (10, 0.7). a. A contract pays off $10 for each year that the S&P500 return is positive in each of the next 10 years. The payoff is then given by P=$10*X. What is the expected payoff on this contract? b. What’s the variance of the payoff? c. What is the probability that 6 out of 10 years the annual return on the S&P500 is positive?

Explanation / Answer

X~Bin(10,0.7)

E[X]=10*0.7=7

V[X]=10*0.7*0.3=2.1

SD=sqrt(2.1)=1.449

a)P=10X

Hence E[P]=10*E[X]=10*7=70

b)V[P]=100V[X]=100*2.1=210

c)P(k=6)=10C6*0.7^6*0.3^4=0.2001

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