Using moment generating function to find standard deviation and expected value

Anna is planning to invest 1000 euros for two years. She will choose between two savings accounts offered by the Nordea Bank of Finland:

- A standard fixed-term account which has a guaranteed interest rate of 5.5% after the two years

- A 'DepositPlus' account, for which the interest rate depends on the stock prices of three companies (Apple, Heineken and Wal-Mart) as follows:

- If the stock prices of all three companies are higher on 16 September 2013 than they were on 5 October 2011, the two-year interest rate is 8.1%.
- If not, the two-year interest rate is 1.1%.

Denote by X the two-year interest rate of the DepositPlus account, and by Y the two-year interest-rate of the standard account. Let pi denote the probability that the condition for the higher interst rate of the DepositPlus account is satisfied at the end of the period.

Calculate the expected value and standard deviation of X and the expected value and standard deviation of Y.

What I needed help with in this question was how to find the probability functions of X and Y, so any help would be greatly appreciated :)

Re: Using moment generating function to find standard deviation and expected value

Hey sakuraxkisu.

The first account won't have a stochastic element at all and the second one will be based on a bernoulli random variable (yes/no).

What I will ask you to do first is write an expression for the random variable for both with the hints that the first one doesn't change (what does that tell you about the variance?) and the second one is based on a yes/no situation (which is a Bernoulli random variable).

Now for the second one, can you write an equation to get the rate so that the variable returns one rate of the bernoulli is 0 and another rate if it is 1?

Re: Using moment generating function to find standard deviation and expected value

So for the first one, would the pmf be p(x)=5.5 ? And would the second one be p(y)=pi when y=1 and p(y)=1-pi when y=0? Thank you

Re: Using moment generating function to find standard deviation and expected value

The probability function will be equal to 1 for the first instrument and you will have P(X = 5.5) = 1 where X is the random variable for the first instrument.

For the second one you are creating a mask. The probability function will be P(Y = 1) = pi and P(Y = 0) = 1 - pi but the price for the second instrument well be Z = 8.1*Y + 1.1*(1-Y) where Y has that distribution.