of historical data, for which the sample mean is 15 percent. In this case the expected value may or may not be 15 percent. In most cases if we say that the expected value is equal to 15 percent, we are making two assumptions: first, we are assuming that the returns in our sample were generated by the same random process over the entire sample period; second, we are assuming that the returns will continue to be generated by this same process in the future. These are very strong assumptions. In finance and risk management, we often assume that the data we are interested in are being generated by a consistent, unchanging process. Testing the validity of this assumption can be an important part of risk management in practice.
The concept of expectations is also a much more general concept than the concept of the mean. Using the expectations operator, we can derive the expected value of functions of random variables. As we will see in subsequent sections, the concept of expectations underpins the definitions of other population statistics (variance, skew, kurtosis), and is important in understanding regression analysis and time series analysis. In these cases, even when we could use the mean to describe a calculation, in practice we tend to talk exclusively in terms of expectations.
Sample Problem
Question:
At the start of the year, you are asked to price a newly issued zero-coupon bond. The bond has a notional value of $100. You believe there is a 20 percent chance that the bond will default, in which case it will be worth $40 at the end of the year. There is also a 30 percent chance that the bond will be downgraded, in which case it will be worth $90 in a year's time. If the bond does not default and is not downgraded, it will be worth $100. Use a continuous interest rate of 5 percent to determine the current price of the bond.
Answer:
We first need to determine the expected future value of the bond, that is, the expected value of the bond in one year's time. We are given the following:
Because there are only three possible outcomes, the probability of no downgrades and no default must be 50 percent:
The expected value of the bond in one year is then:
To get the current price of the bond we then discount this expected future value:
The current price of the bond, in this case $80.85, is often referred to as the present value or fair value of the bond. The price is considered fair because the discounted expected value of the bond is the rational price to pay for the bond, given our knowledge of the world.
The expectations operator is linear. That is, for two random variables, X and Y, and a constant, c, the following two equations are true:
(3.31)
If the expected value of one option, A, is $10, and the expected value of option B is $20, then the expected value of a portfolio containing A and B is $30, and the expected value of a portfolio containing five contracts of option A is $50.
Be very careful, though; the expectations operator is not multiplicative. The expected value of the product of two random variables is not necessarily the same as the product of their expected values:
(3.32)
Imagine we have two binary options. Each pays either $100 or nothing, depending on the value of some underlying asset at expiration. The probability of receiving $100 is 50 percent for both options. Further, assume that it is always the case that if the first option pays $100, the second pays $0, and vice versa. The expected value of each option separately is clearly $50. If we denote the payout of the first option as X and the payout of the second as Y, we have:
(3.33)
It follows that E[X]E[Y] = $50 × $50 = $2,500. In each scenario, though, one option is valued at zero, so the product of the payouts is always zero: $100 · $0 = $0 · $100 = $0. The expected value of the product of the two option payouts is:
(3.34)
In this case, the product of the expected values and the expected value of the products are clearly not equal. In the special case where E[XY] = E[X]E[Y], we say that X and Y are independent.
If the expected value of the product of two variables does not necessarily equal the product of the expectations of those variables, it follows that the expected value of the product of a variable with itself does not necessarily equal the product of the expectations of that variable with itself; that is:
(3.35)
Imagine we have a fair coin. Assign heads a value of +1 and tails a value of –1. We can write the probabilities of the outcomes as follows:
(3.36)
The expected value of any coin flip is zero, but the expected value of X2 is +1, not zero:
(3.37)
As simple as this example is, this distinction is very important. As we will see, the difference between E[X2] and E[X]2 is central to our definition of variance and standard deviation.
Sample Problem
Question:
Given the following equation:
What is the expected value of y? Assume the following:
Answer:
Note that E[x2] and E[x3] cannot be derived from knowledge of E[x]. In this problem, E[x2] ≠ E[x]2. As forewarned, the expectations operator is not necessarily multiplicative. To find the expected value of y, then, we first expand the term (x + 5)3 within the expectations operator:
Because the expectations operator is linear, we can separate the terms in the summation and move the constants outside the expectations operator. We do this in two steps:
At this point, we can substitute in the values for E[x], E[x2], and E[x3], which we were given at the start of the exercise:
This gives us the final answer, 741.
Variance and Standard Deviation
The variance of a random variable measures how noisy or unpredictable that random variable is. Variance is defined as the expected value of the difference between the variable and its mean squared:
(3.38)
where σ2 is the variance of the random variable X with mean μ.
The square root of variance, typically denoted by σ, is called standard deviation. In finance we often refer to