# Expected value

Please open the attached excel and word files for the example.

Assume I am valuing the cash flows of a simple company with contracts 'a' through 'f' that have different varying cash flows over 10 years.

Probability of default for each contract a-f in any given year = 2.50%. Once there is a default, the cash flows cease and there is no ability to recover future cash flows. Defaults are not correlated.

What is the expected value of cash flows for each year? How do I model this?

Do we need to assume a normal distribution?

What happens if I increase the number of contracts?

What if two of the contracts had some correlation of defaults? Three contracts? Assume 25% correlation.

What is the expected value? How do I model this?

#### Solution Preview

This problem can be approached in the following way. For simplicity let's start by assuming that the only contract you have is "a".

At the end of year 1, we'll receive $10 if there is no default, and $0 if there is a default. The probability of a default is 0.025 (so the probability of no default is 0.975). Therefore, the expected value of the cash flow of this contract at the end of year 1 is:

Prob(No Default)*10 + Prob(Default)*0

=Prob(No Default)*10

=0.975*10

=$9.75

Now let's see what's the expected cash flow at year 2. What's the probability that the contract "survives" (doesn't default) until year 2? Since defaults are not correlated through time, if the contract survived through year 1, it still has a probability 0.975 of surviving again through year 2. Therefore the probability that it survives from the beginning through year 2 is:

Prob(No Default on Year 1)*Prob(No Default on Year 2)

=0.975*0.975

=0.975^2

=0.950625

Therefore, since the cash flow under no default in year 2 is $10.20, the expected value of the cash flow in this year is simply:

0.950625*10.20

= $9.696375

The idea should now be clear. The probability that the contract makes it from the beginning through year 3 without default will be 0.975^3; for year 4 ...

#### Solution Summary

This shows how to work with expected value with cash flows and contracts.