Lesson 9 - Uncertainty, Default, and Risk

This lesson introduces uncertainty, default, and risk. The methods for calculating expected returns and risk are introduced, and also how to adjust a rate of return to include default risk. Finally, credit agencies are introduced.

Calculating Expected Returns and Risk

  • Expected returns - average returns given various outcomes
    • Similar to an average or mean
  • Promised returns - what company says will pay in the future
    • Expected returns are always lower because of loan defaults or bankruptcy
  • Return becomes a random variable
    • Random variable - variable has various outcomes and each outcome has a probability associated with it

The Expected Return is:

The expected value

Example 1
Outcome (x) Probability (p)
10 0.4
14 0.2
8 0.2
20 0.2
1.0

Note - the probability always sums to one, because all outcomes are included.  Moreover, all probabilities lie between zero and one.

An example of calculating expected value

The variance is how spread out the outcomes are around the expected return

The variance

An example of calculating the variance

Standard Deviation is how spread out the outcomes are around the expected return

The standard deviation

An example of calculating standard deviation

  • Reward is the expected return of an investment
  • Risk - the variability of the expected return
    • Measured by the standard deviation
    • Variance - the units are squared
    • Standard deviation - the same units as the expected returns
Example 2
Outcome (x) Probability (p)
90 0.5
80 0.3
105 0.1
30 0.1

Expected rate of return is 82.5, variance is 356.25, and standard deviation is 18.87

  • Investors can be defined as
  • Risk neutral - neutral between a sure bet and an expected bet if both are the same
    • Indifferent between taking $100 as a sure bet or $100 if 50% chance he receives $200 and 50% he receives $0
  • Risk averse - investors always takes the sure bet
    • An investor prefers taking $100 as a sure bet as opposed to $100 if 50% chance he receives $200 and 50% he receives $0
  • Risk taker - Investor takes the gamble for the higher payoff
    • An investor prefers taking $100 if 50% chance he receives $200 and 50% he receives $0 as opposed to taking $100 as a sure bet

In this class, all calculations are risk neutral, because they are easier to calculate

 

Default Risk and Other Risks

  • Default - when borrower does not pay back loan and/or interest:
    • Bankruptcy
    • Financial problems
    • Credit risk

Example of default risk - you give a loan to a person for $400 and he promises to pay you back in one year at 10% interest

Outcome Outcome (x) Probability (p) Expected
Return
Pays loan on time $440 0.8 10%
Pays half of loan $200 0.1 -50%
Defaults $0 0.1 -100%
1.0

What is expected return of loan?

Expected payment amount is $440 (0.8) + $200 (0.1) + $0 (0.1) = $372

What is expected return of loan? Expected return is

Calculating the rate of return when borrower defaults

Similar to the last lesson, you compare your return to a comparable U.S. government security and add a default premium to the loan

If a comparable one-year T-bill is 3%, you should expect a higher rate of return.

Outcome Outcome (x) Probability (p) Expected
Return
Pays loan on time $400 (1 + r) 0.8 ???
Pays half of loan $200 0.1 -50%
Defaults $0 0.1 -100%

What if we wanted an expected rate of return of 10%

Calculating the future value given a rate of return

Substitute $440 into the expected payment amount

Calculating the rate of return if borrower defaults

You would charge your friend 31.25% interest. The default premium is 31.25%-3% = 28.25%

All debt is financed by a combination of equity and loans

 

Credit Ratings

  • Two credit agencies for corporations:
    1. Moody's
    2. Standard and Poor's
    3. Note - there are other credit agencies
  • Divide corporations into two classes
    • Investment grade
      • Probability of default is less than 3%
    • Speculative grade (or junk)
      • During recession, default are 10%
      • During economic booms, defaults are 1.5%
Investment Grade Moody's Standard & Poor's
Exceptional Aaa, Aaa1, Aaa2, Aaa3 AAA, AAA-,AA+
Excellent Aa, Aa1, Aa3, Aa3 AA, AA-, A+
Good A, A1, A2, A3 A, A-, BBB+
Adequate Baa, Baa1, Baa2, Baa3 BBB, BBB-, BB+
Speculative Grade (Junk) Ba, Ba1, Ba2, Ba3 BB, BB-, B+
Poor B, B1, B2, B3 B, B-, CCC+
Very Poor Caa, Caa1, Caa2, Caa3 CCC, CCC-, CC+
Extremely Poor Ca, Ca1, Ca2, Ca3 CC, CC-, C+
Lowest C C
 

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