# Level 1 CFA® Exam: Bayes' Formula Explained

CFA Exam: Level 1 / Quantitative Methods / Bayes' Formula

## No Formulas – Just Logical Thinking!

If you have trouble doing questions with Bayes' formula, here is an alternative way of solving this kind of problems in your Level 1 CFA Exam. Using this solution, you need no formulas – just logical thinking.

## Level 1 CFA Exam-Type Question: Bayes' Theorem

PROBLEM:

You have two classes of bonds in your portfolio. 90% of the portfolio consists of bonds with A rating. The rest are junk bonds. The probability of default for bonds with A rating is 3% and for junk bonds it is 20%. What is the probability that a randomly selected bond is a junk bond if it is in default?

SOLUTION:

If I ask you a question:

What is the probability that a randomly selected bond is a junk bond?,

you will say:
It is 10% because in the portfolio there are only two types of bonds: A rating and junk bonds, and A rating bonds constitute 90% of the whole portfolio. Thus, the rest are junk bonds.

However:

The probability that a randomly selected bond is a junk bond IF it is in default is a conditional probability. Why? Because we have a condition here:
We know that a bond is in default.

Intuitively we can see that: The probability that a randomly selected bond is a junk bond IF it is in default is higher than the probability that a randomly selected bond is a junk bond. How should we tackle this problem, though?

### Bayes' Formula

Bayes' formula comes in handy:

$P(A│B)=\frac{P(B│A)}{P(B)}\times P(A)$

where:

$P(A│B)$ – probability of event given the new information,

$P(B│A)$ – probability of the new information given event,

$P(B)$ – unconditional probability of the new information,

$P(A)$ – prior probability of event.

What if we do not remember the formula or we don't know how to use it? What do we do then?

## Bayes' Formula Explained Without Formulas

Assume that we have 1000 bonds in our portfolio.

Because 90% of bonds are A rating bonds, it means that we have 900 A rating bonds in the portfolio.

So, the number of junk bonds is 1000 minus 900, i.e. 100 bonds.

The probability of default for bonds with A rating is 3%, so the number of bonds with A rating that will be in default is 3% times 900 bonds, which gives us 27 bonds.

The probability of default for junk bonds is higher and amounts to 20%. Therefore, the number of junk bonds that will be in default is 20% times 100 bonds, i.e. 20 bonds.

So, in the portfolio we have 27 plus 20 bonds, i.e. 47 bonds that will be in default.

So, the probability that a randomly selected bond is a junk bond IF it is in default is 20 junk bonds in default divided by 47 bonds in default total, i.e. 42.55%.

### Summary and Interpretation

The probability that a randomly selected bond is a junk bond is unconditional probability, and the probability that a randomly selected bond is a junk bond IF it is in default is a conditional probability.

If you are a math lover, you will probably say: “But hey! Using Bayes' formula, I do quite the same calculations...”. That's definitely true – but lots of people prefer the alternative solution.

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