The marginal probability. It is the probability of an event where the occurrence of other events is not important.
The knowledge of another event is important. The probability of A given B, or P(A|B).
P(recession | increase in interest rates) or P(R|I)
The probability that both events occur at the same time. The probability of A and B, or P(AB).
The multiplication rule for probabilities:
\[P(AB) = P(A|B)*P(B)\]
\[P(A|B) = \frac{P(AB)}{P(B)}\] Example:
Assuming that the events an increase in rates and no increase in rates are mutually exclusive and exhaustive, we can calculate
What is the probability of recession and an increase in rates
\[P(RI) = P(R|I)*P(I) = 0.7 \times 0.4 = 0.28\] What is the probability of recession and no increase in rates
\[P(RI^C) = P(R|I^C)*P(I^C) = 0.1 \times 0.6 = 0.06\]
We can now use the total probability rule:
\[P(R) = P(R|I)*P(I) + P(R|I^C)*P(I^C)\] \[P(R) = P(RI)+ P(RI^C)\] \[P(R) = 0.28 + 0.06 = 0.34\]
Given a set of prior probabilities for an event of interest, if you can receive new information, the rule for updating the probability of the event is:
\[\text{Updated Probability} = \]
\[\frac{\text{Probability of new information given event}}{\text{Unconditional probability of new information}} \times \text{Prior probability of event}\]
If you are given \(P(B)\), \(P(A|B)\), and \(P(A|B^C)\), you can then compute \(P(B|A)\).
Example:
Doxa makes dive watches. There’s speculation that Doxa will soon announce a major expansion into overseas markets. Doxa would only do this if its managers estimated the demand to be sufficient to support the sales. If demand is sufficient, Doxa would also be more likely to raise prices. Overseas = “O” and Increase Prices = “I.”
An analyst determines the following probabilities:
The probabilities \(P(I)\) and \(P(I^C)\) are called the priors. Bayes’ formula allows us to compute \(P(I|O)\) where this is the updated probability given new information about “I.”
\[P(I|O) = P(IO)/P(O)\] \[P(IO) = P(O|I)*P(I)\]
Bayes Formula:
\[P(I|O) = \frac{P(O|I)}{P(O)}\times P(I)\]
\[P(O) = P(O|I)*P(I) + P(O|I^C)*P(I^C)\]
\[P(O) = 0.6 * 0.3 + 0.2 * 0.7 = 0.32\]
\[P(I|O) = \frac{0.6}{0.32}\times{0.3} = 0.5625\]
If the new information of “expand overseas” is announced, the prior probability of P(I) = 0.3 must be updated with the new information to give P(I|O) = 0.5625.
\[P(A \text{ or } B)=P(A)+P(B)-P(AB)\]
Prob recession P(R) = 0.34, Prob interest rate hike P(I) = 0.40, P(RI) = 0.28
\[P(R \text{ or } I) = 0.34 + 0.40 - 0.28 = 0.46\]
Knowledge on one event has no influence on the other.
\[P(A|B) = P(A)\] \[P(B|A) = P(B)\]
\[P(A)*P(B) = P(AB)\]
Example:
P(BBB bond default) = 0.03
P(CCC bond default) = 0.45
P(Both default) = 0.03 x 0.45 = 0.0135
Example:
The DJIA closes up on \(2/3\) of all days.
P(up for 5 consecutive days) = \((2/3)^5 = 0.132\)
The probability-weighted average of the possible outcomes of the random variable.
\[E(X) = \sum_{i=1}^n x_i*P(x_i)=x_1*P(x_1)+x_2*P(x_2)+\dotsc+x_n*P(x_n)\]
| Event | \(x_i\): Stock Return | \(P(x_i)\): Prob. of event | \(x_i*P(x_i)\) |
|---|---|---|---|
| Fall short of forecast | -0.03 | 0.20 | -0.0060 |
| Meet forecast | 0.01 | 0.45 | 0.0045 |
| Exceed forecast | 0.04 | 0.35 | 0.0140 |
| Expected Value = | 0.0125 |
A refined forecast that uses additional information.
Suppose that falling short, meeting, or exceeding expectations depends on weather.
\[E(X|good)=-0.03*0.10+0.01*0.50+0.04*0.40=0.018\] \[E(X|bad)=-0.03*0.30+0.01*0.40+0.04*0.30=0.007\]
The expected value of the squared deviations of each observation from the random variable’s expected value.
\[\sigma^2=\sum_{i=1}^n[x_i-E(X)]^2*P(x_i)\]
| Event | \(x_i\) | \(P(x_i)\) | \(x_i*P(x_i)\) | \([x_i-E(X)]^2*P(x_i)\) |
|---|---|---|---|---|
| Fall short of forecast | -0.03 | 0.20 | -0.0060 | 0.000361 |
| Meet forecast | 0.01 | 0.45 | 0.0045 | 0.000003 |
| Exceed forecast | 0.04 | 0.35 | 0.0140 | 0.000265 |
| \(E(X)=\) | 0.0125 | \(\sigma^2=\) 0.000629 |
The positive square root of variance.
\[\sigma = \sqrt{0.000629} = 0.0251 \text{ or } 2.51\%\]
The basic measure of how two assets move together. Covariance is the expected value of the product of the deviations of the two random variables around their respective means.
\[COV(R_i,R_j)= E\{[R_i-E(R_i)]*[R_j-E(R_j)]\}\]
The economy can experience three states. P(boom) = 0.30, p(normal) = 0.50, p(slow) = 0.20. How do the returns on stock A and B move together?
| Event | \(P(S)\) | \(R_A\) | \(R_B\) | \([R_A-E(R_A)]*[R_B-E(R_B)]*P(S)\) |
|---|---|---|---|---|
| Boom | 0.3 | 0.20 | 0.30 | (0.2-0.13)(0.3-0.14)(0.3)=0.00336 |
| Normal | 0.5 | 0.12 | 0.10 | (0.12-0.13)(0.1-0.14)(0.5)=0.00020 |
| slow | 0.2 | 0.05 | 0.000 | (0.05-0.13)(0.0-0.14)(0.2)=0.00224 |
| \(E(R_A)=0.13\) | \(E(R_B)=0.14\) | \(COV(R_A,R_B)\) 0.0058 |
The strength of the linear relationship between two random variables.
\[\rho_{A,B}=\frac{COV(R_A,R_B)}{\sigma(R_A)\sigma(R_B)}\]
\[\sigma(R_A)=(0.0028)^{0.5} = 0.0529\] \[\sigma(R_B)=(0.0124)^{0.5} = 0.1114\]
\[\rho_{A,B}=\frac{0.0058}{0.0529*0.1114}=0.984\]
The expected return of a portfolio composed of \(n\) assets is the weighted average of expected returns:
\[E(R_p) = \sum_{i=1}^n w_i \times E(R_i)\]
The portfolio variance is a function of the asset weights and the covariance of each pair:
\[VAR(R_p) = \sum_{i=1}^n \sum_{j=1}^n w_iw_jCOV(R_i,R_j)\]
Basically, you take each of the \(n^2\) pairs, multiply their portfolio weights and covariance, and sum each result. Keep in mind that the covariance of an asset and itself is just its variance.
For a 3 asset (A,B,C) portfolio:
| Asset 1 | Asset 2 | Result |
|---|---|---|
| A | A | \(w_A^2*VAR(R_A)\) |
| B | B | \(w_B^2*VAR(R_B)\) |
| C | C | \(w_C^2*VAR(R_C)\) |
| A | B | \(w_A*w_B*COV(R_A,R_B)\) |
| B | A | \(w_A*w_B*COV(R_A,R_B)\) |
| A | C | \(w_A*w_C*COV(R_A,R_C)\) |
| C | A | \(w_A*w_C*COV(R_A,R_C)\) |
| B | C | \(w_B*w_C*COV(R_B,R_C)\) |
| C | B | \(w_B*w_C*COV(R_B,R_C)\) |
\[VAR(R_p) = w_A^2*VAR(R_A) + w_B^2*VAR(R_B) + w_C^2*VAR(R_C) + \\ 2w_A*w_B*COV(R_A,R_B) + 2w_A*w_C*COV(R_A,R_C) + \\ 2w_B*w_C*COV(R_B,R_C)\]
Example:
40% in asset A, 60% in asset B
| Joint Probabilities | \(R_B = 0.40\) | \(R_B = 0.20\) | \(R_B = 0.00\) |
|---|---|---|---|
| \(R_A = 0.20\) | 0.15 | 0 | 0 |
| \(R_A = 0.15\) | 0 | 0.60 | 0 |
| \(R_A = 0.04\) | 0 | 0 | 0.25 |
\[E(R_A) = 0.13 = 0.20*0.15 + 0.15*0.60 + 0.04*0.25\] \[E(R_B) = 0.18 = 0.40*0.15 + 0.20*0.60 + 0.00*0.25\]
\[VAR(R_A) = 0.0030 = \\0.15*(0.20-0.13)^2 + 0.6*(0.15-0.13)^2 + 0.25*(0.04-0.13)^2\]
\[VAR(R_B) = 0.0156 = \\0.15*(0.40-0.18)^2 + 0.6*(0.20-0.18)^2 + 0.25*(0.00-0.18)^2\]
\[COV(R_A,R_B) = 0.0066 = \\0.15*(0.20-0.13)*(0.40-0.18) +\\ 0.60*(0.15-0.13)*(0.20-0.18) + \\0.25*(0.04-0.13)*(0.00-0.18)\]
\[E(R_P) = 0.16 = 0.40*0.13 + 0.6*0.18\]
\[VAR(R_P) = 0.009264 = \\0.40^2*0.003 + 0.60^2*0.0156 + 2*0.40*0.60*0.0066\]