Addition Rule of Probability
ANSWER
ADDITION: P(A or B) = P(A) + P(B) - P(AB)
Roy's Safety First Criterion
ANSWER
Safety First Ratio = (E(R) - Rₜ) / σ
Larger ratio is better
If (Rₜ) is risk free rate, then it becomes Sharpe Ratio
Sharpe Ratio
ANSWER
Sharpe Ratio = (E(R) - RFR) / σ
Larger ratio is better
If (Rt) is higher than RFR, then it becomes Safety First Ratio
Central Limit Theorem
ANSWER
If we take samples of a population, with a large enough sample size, the distribution of all
sample means is normal with:
1
,- A mean equal to the population mean
- A variance equal to the population variance divided by sample size (σ² / n)
Standard Error of Sample Mean
ANSWER
σ / n^½
P - Value
ANSWER
Based on a calculated test statistic, rather than a significance level (which is chosen)
p-value = smallest significance level at which an analyst can reject the null hypothesis
one-tailed test - "less than or equal to"
two-tailed test - "equal to"
Binomial Probability
ANSWER
One of two possible outcomes (i.e. success/failure)
Possible outcomes can be demonstrated in binomial tree
Use "nCr" on calculator to solve:
nCr = P(success)^x * P(failure)^(n-x)
2
,Cumulative Distribution Function
ANSWER
Gives the probability that a random variable will have an outcome less than or equal to a
specific value (represented by F(x))
F(x) = probability of an outcome less than or equal to x
Standard normal table (z) shows cumulative probabilities
Effective Annual Yield
ANSWER
EAY = (1 + (i/n))^n - 1
Stated Rate = (EAY^(1/n) - 1) * n
Continuous Compounding
ANSWER
ln(EAY) = continuously compounded stated rate
e^(continuously compounded stated rate) = EAY
Type I Error
ANSWER
Incorrectly rejecting a true null hypothesis
(convicting an innocent person is Type I)
Type II Error
ANSWER
3
, Failure to reject a false null hypothesis
(failure to convict a guilty person is Type II)
Significance Level / Power of a Test
ANSWER
Significance Level = Probability of Type I
Power of a Test = (1 - Probability of Type I)
Covariance (Probability Model)
ANSWER
Covariance of random variables A and B from probability model
On the calculator:
1) Enter returns for set A and joint probabilities for AB; find mean A
2) Enter returns for set B and joint probabilities for AB; find mean B
3) Multiply each joint probability AB by each set's returns minus means
(ex: P(AB1)(A1 - Mean A)(B1 - Mean B) + P(AB2)(A2 - Mean A)(B2 - Mean B) + ... + P(ABn)(An
- Mean A)(Bn - Mean B))
4) The summed total is your covariance
Covariance (Sample)
ANSWER
Covariance of random variables A and B from sample with historical data with n observa-
tions
Bank Discount Yield (Discount basis)
4
ANSWER
ADDITION: P(A or B) = P(A) + P(B) - P(AB)
Roy's Safety First Criterion
ANSWER
Safety First Ratio = (E(R) - Rₜ) / σ
Larger ratio is better
If (Rₜ) is risk free rate, then it becomes Sharpe Ratio
Sharpe Ratio
ANSWER
Sharpe Ratio = (E(R) - RFR) / σ
Larger ratio is better
If (Rt) is higher than RFR, then it becomes Safety First Ratio
Central Limit Theorem
ANSWER
If we take samples of a population, with a large enough sample size, the distribution of all
sample means is normal with:
1
,- A mean equal to the population mean
- A variance equal to the population variance divided by sample size (σ² / n)
Standard Error of Sample Mean
ANSWER
σ / n^½
P - Value
ANSWER
Based on a calculated test statistic, rather than a significance level (which is chosen)
p-value = smallest significance level at which an analyst can reject the null hypothesis
one-tailed test - "less than or equal to"
two-tailed test - "equal to"
Binomial Probability
ANSWER
One of two possible outcomes (i.e. success/failure)
Possible outcomes can be demonstrated in binomial tree
Use "nCr" on calculator to solve:
nCr = P(success)^x * P(failure)^(n-x)
2
,Cumulative Distribution Function
ANSWER
Gives the probability that a random variable will have an outcome less than or equal to a
specific value (represented by F(x))
F(x) = probability of an outcome less than or equal to x
Standard normal table (z) shows cumulative probabilities
Effective Annual Yield
ANSWER
EAY = (1 + (i/n))^n - 1
Stated Rate = (EAY^(1/n) - 1) * n
Continuous Compounding
ANSWER
ln(EAY) = continuously compounded stated rate
e^(continuously compounded stated rate) = EAY
Type I Error
ANSWER
Incorrectly rejecting a true null hypothesis
(convicting an innocent person is Type I)
Type II Error
ANSWER
3
, Failure to reject a false null hypothesis
(failure to convict a guilty person is Type II)
Significance Level / Power of a Test
ANSWER
Significance Level = Probability of Type I
Power of a Test = (1 - Probability of Type I)
Covariance (Probability Model)
ANSWER
Covariance of random variables A and B from probability model
On the calculator:
1) Enter returns for set A and joint probabilities for AB; find mean A
2) Enter returns for set B and joint probabilities for AB; find mean B
3) Multiply each joint probability AB by each set's returns minus means
(ex: P(AB1)(A1 - Mean A)(B1 - Mean B) + P(AB2)(A2 - Mean A)(B2 - Mean B) + ... + P(ABn)(An
- Mean A)(Bn - Mean B))
4) The summed total is your covariance
Covariance (Sample)
ANSWER
Covariance of random variables A and B from sample with historical data with n observa-
tions
Bank Discount Yield (Discount basis)
4
