1
Q
What are three other names for interest rates?
A
Required rate of return, discount rate, opportunity cost
2
Q
What is the real risk free rate and what does it represent?
A
The theoretical interest rate with no inflation or default - represents time preference
3
Q
What is a real rate of return?
A
Investor’s increase in purchasing power after adjusting for inflation
4
Q
What is the formula for nominal risk-free rates?
A
(1+nominal risk free rate) = (1+real risk free rate)(1+expected inflation rate)
Approximated as: nominal risk free rate ≈ real risk free rate + expected inflation rate
5
Q
What are three additional risks which increase the required rate of return and the full formula for this?
A
Default risk
Liquidity risk
Maturity risk
Nominal rate = real risk free rate + inflation + default + liquidity + maturity premiums
6
Q
What is HPR and how do you calculate it for one period and multiple?
A
Percentage increase in the value of an investment
One period: HPR = end/beginning - 1
Multiple: HPR = (1+HPR1)(1+HPR2)(1+HPR3)etc - 1
7
Q
What is the arithmetic mean return and how do you calculate it?
A
Simple average of returns - an unbiased estimator of the true mean of the underlying distribution
Arithmetic mean return - (R1+R2+R3 etc.)/n
8
Q
What is the geometric mean return and how do you calculate it?
A
A compound rate
Geometric mean return = n√(1+r1)(1+r2)(1+r3)etc -1
9
Q
How do you calculate the annual return?
A
As the geometric mean return but n is the number of periods converted into the equivalent number of years
10
Q
How is the harmonic mean calculated and what is it used for?
A
Harmonic mean = N/sum of 1/Xi
Used for average cost of shares purchased over time
11
Q
How do you calculate harmonic mean if there are negative numbers?
A
N / (sum of 1/(1+Xi))
12
Q
What is the relationship between arithmetic, harmonic and geometric means?
A
Arithmetic mean x harmonic mean = (geometric mean)^2
For values that are not all equal: harmonic < geometric < arithmetic
13
Q
What are the appropriate uses for:
1. Arithmetic mean
2. Geometric mean
3. Harmonic mean
4. Trimmed or winsorized mean
A
- Including all values and outliers
- Compounding over multiple periods
- Average share cost from periodic purchases
- Decrease effect of outliers
14
Q
What is the money-weighted return?
A
IRR - rate where NPV is 0
15
Q
What is time-weighted return?
A
Measures compound growth and is the rate at which $1 compounds over a time period
16
Q
How do you calculate time-weighted return?
A
Split into time periods immediately preceding significant additions or withdrawals. Calculate the HPR of each and then the total return.
17
Q
Is time-weighted return affected by the timing of cash flows?
A
No
18
Q
Which method of performance measurement is preferred and why?
A
Time weighted return because managers do not control the timing of deposits and withdrawals
Removes distortions - money weighted return is impacted if funds are put in just before poor/good performance
19
Q
What happens to the money and time weighted returns if funds are contributed to a portfolio just before a period of poor performance?
A
Money weighted return tends to be lower than the time weighted return
(and higher if before good performance)
20
Q
When is the money weighted return the most appropriate performance measure?
A
When the manager has complete control over money flows into and out of the account
21
Q
How do you annualise an HPR?
A
Annualised return = (1+HPR)^(365/days) - 1
22
Q
What is the impact of more frequent compounding?
A
Increases effective interest rate and FV (decreases PV)
23
Q
What is the present value formula?
A
PV = FV/(1+r/m)^mN
24
Q
How do you calculate continuously compounded return?
A
Rcc = ln(1+HPR) = ln(end/beginning)
25
How do you calculated PV and FV with continuous compounding?
FV = PV x e^(rxt)
PV = FV x e^-(r*t)
26
How do you calculate continuously compounded returns over multiple periods?
Sum each period
27
What are gross and net returns?
Gross = total return - commissions
Net = gross - management and admin fees
28
How do you calculate real return from nominal return?
(1+real return) = (1+ real rf)(1+ risk premium incl. inflation risk) / (1+ inflation premium
29
What is a leveraged return?
Return calculated as a percentage of an investor's cash investment
30
How do you calculated leveraged and unleveraged return?
Unleveraged = V0 x r
Leveraged = r(V0+VB) - rB*VB / V0
= (return on asset - rate paid on borrowing) / investor's cash investment
31
What happens when interest rates are negative?
Zero coupon bonds with negative yield are priced at a premium
32
What happens when the coupon rate is equal to, above and below the yield?
Coupon = yield; bond trades at par
Coupon > yield; bond trades at premium
Coupon < yield; bond trades at discount
33
How do you calculate the PV of a perpetual bond?
PV = payment/r
34
What is the difference between fixed-coupon and amortising bonds?
Fixed coupon bonds pay fixed coupon amounts and then all the principal at the end
Amortising bonds pay a level amount each period which includes coupon and part of the principal
35
How do you calculate annuity payments?
r*PV / 1 - (1+r)^-t
36
How do you calculate preferred stock value?
Dp/kp
Dividend per period / market required return
37
What are the three dividend discount models?
1. Assuming a constant future dividend (value the same as a preferred stock)
2. Assume a constant growth rate of dividends (Gordon Growth Model)
3. Assume a changing growth rate (multistage DDM)
38
What is the Gordon Growth model formula?
V0 = D1 / ke-gc
39
How do you calculate the dividend growth rate?
Retention rate x return on equity
40
How do you use the multistage DDM?
Discount short term dividends as individual cash flows and then apply constant growth DDM
41
How do you derive implied forward rates?
Based on the fact that any combination of spot and forward interest rates that cover the same period should have the same cost
ie. (1+S3)^3 = (1+S1)(1+1y1y)(1+2y1y)
42
What does an exchanger ate of 1.416 USD/EUR mean?
1 EUR is worth 1.416 USD
43
How do you calculate the arbitrage free forward exchange rate and how could there be arbitrage?
Forward/spot = (1+interest rate price currency) / (1+interest rate base currency)
Approximation: forward - spot = interest rate price - interest rate base
Could arbitrage by borrowing one currency, converting it, earning interest, and converting it back at the forward rate
44
What 4 things do you need for a binomial model?
1. Value of asset at start
2. Option exercise price
3. Returns resulting from up move and down move
4. Risk freer ate
45
What is the hedge ratio for options and how do you calculate it?
Setting the value of the option in the up and down scenarios to equal and solving for the number of shares is the number of shares they would buy for each call option`
46
What is a trimmed mean?
Excludes the lowest and highest x% of observations
47
What is a winsorised mean?
Replaces the lowest and highest x% with the next biggest/smallest value
48
What is MAD?
The average of the absolute values of the deviations of individual observations from the arithmetic mean
sum of Xi-X / n
49
How do you calculate sample variance?
The sum of each difference between observation and mean squared over n-1
50
What is relative dispersion and how is it measured?
The amount of variability around a reference point or benchmark
Measured by the coefficient of variation = standard deviation of x / average x
51
How do you calculate target downside deviation?
s target = √ sum of Xi-B squared / n-1
52
For a positively skewed distribution, where is the tail and how are the mean, median and mode related?
The tail is to the right
Mode < median < mean
53
When is sample skewness positive?
When a distribution is positively skewed
54
What does kurtosis measure?
The degree to which a distribution is more or less peaked than a normal distribution
55
What is leptokurtic, platykurtic and mesokurtic?
Leptokurtic - distribution is more peaked than normal
Platykurtic - distribution is less peaked than normal
Mesokurtic - same kurtosis as normal
56
What are the features of a leptokurtic distribution?
1. More returns clustered around the mean
2. More returns with large deviations from the mean
3. A greater percentage of small deviations from the mean and extremely large deviations from the mean
57
What is kurtosis and excess kurtosis?
A normal distribution as kurtosis of 3 and excess kurtosis is when the distribution has either more or less than this
58
What is spurious correlation?
Correlation that is a result of chance or associations with a third variable
59
How do you calculate expected value, variance and standard deviation from a probability model?
Expected value = sum of probability x return
Variance = sum of (probability x squared difference between return and expected return)
Standard deviation = √variance
60
How do you calculate expected return and variance of a portfolio?
Expected return = sum of weight x return
Variance = weight^2*variance^2 + weight^2*variance^2 + 2weightAweightBcovariance
61
How do you calculate covariance of two assets in a portfolio?
The product of each asset's return minus its expected return
62
How do you calculate sample covariance?
The sum of each product of each difference between return and expected return divided by n - 1
63
How do you calculate correlation coefficient?
= covariance / SDA*SDB
64
What is shortfall risk?
The probability that a portfolio value or return will fall below a particular target value or return
65
What is Roy's safety-first criterion and how do you calculate it?
What does this formula also calculate?
Maximise E(Rp) - RL / SDp
Number of standard deviations below the mean so the maximum safety first ratio minimises the probability of returns being below the threshold RL
Z-value
66
What kind of distribution does the logarithm of a lognormal distribution have?
Normal distribution
67
How do you calculate asset future price using continuous compounding?
Pt = P0e^r0,T
Where r0,T is the sum of continuously compounded returns over each shorter period
68
What does it mean if returns are independently and identically distribution?
Independently distributed - past returns are not useful for predicting future returns
Identically distributed - mean and variance do not change over time
69
What is monte carlo simulation based on?
The repeated generation of one or more risk factors
70
What is monte carlo simulation used for?
1. Value complex securities
2. Simulate profits/losses from a trading strategy
3. Calculate estimates of VaR
4. Simulate pension fund assets and liabilities over time
5. Value portfolios of assets that have nonnormal return distributions
71
What is an advantage and two disadvantages of monte carlo simulation?
Advantage - inputs are not limited to the range of historical data
Disadvantages - fairly complex, answers only as good as inputs
72
What is bootstrap resampling?
Drawing repeated samples from a full dataset and calculating the sample means. The SD of these sample means is an estimate for the standard error of the sample mean
73
What is simple random sampling?
Selecting a sample in a way that each item has the same likelihood of being selected
74
What is systematic sampling?
Selecting every nth item from a population
75
What is stratified random sampling?
The population is split into smaller groups and a random sample is taken from each in proportion to the size of the group relative to the population
76
What is cluster sampling (one-stage and two-stage)
Using a cluster/group to represent the whole population
One-stage - uses all data from a random sample of clusters
Two-stage - random samples are taken from each randomly sampled cluster
77
What are two nonprobability sampling methods?
Convenience sampling - selecting sample data based on ease of access
Judgemental sampling - selecting data based on experience and judgement
78
What is the central limit theorem?
For simple random samples, the sampling distribution of sample mean approaches normal distribution with mean = sample mean and variance = variance/n when the sample size is around 30
79
What is the standard error of the sample means and how do you calculate it when the population SD is known vs not known?
The standard deviation of the distribution of sample means
= population SD / √n
or
= sample SD / √n
80
What are two other ways to estimate standard error of sample mean using resampling?
Jackknife - calculating the SD of sample means where each sample has one observation removed
Bootstrapping - calculating SD of sample means from bootstrapped samples
81
How do you calculate t stat?
sample stat - hypothesised value / standard error of sample stat
82
What is a type I error and the probability of it?
Rejecting the null hypothesis when it is actually true
P(type I error) = significance level
83
What is a type II error and the probability of it?
Failing to reject the null hypothesis when it is actually false
P(type II error) = 1 - power of the test
84
How does significance level effect the probabilities of types I and II errors?
Lower significance level = lower P(type I) and higher P(type II)
85
What is a p value?
The probability of obtaining a t stat that would lead to the null hypothesis being rejected; can reject null if p value is less than the significance percentage
86
What are the 4 types of hypothesis tests and their degrees of freedom?
T-test (or z-test for large sample) - value of population mean
T-test - equality of two population means
Chi-square test - value of population variance
F-test - equality of two population variances
df is n-1 for one population, n-2 for two populations
87
What should the null and alternative hypotheses be?
Mutually exclusive and exhaustive
88
What are the critical values for
a) 10% 2 tail or 5% 1 tail
b) 5% 2 tail
c) 1% 1 tail
d) 1% 2 tail
a) 1.65
b) 1.96
c) 2.33
d) 2.58
89
What is the decision rule for a two tailed test?
Reject the null hypothesis if t stat is less than -critical value or more than +critical value (tails)
90
What percentage is in each tail of a two tail 5% test?
2.5%
91
What is the difference between paired comparison tests with dependent and independent samples?
Dependent samples - testing whether the mean of one sample is different from the mean of another sample
Null hypothesis is the difference between the two means is 0
Independent samples - testing the mean of the differences between pairs of observations (whether it is 0)
Null hypothesis is the mean difference is 0
92
How do you get the chi-square critical value?
Using the table where the probability indicates to the right of the point only
Note the distribution is bounded by 0
93
What are the hypotheses for a chi-square test?
Null hypothesis is that the variance is the hypothesised value
Alternative hypothesis is that it isn't
94
What are the hypotheses for one and two sided f tests?
One-sided test:
Null hypothesis is that the variance of one population is less/more than or equal to the variance of the other population
Null hypothesis is that the variances of the two populations are the same
Alternative hypothesis is that they are not
95
How do you calculate the f stat?
Only look at upper tail as f stat cant be less than 1
f stat = larger variance/smaller variance
96
How do you calculate critical value for an f stat from the table?
Use the table with degrees of freedom for the numerator on one axis and denominator on the other axis
This will give upper tail value (lower tail value is the reciprocal of this)
97
What is the difference between a parametric and a nonparametric test?
Parametric tests - rely on assumptions regarding the distribution and are specific to population parameters
Non parametric tests - do not consider a particular population parameter or have few population assumptions
98
When are non parametric tests used?
1. Distribution is not normal
2. Data are ranks
3. Hypothesis does not involve the parameters eg. testing whether a variable is normally distributed
99
How do you test whether the correlation coefficient for two variables is different from 0? (using a parametric test)
t-stat = r√n-2 / √1-r^2
critical value uses df of n-2
100
What is the nonparametric test of correlation?
Testing whether two sets of ranks are correlated - calculated the same way as parametric
101
How do you do an independence test using a contingency table?
Chi-square test with degrees of freedom (r-1)*(c-1)
102
What is the purpose of simple linear regression?
To explain the variation in a dependent variable in terms of the variation in a single independent variable
103
What is variation in simple linear regression and how is it calculated?
The degree to which a variable differs from its mean value
= sum of each observation-mean squared
104
What is the simple linear regression model equation?
Yi = b0 + biXi + ei
105
What is the regression line?
The line that minimises the sum of the squared differences (vertical distances) between the actual Y values and the Y values predicted by the line (SSE)
106
What is SSE
Sum of squared errors - sum of squared vertical distances between the estimated and actual y values
107
What is the intercept equation and what does it show?
B0 = Y - B1X
Shows that the regression line passes through a point with coordinates equal to the mean of each variable
108
What are the 4 assumptions of linear regression?
1. A linear relationship exists between the two variables
2. Variance of residual term is constant for all observations
3. Residual term is independently distributed
4. Residual term is normally distributed
109
What is homoskedasticity and heteroskedasticity?
Homoskedasticity - prediction errors all have the same variance
Heteroskedasticity - do not have the same variance or variance changes over time
110
What is ANOVA?
Statistical procedure for analysing the total variability of the dependent variable
111
What is SST?
Total sum of squares - total variation in the dependent variable
SST = sum of squared differences between actual Y values and the mean of Y
SST = SSR + SSE (total variation = explained + unexplained variation)
112
What is SSR?
Sum of squares regression - variation in dependent variable explained by the independent variable
SSR = sum of squared distances between predicted Y values and mean of Y
113
What is MSR?
Mean square regression
MSR = SSR/number of independent variables (MSR = SSR for simple linear regression)
114
What is SSE?
Sum of squared errors - unexplained variation in the dependent variable
SSE = sum of squared distances between actual and predicted Y values
115
What is MSE?
Mean squared error
MSE = SSE / n - 1 - no. of independent variables (MSE = SSE/n-2 for simple linear regression)
116
What is SEE?
Standard error of estimates - standard deviation of residuals
SEE = √MSE
117
What is R^2
Coefficient of determination - percentage of total variation in the dependent variable explained by the independent variable
R^2 = SSR / SST
R^2 = correlation coefficient^2 for simple linear regression
118
What is the f-test in linear regression?
Tests how well a set of independent variables explains the variation in the dependent variable (tests statistical significance of slop coefficient)
Null hypothesis is all slope coefficients is 0
Alternative hypothesis is at least one slope coefficient is not 0
119
What is the F-stat in linear regression?
F = MSR/MSE = SSR/1 / SSE/n-1
120
What does rejecting the null hypothesis of an F-stat in linear regression indicate?
That the independent variable and dependent variable have a significant linear relationship
121
What is the t-test in linear regression and how do you calculate the t stat?
Tests whether the true slope coefficient is equal to the hypothesised value
t stat = sample slope coefficient - hypothesised slope / standard error
122
How do you calculate a confidence interval for predicted values?
Y +/- two tailed critical t value * standard error of the forecast
123
What are three models to transform data to produce a linear relationship?
Log-lin model
Lin-log model
Log-log model
Format: dependent-independent
124
What is the log-lin model?
LnY = b0 + b1X + e
Slope coefficient - relative change in dependent for absolute change in independent
125
What is lin-log model?
Y = b0 + b1ln(X) + e
Slope coefficient - absolute change in dependent for relative change in independent
126
What is log-log model?
LnY = b0 + b1ln(X) + e
Slope coefficient - relative change in dependent for relative change in independent
127
What is big data?
All potentially useful information generated in the economy
128
What is corporate exhaust?
Trail of activities left by online data/transactions eg. bank records, retail scanner data
129
What are the characteristics of big data?
Volume - magnitude
Velocity - how quickly data is communicated
Variety - varying degrees of structure (structured - spreadsheets, semistructured - photos, unstructured - video)
130
What are 5 data processing methods?
Capture - collecting data
Curation - adjusting for bad or missing data
Storage - archiving and accessing data
Search - examining stored data to find needed information
Transfer - moving data from source/storage to where needed
131
What are neural networks?
An example of AI programmed to process information in a way similar to the human brain
132
What is the machine learning process?
Training dataset - looks for relationships
Validation dataset - refines relationship models
Test dataset - analyses predictive ability
133
What is supervised vs unsupervised learning?
Supervised - input and output data are labelled
Unsupervised - input data not labelled
134
What is deep learning?
Uses layers of neutral networks to identify patterns
135
What is overfitting and underfitting?
Overfitting - learns data too exactly, model is too complex
Underfitting - fails to identify actual patterns and relationships, model not complex enough
136
What is text analytics?
Analysis of unstructured data in text or voice forms
137
What is natural language processing?
Using computers and AI to interpret human language eg. speech recognition and language translation
138
What is algorithmic trading?
Computerised securities trading based on a predetermined set of rules
139
What is high-frequency trading?
A type of algorithmic trading which identifies and takes advantage of intraday securities mispricing
140
What is the z value and how do you calculate it?
Measures distance as the number of standard deviations away from the mean
Return - mean / SD
141
What percentage of observations lie within 1 SD of the mean?
68%
142
90% of observations lie within how many SD of the mean?
1.645
143
95% of observations lie within how many SD of the mean?
1.96
144
99% of observations lie within how many SD of the mean?
2.58
145
How do you use a z table?
For a z value of 0.XY - find the row with X and the column with Y
Number gives the probability to the left of that point
146
How are stock prices best modelled and why?
As a lognormal distribution as it is bounded by 0 and has positive skew
147
How are daily continuously compounded returns scaled up?
Added so multiply by the number of days
148
How is daily standard deviation scaled up?
Multiplied by the square root of time
CFA Level I
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