Quant Flashcards
1
Q
An interest rate, r, can have three interpretations:
A
(1) a required rate of return, (2) a discount rate, or (3) an opportunity cost. An interest rate reflects the relationship between differently dated cash flows.
2
Q
The time value of money establishes WHAT?
A
the equivalence between cash flows occurring on different dates. As cash received today is preferred to cash promised in the future, we must establish a consistent basis for this trade-off to compare financial instruments in cases in which cash is paid or received at different times.
3
Q
required rates of return—
A
that is, the minimum rate of return an investor must receive to accept an investment.
4
Q
True or False : we use the terms “interest rate” and “discount rate” almost interchangeably.
A
True
5
Q
opportunity costs
A
An opportunity cost is the value that investors forgo by choosing a course of action.
6
Q
r = Real risk-free interest rate + … +
A
r = Real risk-free interest rate + Inflation premium + Default risk premium + Liquidity premium + Maturity premium.
7
Q
The real risk-free interest rate is :
A
the single-period interest rate for a completely risk-free security if no inflation were expected.
8
Q
The inflation premium :
A
The inflation premium compensates investors for expected inflation and reflects the average inflation rate expected over the maturity of the debt. Inflation reduces the purchasing power of a unit of currency—the amount of goods and services one can buy with it.
9
Q
What does the time value of money establish?
A
The equivalence between cash flows occurring on different dates.
10
Q
Why is cash received today preferred to cash promised in the future?
A
Because there is a preference for immediate receipt, requiring a basis to compare cash flows at different times.
11
Q
Define an interest rate (or yield).
A
It is a rate of return that reflects the relationship between differently dated cash flows.
12
Q
If USD 9,500 today is equivalent to USD 10,000 in one year, what is the implied interest rate?
A
5.26% (USD 500/USD 9,500).
13
Q
What are the three ways interest rates can be thought of?
A
As required rates of return, discount rates, and opportunity costs.
14
Q
What is the required rate of return?
A
The minimum rate an investor must receive to accept an investment.
15
Q
How do interest rates function as discount rates?
A
They equate the value of future cash flows to their present value.
16
Q
What is the opportunity cost in the context of interest rates?
A
The value forgone by choosing one action over another, such as consuming instead of saving.
17
Q
How are interest rates determined in the market?
A
By the forces of supply and demand for funds.
18
Q
What is the formula for interest rates incorporating risk factors?
A
A:
𝑟
=
Realrisk-freerate
+
Inflationpremium
+
Defaultriskpremium
+
Liquiditypremium
+
Maturitypremium
r=Realrisk-freerate+Inflationpremium+Defaultriskpremium+Liquiditypremium+Maturitypremium.
19
Q
What does the real risk-free interest rate represent?
A
The single-period interest rate for a completely risk-free security with no expected inflation.
20
Q
What does the inflation premium compensate investors for?
A
Expected inflation over the maturity of the debt.
21
Q
What is the default risk premium?
A
Compensation for the possibility of a borrower failing to make a promised payment.
22
Q
What does the liquidity premium reflect?
A
The risk of loss if an investment needs to be quickly converted to cash.
23
Q
What is the nominal risk-free interest rate?
A
The sum of the real risk-free rate and the inflation premium.
24
Q
What are the two primary ways financial assets generate returns?
A
A: Through periodic income (e.g., dividends, interest) and capital gains or losses from price changes.
25
Q: Why is the geometric mean return often preferred for multi-period return calculations?
A: It accounts for compounding, providing a more accurate representation of portfolio growth.
26
Q: What is the relationship between arithmetic mean and geometric mean returns?
A: The geometric mean is always less than or equal to the arithmetic mean unless all returns are identical.
27
Q: What is the harmonic mean, and when is it used?
A: The harmonic mean averages rates or ratios and is used when data involves quantities like P/E ratios or costs per unit.
28
Q: What is the key difference between trimmed and winsorized means?
A: Trimmed means exclude extreme values, while winsorized means replace them with the nearest non-extreme values.
29
Q: Why might the arithmetic mean be biased upward in return calculations?
A: It does not account for compounding or variability in returns.
30
Q: What is the main application of the geometric mean in investment?
A: To measure the compound annual growth rate of an investment over multiple periods.
31
Q: In which situation is the harmonic mean particularly useful?
A: When averaging prices or ratios in cost averaging strategies.
32
Q: How is the bias of the arithmetic mean affected by return variability?
A: The greater the variability in returns, the larger the difference between the arithmetic and geometric means.
33
Q: What formula connects arithmetic, geometric, and harmonic means?
A: Arithmetic Mean
×
× Harmonic Mean = (Geometric Mean)
2
2
.
34
In fact, the geometric mean is always less than or equal to the arithmetic mean with one exception:
the two means will be equal is when there is no variability in the observations—that is, when all the observations in the series are the same.
35
the trimmed mean
removes a small defined percentage of the largest and smallest values from a dataset containing our observation before calculating the mean by averaging the remaining observations.
36
The winsorized mean is :
calculated after replacing extreme values at both ends with the values of their nearest observations, and then calculating the mean by averaging the remaining observations.
37
Flashcard 1
Q: Why don't arithmetic and geometric return computations account for portfolio cash flow timing?
A: They don't consider the timing of cash inflows and outflows, which can significantly impact returns depending on when investments are made or withdrawn.
38
What is the money-weighted return?
A: It accounts for the timing and amount of money invested, similar to the internal rate of return (IRR), showing the actual return earned by the investor.
39
Q: How is the internal rate of return (IRR) calculated in relation to cash flows?
A: It is the discount rate that equates the present value of all cash inflows and outflows to zero.
40
How does a positive cash flow differ from a negative one?
A: Positive cash flow is money received (inflows), while negative cash flow is money spent (outflows).
41
Q: What is the IRR for an investment with the cash flows
−
100
,
−
950
,
+
350
,
+
1270
−100,−950,+350,+1270 over 3 years?
A: 26.11%.
42
How does the timing of cash flows affect the money-weighted return?
A: It gives greater weight to periods with larger investments, impacting the overall return calculation.
43
Q: How is the time-weighted return calculated?
A: By breaking the period into subperiods, calculating each subperiod’s return, and linking them geometrically.
44
Q: What is the main advantage of the time-weighted return?
A: It neutralizes the effects of cash inflows and outflows, making it suitable for comparing portfolio managers.
45
Q: How does daily portfolio valuation improve time-weighted return accuracy?
A: Frequent valuation minimizes the impact of cash flow timing on return approximation.
46
Q: In a dividend-paying stock example, what is the IRR when cash flows are
−
200
,
−
220
,
+
480
−200,−220,+480?
A: 9.39%.
47
Q: What was the mean holding period return for a portfolio with yearly returns of 15% and 6.67%?
A: 10.84%.
48
Q: Why might two investors in the same fund have different money-weighted returns?
A: Differences in the timing and amounts of their investments lead to varying individual returns.
49
Q: What distinguishes the money-weighted return from the time-weighted return?
A: The money-weighted return considers the timing and amount of cash flows, while the time-weighted return eliminates their effects.
50
True or False : The arithmetic and geometric return computations do account for the timing of cash flows into and out of a portfolio
False : The arithmetic and geometric return computations do not account for the timing of cash flows into and out of a portfolio
51
Q: Why is it important to annualize returns?
A: Annualizing returns facilitates comparison by converting daily, weekly, monthly, or quarterly returns into a standard annualized rate.
52
What formula is commonly used for option pricing that requires annualized returns?
A: The Black–Scholes option-pricing model.
53
What is a limitation of annualizing short-term returns?
.
A: It assumes returns can be reinvested repeatedly at the same rate, which may not be realistic
54
How do continuously compounded returns differ from holding period returns?
.
A: Continuously compounded returns are slightly smaller than holding period returns but allow for additive properties in calculations
55
That is the significance of
𝑐 in the annualizing formula?
A:
𝑐
c represents the number of compounding periods in a year, such as 12 for monthly, 52 for weekly, or 365 for daily returns.
56
What does gross return measure?
A: Gross return is the return on assets managed, excluding management fees, custody fees, and taxes, but including trading expenses and commissions.
57
Q: How is net return different from gross return?
A: Net return accounts for all managerial and administrative expenses, reflecting what the investor actually receives.
58
Q: What does pre-tax nominal return represent?
A: It is the return without adjustments for taxes or inflation.
59
Q: How is after-tax nominal return calculated?
A: By deducting taxes on dividends, interest, and realized gains from the total return.
60
How does leverage impact returns?
A: Leverage amplifies both gains and losses; if portfolio returns exceed borrowing costs, it increases returns, otherwise it decreases them.
61
Q: What is the risk premium for an asset?
A: The return earned above the risk-free rate for taking on additional risk.
62
Q: Why might small mutual funds waive part of their expenses?
A: To remain competitive due to their limited ability to spread fixed administrative costs over a large asset base.
63
Q: Why are real returns useful for international comparisons?
A: They account for inflation differences, allowing for consistent comparisons across currencies and time periods.
64
Q: What are the three general cash flow patterns associated with fixed-income instruments?
A: 1. Discount instruments (single principal cash flow at maturity).
2. Periodic interest instruments (periodic payments and principal at maturity).
3. Level payments (uniform periodic payments combining interest and principal).
65
How do interest rate changes affect bond prices?
A: Bond prices move inversely to changes in interest rates; an increase in rates lowers bond prices, while a decrease raises them.
66
Q: What is Yield-to-Maturity (YTM) and how is it related to bond valuation?
A: YTM is the discount rate at which the present value of all future cash flows equals the bond's current price. It represents the bond’s expected annual return if held to maturity.
67
Q: What is the implied return for a fixed-income instrument?
A: The implied return is the discount rate (YTM) that equates the present value of an instrument’s cash flows to its price.
68
How is YTM for a coupon bond calculated?
A: YTM is the discount rate that equates the present value of all coupon and principal payments to the bond's price.
69
Q: What does the price-to-earnings (P/E) ratio indicate?
A: The P/E ratio measures the market price per share relative to earnings per share, indicating how much investors are willing to pay for each dollar of earnings.
70
Q: How does the dividend payout ratio affect the forward P/E ratio?
A: A higher dividend payout ratio increases the forward P/E ratio, assuming constant required return and growth rate.
71
Q: What should an investor do if the required return exceeds the expected return?
A: The stock may be overvalued, and the investor should consider avoiding or selling the position.
72
Q: What is the cash flow additivity principle?
A: It states that the present value of any future cash flow stream, indexed at the same point in time, equals the sum of the present values of its individual cash flows.
73
Q: How does the cash flow additivity principle ensure no-arbitrage?
A: By ensuring market prices reflect the true value of combined cash flows, it prevents the possibility of earning riskless profits without transaction costs.
74
Q: How is the cash flow additivity principle used to calculate implied forward interest rates?
A: It equates the present value of investing at a long-term rate to the compounded returns of sequential short-term rates, ensuring no arbitrage.
75
Q: How does cash flow additivity apply to forward exchange rates?
.
A: It ensures no arbitrage between different currencies by equating the returns from domestic and foreign investments adjusted for the forward rate
76
Q: How is the cash flow additivity principle used in option pricing?
A: By constructing replicating portfolios that match the option's cash flows in all future scenarios, ensuring no-arbitrage pricing.
77
deleteQ: What is the hedge ratio for a put option in the example?
A: The hedge ratio is 0.75, representing the number of asset units needed to replicate the option’s payoff.
78
Q: In a one-period binomial model, how are option prices determined?
A: By equating the present value of a replicating portfolio’s cash flows to its payoff under different scenarios.
79
Q: What happens if the forward rate deviates from the no-arbitrage level?
A: Investors can exploit the difference by arbitrage, earning riskless profits until equilibrium is restored.
80
What is a measure of central tendency?
A: It specifies where the data are centered and shows the "expected" value based on the observed sample, such as the mean, median, or mode.
81
: What is the arithmetic mean?
A: The arithmetic mean is the sum of all observations in a dataset divided by the number of observations.
82
What is the median, and why might it be preferred over the mean?
A: The median is the middle value of a dataset sorted in order. It is less affected by outliers compared to the mean.
83
Q: What is the mode in a dataset?
A: The mode is the most frequently occurring value. A dataset can be unimodal, bimodal, or have no mode.
84
Q: What are the three methods for dealing with outliers?
A: 1) Do nothing, 2) Delete outliers (e.g., trimmed mean), 3) Replace outliers (e.g., winsorized mean).
85
: What is a trimmed mean?
A: It is an arithmetic mean calculated after excluding a small percentage of the lowest and highest values in a dataset.
86
Q: What is a winsorized mean?
A: It replaces extreme values with the nearest specified percentile values, such as the 2.5th and 97.5th percentiles for a 95% winsorized mean.
87
Q: What are quantiles, and how are they categorized?
A: Quantiles divide a dataset into equal parts. Categories include quartiles (4 parts), quintiles (5 parts), deciles (10 parts), and percentiles (100 parts).
88
Q: What is the interquartile range (IQR)?
A: The IQR is the difference between the third quartile (Q3) and the first quartile (Q1):
IQR=𝑄3 −𝑄1
IQR=Q3−Q1.
89
How is the median identified in a histogram?
A: The median corresponds to the bin that contains the 50th percentile of the observations.
90
Q: How can quantiles assist in investment practice?
A: Quantiles help rank performance (e.g., fund rankings) and analyze characteristics like asset returns across different subsets of data.
91
What does the mean return in investments represent?
A: The mean return represents the reward of an investment.
92
Q: What does dispersion in investment returns address?
A: Dispersion addresses risk and uncertainty by showing how returns are distributed around the mean.
93
Q: How is the range of a dataset calculated?
A: Range = Maximum value − Minimum value.
94
Q: What is a key advantage and disadvantage of the range as a measure of dispersion?
A: Advantage: Easy to compute.
Disadvantage: Only uses the maximum and minimum values, ignoring the rest of the dataset
95
What does the Mean Absolute Deviation (MAD) measure?
.
A: MAD measures the average of the absolute deviations from the mean
96
Why is sample variance a preferred measure over MAD?
A: Sample variance is easier to manipulate mathematically because it uses squared deviations instead of absolute deviations.
97
What is the key difference between variance and standard deviation?
A: Variance is in squared units, while standard deviation is the square root of variance and shares the same units as the original data.
98
What is the target downside deviation?
A: It measures the dispersion of observations below a specified target, often used to assess downside risk.
99
Why is the coefficient of variation (CV) useful?
A: It measures relative dispersion, allowing comparison across datasets with different means or units of measurement.
100
How does standard deviation relate to downside deviation?
A: Standard deviation considers both upside and downside variations, whereas downside deviation focuses only on variations below a specified target.
101
Q: What does the mean return represent in investments?
A: The mean return represents the expected reward of an investment
102
What is dispersion in the context of investment returns?
A: Dispersion refers to the variability or spread of returns around the mean, addressing risk and uncertainty.
103
What do mean and variance fail to describe in an investment's return distribution?
A: They do not indicate whether large deviations are likely to be positive or negative.
104
Q: What is a symmetrical distribution?
A: A distribution where each side is a mirror image of the other, with equal loss and gain intervals having the same frequencies.
105
Q: What are the characteristics of a normal distribution?
A: The mean, median, and mode are equal, and it is completely described by its mean and variance.
106
Q: What does a positively skewed distribution indicate?
A: Frequent small losses and a few extreme gains, with a long tail on the right sid
107
Q: What does a negatively skewed distribution indicate?
A: Frequent small gains and a few extreme losses, with a long tail on the left side.
108
Q: How is skewness calculated?
A: By averaging the cubed deviations from the mean, standardized by dividing by the standard deviation cubed.
109
Q: What is kurtosis, and what does it measure?
A: Kurtosis measures the combined weight of the tails of a distribution relative to the rest of the distribution.
110
Q: What are the characteristics of leptokurtic distributions?
A: They are fat-tailed, generating more frequent extreme deviations from the mean than normal distributions.
111
Q: What does a kurtosis value of 3.0 indicate?
A: A mesokurtic distribution with tails similar to the normal distribution.
112
Q: What is excess kurtosis?
A: The kurtosis relative to the normal distribution, where a normal distribution has excess kurtosis of 0.
113
Q: What conclusion can be drawn if a stock's returns have an excess kurtosis of -0.75?
A: The distribution is thin-tailed compared to the normal distribution.
114
Q: What is indicated by a positive skewness in a cross-section of annual returns?
A: The returns are positively skewed, meaning the distribution has more extreme positive deviations than negative ones.
115
Q: What is the primary purpose of a scatter plot?
A: To display and understand potential relationships between two variables.
116
Q: How does tight clustering in a scatter plot indicate the strength of a relationship?
A: Tight clustering signals a stronger relationship, while loose clustering signals a weaker relationship.
117
Q: What does a scatter plot with data points tightly clustered along a positively sloped line suggest?
A: A strong positive linear relationship between two variables.
118
Q: What is sample covariance?
A: A measure of how two variables in a sample move together, calculated as the average product of their deviations from the mean.
119
Q: Why is covariance difficult to interpret?
A: It involves squared units, making its magnitude dependent on the variables' scale.
120
Q: What is the sample correlation coefficient?
A: A standardized measure of how two variables move together, calculated as the ratio of sample covariance to the product of standard deviations.
121
Q: What are the possible values for the correlation coefficient?
A: Between −1 and +1.
122
Q: What does a correlation coefficient of 0 indicate?
A: No linear relationship between the two variables.
123
Q: What are potential problems with interpreting a correlation coefficient?
A: Outliers, spurious correlation, and non-linear relationships.
124
Q: What is spurious correlation?
A: A correlation that arises by chance, from a calculation involving a third variable, or from relationships with a third variable.
125
Q: Why is it essential to avoid assuming causation from correlation?
A: Correlation does not imply that one variable causes changes in another; the relationship could be coincidental or influenced by a third variable.
126
ront: How does the expected value differ from the historical or sample mean?
Back:
Expected value: A forecast of the "true" mean (population mean), used to predict future outcomes.
Sample mean: A historical measure summarizing observed data, calculated as an equally weighted average of past observations.
127
Front: How do variance and standard deviation differ?
Back:
Variance
𝜎^2(𝑋)σ^2
(X): The dispersion of outcomes, expressed in squared units of the random variable.
Standard deviation
𝜎(𝑋)
σ(X): The square root of variance, expressed in the same units as the random variable.
128
Front: How does a probability tree help in investment analysis?
Back:
A probability tree represents scenarios and their associated probabilities visually. It aids in:
Structuring mutually exclusive and exhaustive outcomes.
Calculating conditional expectations for different scenarios.
Working backward to compute the unconditional expected value.
129
Front: What is the purpose of conditional variance?
Back:
Conditional variance measures the variability or risk of outcomes given a specific scenario. It provides insight into the dispersion of a random variable around its conditional mean under that scenario.
130
Front: Why is consistency between unconditional and conditional expectations important?
Back:
Consistency ensures that all probabilities and expected values align logically. Inconsistent calculations can lead to flawed investment decisions and missed opportunities.
131
Front: How are mutually exclusive and exhaustive events used in probability trees?
Back:
Mutually exclusive: Events cannot occur simultaneously.
Exhaustive: The set of events covers all possible outcomes.
These properties ensure all probabilities in a tree diagram sum to 1.
132
Front: What is the role of scenarios in investment decision-making?
Back:
Scenarios represent key factors or events that influence outcomes. By analyzing probabilities and expectations under different scenarios, investors can make informed decisions about risk and return.
133
Front: Why is Bayes’ formula important in investment decision-making?
Back:
It helps update beliefs and probabilities as new information becomes available, allowing investors to make rational decisions under uncertainty.
134
Front: How does Bayes' formula reverse the “given that” information?
Back:
Instead of calculating the likelihood of an observation given a scenario, it calculates the likelihood of a scenario given the observation.
135
Front: What are prior probabilities in Bayes’ formula?
Back:
Prior probabilities represent beliefs about the likelihood of an event before receiving new information.
136
Front: What are conditional probabilities or likelihoods?
Back:
These are the probabilities of observing specific information given a particular event or scenario has occurred.
137
Front: What is the posterior probability in Bayes’ formula?
Back:
The posterior probability is the updated likelihood of an event after considering new information. It reflects the combined impact of priors and observations.
138
Front: What does it mean if the posterior probability is higher than the prior probability?
Back:
It means the new information makes the event more likely than initially thought, reflecting a positive revision in beliefs.
139
Front: Why is Bayes' formula described as a learning mechanism in probability theory?
Back:
It allows for continuous updating of beliefs as new data arrives, adapting predictions and decisions dynamically.
140
Q: For a portfolio with weights (0.50, 0.25, 0.25) and expected returns (13%, 6%, 15%), what is
𝐸(𝑅𝑝)?
A:
𝐸(𝑅𝑝)=0.50(13)+0.25(6)+0.25(15)=11.75%
141
For weights (0.50, 0.25, 0.25) and a covariance matrix, what is
𝜎2(𝑅𝑝)?
A: Using variances and covariances:
𝜎2(𝑅𝑝)=195.875
Standard deviation:
𝜎(𝑅𝑝)=14%
142
Q: What is the impact of decreasing covariance on portfolio risk?
A: Decreasing covariance reduces portfolio variance, thus lowering portfolio risk.
143
Q: What are the bounds of the correlation coefficient?
A: The correlation coefficient ranges from −1 to +1.
144
Given a portfolio of five stocks, how many unique covariance terms, excluding variances, are required to calculate the portfolio return variance?
a 10
b 20
c 25
A is correct. A covariance matrix for five stocks has 5 × 5 = 25 entries. Subtracting the 5 diagonal variance terms results in 20 off-diagonal entries. Because a covariance matrix is symmetrical, only 10 entries are unique (20/2 = 10).
145
Which of the following statements is most accurate? If the covariance of returns between two assets is 0.0023, then the:
a assets’ risk is near zero.
b asset returns are unrelated.
c asset returns have a positive relationship.
C is correct. The covariance of returns is positive when the returns on both assets tend to be on the same side (above or below) their expected values at the same time.
146
Q: What is covariance in portfolio returns?
A: Covariance measures the degree to which two securities' returns move together. Positive covariance indicates that returns move in the same direction, while negative covariance shows they move in opposite directions.
147
Flashcard 3
Q: What is the relationship between independence and the joint probability function?
A: Two random variables are independent if P(X,Y)=P(X)P(Y). This means the joint probability is the product of individual probabilities.
148
deleteWhat insights did Isabel Vasquez gain from the equity and bond class data?
A: Vasquez observed that equity classes had higher variances and covariances than bond classes, but their correlation was lower. Long-term bonds had higher volatility than intermediate-term bonds but still showed high correlation.
149
Q: Why might historical covariance and correlation estimates be used?
A: Historical data provides a basis for forecasting future relationships between asset returns, though alternative methods like market model regressions can also be used.
150
Q: How are deviations used in covariance calculations?
A: Deviations are the differences between actual returns and their expected values. These deviations are multiplied and weighted by joint probabilities to compute covariance.
151
Q: What is shortfall risk?
A: Shortfall risk is the risk that a portfolio's value or return will fall below a minimum acceptable level over a specified time horizon.
152
What is Roy’s safety-first criterion?
A: Roy’s safety-first criterion states that the optimal portfolio minimizes the probability that portfolio return will fall below a specified threshold return RL .
153
How do you determine the optimal portfolio using the SFRatio?
A: Calculate the SFRatio for each portfolio and select the portfolio with the highest SFRatio, as it has the lowest probability of falling below the shortfall level.
154
How does the SFRatio compare to the Sharpe ratio?
A: The SFRatio is similar to the Sharpe ratio, but substitutes the shortfall level
𝑅𝐿 for the risk-free rate 𝑅𝐹. Maximizing the Sharpe ratio minimizes the probability of returns falling below 𝑅𝐹.
155
Why does the normal distribution play a role in portfolio risk analysis?
A: Normal distribution simplifies calculations for probabilities, including shortfall risks and measures like VaR, under the assumption of normally distributed returns.
156
What are VaR and stress testing used for in financial risk management?
A: VaR measures the minimum expected losses over a specified time and probability level, while stress testing estimates losses under extreme scenarios.
157
Q: What is the relationship between a lognormal distribution and a normal distribution?
A: A random variable
𝑌 follows a lognormal distribution if ln(𝑌)ln(Y) is normally distributed. Conversely, if ln(𝑌)Ln(Y) is normal, Y is lognormal.
158
Why is the lognormal distribution used to model asset prices?
A: Asset prices are bounded below by 0 and exhibit right skewness, characteristics well captured by the lognormal distribution.
159
What are the two key properties of the lognormal distribution?
A: It is bounded below by 0 and has a long right tail, making it suitable for modeling asset prices.
160
Q: How are the parameters of a lognormal distribution determined?
A: They are derived from the mean (𝜇) and variance (𝜎2) of the associated normal distribution of ln(𝑌)ln(Y).
161
How does the assumption of normality in continuously compounded returns affect the modeling of asset prices?
A: If continuously compounded returns are normally distributed, future asset prices are lognormally distributed.
162
Q: What assumptions are critical when using the lognormal distribution to model asset prices?
A: Returns are assumed to be independently and identically distributed (i.i.d.), with a constant mean and variance.
163
Q: Why is the lognormal distribution a good approximation for asset prices even if returns are not perfectly normal?
A: Due to the central limit theorem, sums of returns tend to normality, supporting the lognormal model for prices.
164
Q: What is the relationship between volatility and the standard deviation of continuously compounded returns?
A: Volatility is the annualized standard deviation of continuously compounded returns, often estimated using historical data.
165
The two most noteworthy observations about the lognormal distribution are :
Are that it is bounded below by 0 and it is skewed to the right (it has a long right tail).
166
What is Monte Carlo simulation?
A: It is a technique that generates a large number of random samples from specified probability distributions to obtain the likelihood of a range of results.
167
: How is Monte Carlo simulation used in investment applications?
A: It estimates portfolio risk and return by simulating profit and loss performance over a specified time horizon and derives performance and risk measures from a frequency distribution of portfolio returns.
168
Q: Why is Monte Carlo simulation useful for valuing complex securities?
A: It is helpful for pricing securities without an analytic formula, such as mortgage-backed securities, by modeling their sensitivity to changes in assumptions.
169
What are the key steps involved in Monte Carlo simulation?
A:
Specify the quantity of interest and underlying variables.
Set a time grid and divide the time horizon into subperiods.
Define the method for generating data based on distributional assumptions.
Generate random values for key risk factors.
Use simulated values to calculate outcomes.
Repeat trials to produce summary statistics.
170
How does Monte Carlo simulation value contingent claims, such as an Asian option?
A: By simulating stock prices over time, calculating the average price during the claim's life, and determining the payoff as the difference between the final and average stock prices, if positive.
171
Q: What are the strengths of Monte Carlo simulation compared to analytical methods?
A: It allows testing model sensitivity to assumptions and can handle complex securities and scenarios where no analytic solutions exist.
172
Q: What are the weaknesses of Monte Carlo simulation?
A: It provides statistical estimates rather than exact results and cannot independently establish cause-and-effect relationships.
173
Q: How can Monte Carlo simulation test the sensitivity of models in investment applications?
A: By modifying key assumptions, such as distributions of risk factors, and observing how changes affect outcomes.
174
Compared with analytical methods, what are the strengths and weaknesses of using Monte Carlo simulation for valuing securities?
Solution:
Strengths: Monte Carlo simulation can be used to price complex securities for which no analytic expression is available, particularly European-style options.
Weaknesses: Monte Carlo simulation provides only statistical estimates, not exact results. Analytic methods, when available, provide more insight into cause-and-effect relationships than does Monte Carlo simulation.
175
A Monte Carlo simulation can be used to:
directly provide precise valuations of call options.
simulate a process from historical records of returns.
test the sensitivity of a model to changes in assumptions—for example, on distributions of key variables.
Solution:
C is correct. A characteristic feature of Monte Carlo simulation is the generation of a large number of random samples from a specified probability distribution or distributions to represent the role of risk in the system. Therefore, it is very useful for investigating the sensitivity of a model to changes in assumptions—for example, on distributions of key variables.
176
A limitation of Monte Carlo simulation is:
its failure to do “what if” analysis.
that it requires historical records of returns.
its inability to independently specify cause-and-effect relationships.
Solution:
C is correct. Monte Carlo simulation is a complement to analytical methods. Monte Carlo simulation provides statistical estimates and not exact results. Analytical methods, when available, provide more insight into cause-and-effect relationships.
177
Q: What is bootstrap resampling?
A: A computational method that repeatedly draws samples from an observed dataset (with replacement) to estimate population parameters like mean, variance, skewness, and kurtosis.
178
How does bootstrap mimic sampling from a population?
A: By treating the observed sample as a proxy for the population, it simulates sampling by resampling with replacement from the original data.
179
: What are the steps in bootstrap resampling for investment simulations?
A:
Specify the quantity of interest and underlying variables.
Set a time grid consistent with the data's periodicity.
Use observed data as the empirical distribution for simulations.
Generate random values via resampling and simulate outcomes.
Calculate outcomes like contingent claim value.
Repeat trials and derive summary statistics.
180
Q: How is bootstrap resampling different from Monte Carlo simulation?
A: Bootstrap uses observed data to create an empirical distribution, while Monte Carlo simulation generates random data from predefined probability distributions.
181
What are the strengths of bootstrap resampling?
A:
Simple to perform.
Provides a good representation of population features based on observed data.
Avoids reliance on analytical formulas like z- or t-statistics.
182
What are the weaknesses of bootstrap resampling?
A:
Provides statistical estimates, not exact results.
Results depend on the quality and representativeness of the observed sample.
183
What are the main strengths and weaknesses of bootstrapping?
Solution:
Strengths:
Bootstrapping is simple to perform.
Bootstrapping offers a good representation of the statistical features of the population and can simulate sampling from the population by sampling from the observed sample.
Weaknesses:
Bootstrapping provides only statistical estimates, not exact results.
184
Front: Why do analysts use sampling instead of examining the entire population?
Back: Analysts use sampling to save time and money when examining the entire population is infeasible or inefficient.
185
Front: What is the key distinction between probability and non-probability sampling?
Back:
Probability Sampling: Every population member has an equal chance of being selected, producing representative samples with lower sampling error.
Non-Probability Sampling: Relies on convenience or judgment, with a higher risk of non-representative samples and potential bias.
186
Front: What is simple random sampling, and when is it appropriate?
Back:
Definition: A subset where every population member has an equal probability of selection.
Use Case: Best for homogeneous populations with broadly similar characteristics.
187
Front: How does stratified random sampling differ from simple random sampling?
Back:
Stratified Random Sampling: Divides the population into strata (subpopulations) and samples proportionally from each stratum.
Advantages: Ensures representation of all subpopulations and yields more precise parameter estimates.
188
Front: What is cluster sampling, and what are its pros and cons?
Back:
Definition: Divides the population into clusters, then randomly selects entire clusters or subsamples within clusters.
Pros: Cost-efficient for large, dispersed populations.
Cons: Yields less accurate results than other probability sampling methods due to intra-cluster similarity.
189
Front: What is convenience sampling, and what are its limitations?
Back:
Definition: Selects samples based on ease of access.
Limitations: May produce biased, non-representative samples, limiting accuracy.
190
Front: What is judgmental sampling, and when might it be used?
Back:
Definition: Relies on the researcher's expertise to select samples.
Use Case: When expert knowledge is critical to target specific, representative samples quickly (e.g., auditing).
191
Front: How does sampling error arise, and what is its impact?
Back:
Cause: Arises when only a subset of the population is sampled, leading to differences between sample statistics and population parameters.
Impact: Sampling error decreases with larger, well-chosen samples, especially in probability sampling methods.
192
Q: What does the Central Limit Theorem (CLT) state about the distribution of the sample mean?
A: For a population with mean 𝜇 and finite variance 𝜎2, the sampling distribution of the sample mean 𝑋ˉwill be approximately normal with mean 𝜇 and variance 𝜎2/𝑛 when the sample size
𝑛 is large, regardless of the population's distribution.
193
Q: Why is the Central Limit Theorem important in statistics?
A: The CLT enables us to make probability statements about the population mean using the sample mean, regardless of the population's distribution, as long as the sample size is sufficiently large. This is crucial for constructing confidence intervals and hypothesis testing.
194
Q: Are the following statements about sampling correct?
Sample means from large samples are approximately normally distributed.
The population must be normally distributed for the sample mean to be normally distributed.
A: Statement 1 is true; Statement 2 is false. The CLT ensures the sample mean is approximately normal for large sample sizes, regardless of the population's distribution.
195
Which of the following best describes the validity of the analyst’s statements?
Statement 1 is false; Statement 2 is true.
Both statements are true.
Statement 1 is true; Statement 2 is false.
Solution:
C is correct. According to the central limit theorem, Statement 1 is true. Statement 2 is false because the underlying population does not need to be normally distributed in order for the sample mean to be normally distributed.
196
: Are the following statements about sampling correct?
Sample means from large samples are approximately normally distributed.
The population must be normally distributed for the sample mean to be normally distributed.
A: Statement 1 is true; Statement 2 is false. The CLT ensures the sample mean is approximately normal for large sample sizes, regardless of the population's distribution.
197
Q: What is the standard error of the sample mean?
.
A: The standard error of the sample mean is a measure of the variability of sample means around the true population mean, often estimated using resampling methods like bootstrap
198
What is bootstrap resampling?
A: Bootstrap is a resampling method where samples are repeatedly drawn with replacement from the original data to approximate the sampling distribution of a statistic, such as the mean.
199
How are bootstrap samples generated?
A: Each resample is the same size as the original sample and is created by randomly drawing items with replacement, allowing some items to appear multiple times and others to be excluded.
200
: What is the purpose of the bootstrap sampling distribution?
A: It approximates the true sampling distribution and is used to estimate the standard error and other statistics without relying on analytical formulas.
201
: How does bootstrap compare to jackknife resampling?
A: Bootstrap resamples with replacement, producing variability across runs, while jackknife excludes one observation at a time, providing consistent results across runs.
202
Q: When is bootstrap especially useful?
A: Bootstrap is beneficial for estimating statistics of complex estimators or when no analytical formula is available. It is widely used in finance for tasks like historical simulations and performance benchmarking.
203
Q: What is hypothesis testing?
A: Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It determines whether the sample data supports a stated hypothesis about a population parameter.
204
What are the two hypotheses in hypothesis testing?
A: The null hypothesis (H₀) is the hypothesis being tested, typically suggesting no effect or no difference. The alternative hypothesis (Hₐ) is what we accept if we reject H₀, indicating an effect or difference.
205
Q: What are the six steps in the hypothesis testing process?
A:
State the hypotheses (H₀ and Hₐ).
Identify the test statistic and its distribution.
Specify the significance level (α).
State the decision rule.
Collect data and calculate the test statistic.
Make a decision (reject or fail to reject H₀).
206
Q: What is the level of significance (α)?
A: The level of significance is the probability of rejecting a true null hypothesis (Type I error). Common values are 0.01, 0.05, and 0.10, corresponding to 99%, 95%, and 90% confidence levels.
207
Q: What is a Type I error in hypothesis testing?
A: A Type I error occurs when the null hypothesis (H₀) is true, but we mistakenly reject it. Its probability is equal to the significance level (α).
208
What is a Type II error in hypothesis testing?
A: A Type II error occurs when the null hypothesis (H₀) is false, but we fail to reject it. Its probability is denoted by β.
209
Q: What is the power of a test?
A: The power of a test is the probability of correctly rejecting the null hypothesis when it is false. It is calculated as 1−𝛽
210
How does specifying a smaller significance level (α) affect Type I and Type II errors?
A: A smaller α reduces the likelihood of a Type I error but increases the probability of a Type II error, assuming the sample size remains constant.
211
Q: In hypothesis testing, what determines whether to reject or fail to reject H₀?
A: Compare the test statistic to critical values or compare the p-value to the significance level (α). Reject H₀ if the test statistic is more extreme than the critical value or if the p-value is less than α.
212
Q: What is the p-value in hypothesis testing?
A: The p-value is the smallest level of significance at which the null hypothesis can be rejected. A smaller p-value indicates stronger evidence against H₀.
213
Q: For the test H₀: μ = 10 versus Hₐ: μ ≠ 10 with a calculated t-statistic of 2.05 and critical t-values of ±1.984, what is the conclusion?
A: Reject H₀ because the test statistic (2.05) exceeds the critical value (+1.984), indicating statistical significance.
214
Q: What are the null (H₀) and alternative (Hₐ) hypotheses in hypothesis testing?
A:
The null hypothesis (H₀) is the default assumption that there is no effect or no difference.
The alternative hypothesis (Hₐ) is the statement that contradicts H₀, suggesting an effect or difference exists.
215
Q: What is the difference between a Type I error and a Type II error in hypothesis testing?
A:
Type I error: Rejecting the null hypothesis (H₀) when it is actually true (false positive).
Type II error: Failing to reject the null hypothesis when the alternative hypothesis (Hₐ) is true (false negative).
216
What is the significance level, and how does it relate to the power of a test?
A:
The significance level (α) is the probability of making a Type I error. Commonly, α = 0.05.
The power of a test is the probability of correctly rejecting H₀ when Hₐ is true (1 − β). It increases with a larger sample size or effect size.
217
Q: When is the t-distribution used instead of the normal (z) distribution in hypothesis testing?
A:
The t-distribution is used when the population standard deviation is unknown, and the sample size is relatively small. It accounts for additional variability due to estimating the standard deviation.
218
Q: How are confidence intervals related to hypothesis tests?
A:
A confidence interval provides a range of plausible values for a population parameter.
If the hypothesized value (e.g., mean or variance) falls outside the confidence interval, H₀ is rejected at the corresponding significance level.
219
Why might an analyst test variances, and what test statistic is commonly used?
A:
Analysts test variances to assess the consistency or volatility of data (e.g., returns before and after an event).
The chi-square statistic is used for single population variance tests, while the F-test is used to compare two variances.
220
What is a parametric test?
A: A parametric test is concerned with parameters of distributions (e.g., mean, variance) and depends on specific assumptions about the population distribution.
221
Q: What is a nonparametric test?
A: A nonparametric test is either not concerned with parameters or makes minimal assumptions about the population distribution, often used with ranked or ordinal data.
222
When is it appropriate to use a nonparametric test?
A: Nonparametric tests are appropriate when:
Data do not meet distributional assumptions.
There are outliers.
Data are given in ranks or use an ordinal scale.
The hypothesis does not concern a parameter.
223
Q: What are examples of nonparametric alternatives to parametric tests?
A: Examples include:
Wilcoxon signed-rank test (for a single mean or paired comparisons).
Mann–Whitney U test (for differences between means).
Sign test (for mean differences in paired data).
224
Q: Why are parametric tests preferred over nonparametric tests when assumptions are met?
A: Parametric tests are preferred because they generally have more statistical power, meaning they are better at rejecting a false null hypothesis when assumptions are satisfied.
225
Nonparametric procedures are primarily used in four situations:
(1) when the data do not meet distributional assumptions, (2) when there are outliers, (3) when the data are given in ranks or use an ordinal scale, or (4) when the relevant hypotheses do not concern a parameter.
226
An analyst suspects that, in the most recent year, excess returns on stocks have fallen below 5%. She wants to study whether the excess returns are less than 5%. Designating the population mean as μ, which hypotheses are most appropriate for her analysis?
A.H0: µ = 5% versus Ha: µ ≠ 5%
B.H0: µ> 5% versus Ha: µ < 5%
C.H0: µ< 5% versus Ha: µ > 5%
B is correct. The null hypothesis is what she wants to reject in favor of the alternative, which is that population mean excess return is less than 5%. This is a one-sided (left-side) alternative hypothesis.
227
Front: How does increasing the sample size
𝑛
n affect the t-test for correlation?
Back:
As
𝑛 increases, the t-statistic's magnitude increases.
A larger sample size makes it more likely to reject the null hypothesis if it is false.
228
Flashcard 1: Q: What is the null hypothesis (𝐻0) for a test of independence using a contingency table?
A:
𝐻0 : The two classifications are independent (e.g., dividend and financial leverage groups are not related).
229
Q: What is the alternative hypothesis (𝐻𝑎) for a test of independence using a contingency table?
A: 𝐻𝑎: The two classifications are not independent (e.g., dividend and financial leverage groups are related).
230
Q: What are dependent and independent variables in regression analysis?
A: The dependent variable (Y) is the one being explained, while the independent variable (X) is used to explain the variation in Y.
231
Q: What does a scatter plot show in linear regression?
A: It shows the relationship between the dependent variable (Y) and independent variable (X), with each point representing a paired observation.
232
Q: What is the goal of linear regression?
A: To fit a line to data that minimizes the sum of squared residuals, representing the best fit to the observed data.
233
Q: How are slope and correlation related?
A: Slope uses variance of X, while correlation uses the product of standard deviations of X and Y. Both depend on the covariance.
234
Q: What is the purpose of Ordinary Least Squares (OLS)?
A: OLS minimizes the sum of squared residuals to estimate the best-fitting regression line.
235
Q: When might the intercept (𝑏0) lack practical meaning?
A: If the independent variable (X) cannot realistically take a value of zero, the intercept may not have a meaningful interpretation.
236
What are the four key assumptions of a simple linear regression model?
A: Linearity, homoskedasticity, independence, and normality of residuals.
237
What does the assumption of linearity mean in simple linear regression?
A: The relationship between the dependent variable
𝑌and the independent variable
𝑋 is linear, and
𝑋 is non-random.
238
How can residual plots indicate a violation of the linearity assumption?
A: If residuals show a non-random pattern (e.g., curved or systematic trends), the linearity assumption may be violated.
239
What is homoskedasticity in the context of regression?
A: The variance of the residuals is constant for all values of the independent variable
𝑋.
240
What does a residual plot look like when the homoskedasticity assumption is violated?
A: Residuals may fan out or cluster at different levels of
𝑋
X, indicating heteroskedasticity.
241
What does the independence assumption mean in regression?
A: Observations (pairs of
𝑋
X and
𝑌
Y) are uncorrelated, meaning residuals are not autocorrelated.
242
How can you detect a violation of the independence assumption?
A: If residuals exhibit patterns (e.g., cycles or trends over time), the independence assumption may be violated.
243
What is the role of the normality assumption in regression?
A: Residuals must be normally distributed to perform valid hypothesis testing on regression coefficient
244
What is the purpose of ANOVA in regression analysis?
A: ANOVA decomposes the total variability of the dependent variable into variability explained by the regression (SSR) and the error (SSE). It evaluates the goodness of fit of the model using measures like the F-statistic.
245
Q: How do you interpret the F-statistic in ANOVA for regression analysis?
A: The F-statistic tests if the slope coefficient is significantly different from zero, indicating that the independent variable explains variability in the dependent variable.
246
Q: What is fintech in its broadest sense?
A: Fintech refers to technology-driven innovation in the financial services industry, including the design and delivery of financial products, companies developing these technologies, and the sector comprising such companies.
247
What are the key characteristics of Big Data?
A: Big Data is characterized by the "3Vs":
Volume: Large quantities of data.
Velocity: Rapid speed of data generation and transmission.
Variety: Data in diverse formats, such as structured, semi-structured, and unstructured data.
248
Q: What is alternative data, and why is it significant in investment analysis?
A: Alternative data is non-traditional data (e.g., social media, sensor data) used to gain insights into consumer behavior, firm performance, and trends. It enhances investment models by providing new factors to assess security prices and trade execution.
249
What are the three main sources of alternative data?
A:
Individuals: Data from social media, web searches, and e-commerce activities.
Business Processes: Data from transactions, corporate financials, and supply chains.
Sensors: Data from devices like smartphones, satellites, and the Internet of Things (IoT).
250
What are some challenges associated with Big Data in investment analysis?
A: Challenges include ensuring data quality, addressing selection bias, handling missing data, and processing unstructured or large datasets, often requiring AI and machine learning techniques.
251
What is the "fourth V" of Big Data, and why is it important?
A: The fourth V is Veracity, which relates to the credibility and reliability of data sources. Veracity is critical to ensuring accurate and trustworthy investment analysis.
252
Q: What is artificial intelligence (AI)?
A: AI refers to computer systems capable of performing tasks that traditionally require human intelligence, such as cognitive and decision-making abilities.
253
What is the main goal of machine learning (ML)?
A: The goal of ML is to automate decision-making processes by identifying patterns in data without human intervention or assumptions about the data's underlying probability distribution.
254
Q: What is the difference between supervised and unsupervised learning?
A: In supervised learning, inputs and outputs are labeled, allowing the algorithm to predict outcomes. In unsupervised learning, no labels are provided, and the algorithm identifies data structures or patterns independently.
255
Q: What is deep learning, and how does it relate to neural networks?
A: Deep learning involves using neural networks with multiple hidden layers to perform multistage, non-linear data processing, enabling advanced pattern and relationship recognition.
256
Q: What are overfitting and underfitting in machine learning?
A: Overfitting occurs when a model learns the training data too precisely, including noise, reducing its predictive power on new data. Underfitting happens when a model is too simplistic and fails to identify underlying data patterns.
257
Q: How is AI applied in finance and other fields?
A: In finance, AI is used for fraud detection, Big Data analysis, and predicting market trends. Outside finance, AI powers virtual assistants, product recommendations, and victories in games like Go and poker.
258
What is data science?
A: Data science is an interdisciplinary field that combines advances in computer science, statistics, and other disciplines to extract information from Big Data or data in general.
259
What are the five key data processing methods for managing Big Data?
A: The five key data processing methods are:
Capture: Collecting and transforming data into a usable format.
Curation: Ensuring data quality and accuracy through cleaning.
Storage: Recording, archiving, and accessing data based on its structure.
Search: Querying and reviewing large datasets.
Transfer: Moving data to analytical tools via feeds or other methods.
260
Q: Why is data visualization important for Big Data?
A: Data visualization helps format, display, and summarize Big Data in graphical forms, enabling users to identify trends, uncover relationships, and analyze data more effectively, even when dealing with unstructured datasets.
261
Q: What is text analytics, and how is it applied in investment management?
A: Text analytics involves analyzing large, unstructured text datasets to derive meaning, such as identifying future performance indicators from company filings, earnings calls, and social media.
262
Q: What is NLP, and what are its key applications in finance?
A: NLP is a field combining computer science, AI, and linguistics to analyze human language. Applications include sentiment analysis, topic analysis, fraud detection, and compliance monitoring in finance.
263
Q: Which programming languages are commonly used in data science?
A: Common programming languages include Python (user-friendly for fintech), R (statistical analysis), Java (internet applications), C/C++ (high-frequency trading), and Excel VBA (customized automation).
264
Q: How can NLP and ML improve investment decision-making?
A: NLP and ML can analyze large datasets like earnings reports, analyst commentary, and social media posts to identify sentiment, detect trends, and forecast market events, enhancing investment performance.
265
The value of an investment increases 5% before commissions and fees. This 5% increase represents:
A)
the investment’s net return.
B)
the investment’s gross return.
C)
neither the investment’s gross return nor its net return.
Explanation
C) Gross return is the total return after deducting commissions on trades and other costs necessary to generate the returns, but before deducting fees for the management and administration of the investment account. Net return is the return after management and administration fees have been deducted. (Module 1.3, LOS 1.e)
266
For an equity share with a constant growth rate of dividends, we can estimate its:
A)
value as the next dividend discounted at the required rate of return.
B)
growth rate as the sum of its required rate of return and its dividend yield.
C)
required return as the sum of its constant growth rate and its dividend yield.
Correct Answer
Explanation
Using the constant growth dividend discount model, we can estimate the required rate of return as
k
e
=
D
1
V
0
+ g
c
. The estimated value of a share is all of its future dividends discounted at the required rate of return, which simplifies to
V
0
=
D
1
k
e
- g
c
if we assume a constant growth rate. We can estimate the constant growth rate as the required rate of return minus the dividend yield. (Module 2.2, LOS 2.b)
267
A dataset has 100 observations. Which of the following measures of central tendency will be calculated using a denominator of 100?
A)
The winsorized mean, but not the trimmed mean.
r
B)
C)
Neither the trimmed mean nor the winsorized mean.
Incorrect Answer
Explanation
A The winsorized mean substitutes a value for some of the largest and smallest observations. The trimmed mean removes some of the largest and smallest observations. (Module 3.1, LOS 3.a)
268
1.3% complete
Question
A graphical depiction of a continuous distribution shows the left tail to be longer than the right tail. The distribution is best described as having:
A.leptokurtosis.
B.positive skewness.
C.negative skewness.
Solution
Incorrect. Kurtosis refers to relative peakedness of a distribution; leptokurtosis means more peaked than normal.
Incorrect. A negatively skewed distribution appears as if the left tail has been pulled away from the mean.
Correct. A negatively skewed distribution appears as if the left tail has been pulled away from the mean. The average magnitude of negative deviations from the mean is larger than the average magnitude of positive deviations.
269
13.6% complete
Question
If a stock's continuously compounded return is normally distributed, the future stock price is most likely:
A.normally distributed.
B.uniformly distributed.
C.lognormally distributed.
Solution
Incorrect because the stock price will be distributed in a lognormal manner, not in a normal or uniform manner.
Incorrect because the stock price will be distributed in a lognormal manner, not in a normal or uniform manner.
Correct because the relationship between normal and lognormal distributions is if a stock's continuously compounded return is normally distributed, then future stock price is necessarily lognormally distributed.
Simulation Methods
explain the relationship between normal and lognormal distributions and why the lognormal distribution is used to model asset prices when using continuously compounded asset returns
270
Question
An investor invests a fixed amount of money into a fund each year for three years as follows:
Year Price per Share
1 €14.00
2 €12.00
3 €17.00
The investor's average cost per share is closest to:
A.€14.05.
B.€14.33.
C.€14.63.
Solution
Correct because the harmonic mean is used to determine the average price paid per share when using cost averaging;3114+112+117=14.05
. The weighted mean formula could also be used, where the weights would be the proportion of the total number of shares purchased. However, in order to use this method a fixed investment amount would need to be created.
271
