Quantative Methods Flashcards
Interpret and understand the different rate of return calculations and when to use them
What is the arithmetic rate of return, and how is it calculated?
The arithmetic rate of return is the simple average of a series of periodic returns. It is calculated as:
ArithmeticRateofReturn
ArithmeticRateofReturn=
∑𝑅𝑡 / n
Where:
𝑅𝑡 is the return for each period, and
𝑛 n is the number of periods.
Why is the arithmetic rate of return considered a simple average of returns?
It treats each period’s return equally, assuming they are independent and do not compound over time. This makes it straightforward to compute and interpret as a snapshot of average performance.
What assumptions does the arithmetic rate of return make about the reinvestment of returns over time?
The arithmetic rate assumes that returns are not reinvested and do not compound. It ignores the impact of volatility and time on investment growth.
How does the arithmetic rate of return differ from the geometric rate of return in measuring investment performance?
The arithmetic rate provides the simple average of returns, while the geometric rate accounts for compounding. The geometric rate of return reflects the actual average growth rate per period, making it more accurate for long-term performance evaluation.
In what scenarios might the arithmetic rate of return provide misleading insights into an investment’s long-term performance?
It can be misleading for volatile investments because it does not account for compounding or the sequence of returns. For example, large negative returns can have a disproportionate effect on actual portfolio value, which the arithmetic rate does not capture.
Why might the arithmetic rate of return overestimate the average performance of a volatile investment?
It ignores the negative compounding effects of volatility. For example, if an investment gains 50% in one year and loses 50% the next, the arithmetic return is 0%, but the actual portfolio value is down 25%.
How does the absence of compounding in the arithmetic rate of return calculation affect its usefulness for decision-making?
It makes the arithmetic rate less suitable for evaluating long-term growth. Investors might overestimate returns, especially when there is significant volatility.
If an investment returns 8%, -4%, and 12% over three years, what is the arithmetic rate of return?
ArithmeticRateofReturn=
{ 8+(−4)+12 }/ 3 = 16/3
=5.33%
The arithmetic rate of return is 5.33%.
How would you interpret the arithmetic rate of return in the context of annualised returns for an investor?
The arithmetic rate provides an average measure of the returns over individual periods but does not reflect the true annual growth of the investment. It is useful for short-term analysis but can overstate returns for longer horizons.
Consider two investments with the same arithmetic rate of return but different levels of volatility. How would this affect an investor’s perception of risk and potential returns?
Higher volatility increases the risk of negative compounding, making the geometric rate of return lower than the arithmetic rate. Investors might prefer the less volatile investment despite similar arithmetic averages.
In what situations would a financial analyst use the arithmetic rate of return instead of the geometric rate of return?
It is used when estimating expected returns over a single period or when constructing portfolio returns for risk and return analysis. For example, it is used in the Capital Asset Pricing Model (CAPM) to estimate expected returns.
How can the arithmetic rate of return be used in portfolio optimisation and asset allocation decisions?
The arithmetic return is often used as an input for estimating expected returns when optimising portfolios under the mean-variance framework. However, adjustments are made for volatility and compounding to ensure realistic projections.
Calculate the arithmetic rate of return for a portfolio with the following yearly returns: 10%, 5%, -2%, and 15%.
ArithmeticRateofReturn=
10+5+(−2)+15 / 4 = 28/4 =
=7%
The arithmetic rate of return is 7%.
If the same portfolio had a geometric rate of return of 6%, what conclusions can you draw about the difference between these two measures?
The arithmetic rate of return (7%) is higher than the geometric rate (6%) because the latter accounts for compounding and volatility. This difference highlights the impact of fluctuations on actual investment performance.
What is the geometric rate of return, and how is it calculated?
The geometric rate of return is the compounded average rate of return over multiple periods. It reflects the actual growth rate of an investment, considering the effects of compounding.
GeometricRateofReturn= ||FIND EQUATION ELSEWHERE||
How does the geometric rate of return differ from the arithmetic rate of return?
The geometric rate accounts for compounding and reflects the actual average annual growth rate of an investment. In contrast, the arithmetic rate is a simple average and does not consider compounding or the sequence of returns.
Why is the geometric rate of return considered more accurate for evaluating long-term performance?
It accounts for compounding, which reflects the true value growth over time. It also incorporates the effects of volatility, making it more representative of an investment’s performance over multiple periods.
What are the limitations of the geometric rate of return?
Calculating it is more complex than the arithmetic rate. It also assumes reinvestment of all returns and does not reflect the impact of external cash flows like withdrawals or additional contributions.
How does volatility affect the geometric rate of return?
Volatility reduces the geometric return relative to the arithmetic return. The greater the variability in returns, the larger the gap between the arithmetic and geometric rates.
What happens to the geometric rate of return when all periodic returns are identical?
When all returns are identical, the geometric and arithmetic rates of return are equal, as there is no impact from compounding or volatility.
How does the geometric rate of return relate to risk-adjusted performance measures?
The geometric rate is often used in risk-adjusted metrics like the Sharpe ratio because it provides a realistic measure of return that incorporates the effects of volatility.
If an investment returns 8%, -4%, and 12% over three years, what is the geometric rate of return?
GeometricRateofReturn=((1+0.08)(1−0.04)(1+0.12))^1/3 - 1
=(1.08×0.96×1.12)^1/3 - 1
=(1.16006)^1/3 - 1
=0.0507or5.07%
How would you interpret the geometric rate of return for a long-term investment?
The geometric rate reflects the annualised growth rate of the investment, accounting for compounding. It represents what an investor would effectively earn per year if the returns were constant.
How does the geometric rate of return handle the sequence of returns differently from the arithmetic rate of return?
The geometric rate adjusts for the sequence of returns by compounding them. Poor returns in one period reduce the base for subsequent growth, which is not reflected in the arithmetic rate.
Why is the geometric rate of return always less than or equal to the arithmetic rate of return?
The geometric return accounts for compounding and the effects of volatility. When returns vary, negative compounding reduces the geometric return relative to the arithmetic return. They are equal only when returns are constant.
If an investment has a geometric return of 5% over 10 years, what does this imply about the overall growth?
It implies the investment grows by approximately 5% per year on a compounded basis. Over 10 years, the total growth factor would be:
TotalGrowthFactor=(1+0.05)^10 = 1.629
The investment grows by approximately 62.9% over 10 years.
How is the geometric rate of return used in evaluating fund or portfolio performance?
It provides a realistic measure of long-term performance, accounting for compounding. It is used in comparing investment strategies, benchmarking fund performance, and calculating risk-adjusted returns.
What is the Harmonic mean?
The harmonic mean, 𝑋𝐻
, is another measure of central tendency. The harmonic mean is appropriate in cases in which the variable is a rate or a ratio. The terminology “harmonic” arises from its use of a type of series involving reciprocals known as a harmonic series.
Give an example of when you would use the Harmonic Mean
The harmonic mean is used most often when the data consist of rates and ratios, such as P/Es. Suppose three peer companies have P/Es of 45, 15, and 15. The arithmetic mean is 25, but the harmonic mean, which gives less weight to the P/E of 45, is 19.3.
Suppose an investor invests EUR1,000 each month in a particular stock for n = 2 months. The share prices are EUR10 and EUR15 at the two purchase dates. What was the average price paid for the security?
Purchase in the first month = EUR1,000/EUR10 = 100 shares.
Purchase in the second month = EUR1,000/EUR15 = 66.67 shares.
The investor purchased a total of 166.67 shares for EUR2,000, so the average price paid per share is EUR2,000/166.67 = EUR12.
The average price paid is in fact the harmonic mean of the asset’s prices at the purchase dates. Using Equation 5, the harmonic mean price is 2/[(1/10) + (1/15)] = EUR12. The value EUR12 is less than the arithmetic mean purchase price (EUR10 + EUR15)/2 = EUR12.5.
How does the harmonic mean differ from the arithmetic and geometric means?
The harmonic mean gives greater weight to smaller values, making it lower than both the arithmetic and geometric means when the data includes variability. It is most suitable for averaging rates (e.g., speeds or yields) or ratios.
When is it most appropriate to use the harmonic mean?
It is used when dealing with ratios or rates, such as in scenarios involving average speed, financial ratios, or price-earnings (P/E) ratios in finance. For example, it is ideal for calculating the average rate of return when equal amounts are invested in each period.
What are the limitations of the harmonic mean?
It is sensitive to very small values, which can disproportionately reduce the result. It is not suitable for datasets containing zeros, as dividing by zero makes the calculation undefined.
Why is the harmonic mean commonly used for calculating average price-earnings (P/E) ratios in finance?
The harmonic mean provides a more accurate reflection of the overall P/E ratio of a portfolio, particularly when weights are equally distributed. It avoids the distortions caused by outliers in high P/E ratios that might skew the arithmetic mean.
If a car travels at speeds of 40 km/h and 60 km/h for equal distances, what is the average speed?
The average speed is the harmonic mean because the distances are the same:
2 ÷ (1/40+1/60) = 48km/h
The average speed is 48 km/h.
Why is the harmonic mean preferred over the arithmetic mean in averaging speeds for equal distances?
The harmonic mean accurately accounts for the relationship between speed and time. Using the arithmetic mean would incorrectly assume equal time intervals rather than equal distances.
What is the trimmed mean?
The trimmed mean is a measure of central tendency calculated by removing a specified percentage of the smallest and largest values in a dataset before calculating the arithmetic mean of the remaining values. It is often used to reduce the influence of outliers or extreme values in a dataset. For example, a 10% trimmed mean removes the lowest 10% and highest 10% of values and then averages the rest.
Calculate the 20% Winsorised mean for the dataset: 3, 7, 8, 12, 15, 18, 24.
- Sort the data: 3, 7, 8, 12, 15, 18, 24.
- Replace the lowest 20% (3) with the next smallest value (7), and the highest 20% (24) with the next largest value (18): 7, 7, 8, 12, 15, 18, 18.
- Calculate the mean of the adjusted dataset:
WinsorisedMean=
(7+7+8+12+15+18+18)
/7
=
85
/7
≈12.14
The 20% Winsorised mean is approximately 12.14.
How does the Winsorised mean reduce the impact of outliers compared to the arithmetic mean?
Instead of removing outliers, it limits their influence by replacing extreme values with less extreme ones. This ensures all data points contribute to the mean while still mitigating distortion caused by outliers.
What is the Winsorised mean?
The Winsorised mean is a measure of central tendency where a specified percentage of the smallest and largest values in a dataset are replaced with the nearest remaining values. Unlike the trimmed mean, which removes extreme values, the Winsorised mean retains all data points but reduces the impact of outliers by capping their values. For example, in a 10% Winsorised mean, the lowest 10% of values are replaced with the smallest remaining value, and the highest 10% are replaced with the largest remaining value.
How to compute the exact time-weighted rate of return on a portfolio?
Price the portfolio immediately prior to any significant addition or withdrawal of funds. Break the overall evaluation period into subperiods based on the dates of cash inflows and outflows.
Calculate the holding period return on the portfolio for each subperiod.
Link or compound holding period returns to obtain an annual rate of return for the year (the time-weighted rate of return for the year). If the investment is for more than one year, take the geometric mean of the annual returns to obtain the time-weighted rate of return over that measurement period.
How would you annualise an 18-month holding period return?
Because one year contains two-thirds of 18-month periods, c = 2/3 in the above equation. For example, an 18-month return of 20 percent can be annualized as follows:
𝑅𝑎𝑛𝑛𝑢𝑎𝑙=(1+𝑅18𝑚𝑜𝑛𝑡ℎ)^2/3−1=(1+0.20)^2/3−1=0.1292=12.92%
.
What is the concept continuously compounded return?
The continuously compounded return associated with a holding period return is the natural logarithm of one plus that holding period return, or equivalently, the natural logarithm of the ending price over the beginning price (the price relative).
Greek Coupon Bond Implied Return
Recall from Example 2 that seven-year Greek government bonds issued in 2020 with a 2.00 percent annual coupon had a price of EUR93.091 per EUR100 principal two years later.
What is the implied two-year return for an investor able to reinvest periodic interest at the original YTM of 2.00 percent?
Solution:
−1.445 percent
Using Equation 18 as for the discount bond, we must first calculate the future value (FV) after two years including the future price of 93.091 and all cash flows reinvested to that date:
FV2 = PMT1(1 + r) + PMT2 + PV2
FV2 = 97.13 = 2 × (1.02) + 2 + 93.091
𝑟= −1.445%=(97.13100)12−1
.
This negative annualized return was due to a EUR6.91 (=100 − 93.091) capital loss which exceeded the periodic interest plus reinvestment proceeds of EUR4.04 (=2 × (1.02) + 2). The fall in the price of the bond to EUR93.091 in 2022 was a result of a rise in Eurozone inflation over the period.
For an investor purchasing the 2 percent coupon bond in 2022 at a price of EUR93.091, recall that we solved for an r (or YTM) of 3.532 percent. Note that this YTM calculation for the remaining five years assumes all cash flows can be reinvested at 3.532 percent through maturity. The rate(s) at which periodic interest can be reinvested is critical for the implied return calculation for coupon bonds as shown below. YTM is computed assuming that the bond is held to maturity.
Suppose Coca-Cola stock trades at a forward price to earnings ratio of 28, its expected dividend payout ratio is 70 percent, and analysts believe that its dividend will grow at a constant rate of 4 percent per year. Calculate Coca-Cola’s required return.
6.50 percent
Using Equation 24,
28= 0.7/ (𝑟−0.04)
.
Solving this equation for r, Coca-Cola’s required return is 6.50 percent.
Given the above result for Coca-Cola’s required return, it should come as no surprise that if we instead assume that the required return on Coca-Cola stock is 6.50 percent, then we would find that Coca-Cola’s implied growth rate is 4 percent, and this result is confirmed in question 2.
What are the main benefits of using the cash flow additivity principle?
What is Skewness?
What is Kurtosis?
What is a leptokurtic or fat-tailed distribution?
What is a platykurtic or thin-tailed distribution?
What is a mesokurtic distribution?
Continuously Compounded Return
The natural logarithm of 1 plus the holding period return, or equivalently, the natural logarithm of the ending price over the beginning price.
Cost Averaging
The periodic investment of a fixed amount of money
Default Risk Premium
An extra return that compensates investors for the possibility that the borrower will fail to make a promised payment at the contracted time and in the contracted amount.
Harmonic Mean
A type of weighted mean computed as the reciprocal of the arithmetic average of the reciprocals.
Holding Period Return
The single-period internal rate of return for a real estate property that includes property income and the change in property value over the period.
Inflation premium
An extra return that compensates investors for expected inflation.
Interest Rate
A rate of return that reflects the relationship between differently dated cash flows; a discount rate.
Internal Rate of Return
The discount rate that makes net present value equal 0; the discount rate that makes the present value of an investment’s costs (outflows) equal to the present value of the investment’s benefits (inflows).
Leverage
A measure for identifying a potentially influential high-leverage point.
Liquidity premium
The compensation for liquidity risk that increases in proportion to the investment’s illiquidity.
Maturity premium
An extra return that compensates investors for the increased sensitivity of the market value of debt to a change in market interest rates as maturity is extended.
Money-weighted return
The internal rate of return on a portfolio, taking account of all cash flows.
Opportunity Cost
The value that investors forgo by choosing a particular course of action; the value of something in its best alternative use.
Real risk-free interest rate
The single-period interest rate for a completely risk-free security if no inflation were expected.
Time-weighted rate of return
The compound rate of growth of one unit of currency invested in a portfolio during a stated measurement period; a measure of investment performance that is not sensitive to the timing and amount of withdrawals or additions to the portfolio.
Trimmed Mean
A mean computed after excluding a stated small percentage of the lowest and highest observations.
Winsorized mean
A mean computed after assigning a stated percentage of the lowest values equal to one specified low value and a stated percentage of the highest values equal to one specified high value.
What does the geometric mean measure?
The geometric mean measures an investment’s compound rate of growth over multiple periods.
Contingent Claim
A type of derivative in which one of the counterparties determines whether and when the trade will settle. An option is a common type of contingent claim.
Define Monte Carlo simulation and explain its use in investment management.
A Monte Carlo simulation generates of a large number of random samples from a specified probability distribution (or distributions) to represent the role of risk in the system. Monte Carlo simulation is widely used to estimate risk and return in investment applications. In this setting, we simulate the portfolio’s profit and loss performance for a specified time horizon. Repeated trials within the simulation produce a simulated frequency distribution of portfolio returns from which performance and risk measures are derived. Another important use of Monte Carlo simulation in investments is as a tool for valuing complex securities for which no analytic pricing formula is available. It is also an important modeling resource for securities with complex embedded options.
Resampling
A statistical method that repeatedly draws samples from the original observed data sample for the statistical inference of population parameters.
Bootstrap
A resampling method that repeatedly draws samples with replacement of the selected elements from the original observed sample. Bootstrap is usually conducted by using computer simulation and is often used to find standard error or construct confidence intervals of population parameters.
stratified random sampling
A procedure that first divides a population into subpopulations (strata) based on classification criteria and then randomly draws samples from each stratum in sizes proportional to that of each stratum in the population.
What is the Sharpe Ratio?
The Sharpe ratio is the average return in excess of the risk-free rate divided by the standard deviation of returns. This ratio measures the return of a fund or a security above the risk-free rate (the excess return) earned per unit of standard deviation of return.
Central Limit Theorem
The theorem that states the sum (and the mean) of a set of independent, identically distributed random variables with finite variances is normally distributed, whatever distribution the random variables follow.
What is Jackknife?
A resampling method that repeatedly draws samples by taking the original observed data sample and leaving out one observation at a time (without replacement) from the set.
what does this symbol mean ∩ ?
In mathematics, the symbol “∩” represents the “intersection” of two sets, meaning it denotes the collection of elements that are common to both sets. (imagine the inside of a Venn diagram)
What is the present value of an annuity formula?
PV = 𝐴[1−1[1+(𝑟𝑠/𝑚)]𝑚𝑁𝑟𝑠/𝑚]
Cash flow Additivity Principle
The principle that dollar amounts indexed at the same point in time are additive.
Dividend Payout Ratio
The ratio of cash dividends paid to earnings for a period.
Forward price-to-earnings ratio
A P/E calculated on the basis of a forecast of EPS; a stock’s current price divided by next year’s expected earnings.
Hedge Ratio
The proportion of an underlying that will offset the risk associated with a derivative position.
Perpetual Bonds
Bonds with no stated maturity date.
Perpetuity
A perpetual annuity, or a set of never-ending level sequential cash flows, with the first cash flow occurring one period from now.
Premium
In the case of bonds, premium refers to the amount by which a bond is priced above its face (par) value. In the case of an option, the amount paid for the option contract.
Price to Earnings ratio (P/E)
The ratio of share price to earnings per share.
Terminal Value
The expected value of a share at the end of the investment horizon—in effect, the expected selling price.
Zero-coupon bonds
Bonds that do not pay interest during their life. They are issued at a discount to par value and redeemed at par. Also called pure discount bond.
Time Value of Money
The principles governing equivalence relationships between cash flows with different dates.
Opportunity Cost
The value that investors forgo by choosing a particular course of action; the value of something in its best alternative use.
The expression for the future value of a sum in N years with continuous compounding is
FV𝑁=PV^𝑒𝑟𝑠𝑁
What is the general annuity formula?
FV𝑁=𝐴[(1+𝑟)^𝑁−1+(1+𝑟)^𝑁−2+(1+𝑟)^𝑁−3+…+(1+𝑟)1+(1+𝑟)0]
simplified to: FV𝑁=𝐴[(1+𝑟)^𝑁−1𝑟]
What is the Mean Absolute Deviation (MAD)?
The mean absolute deviation of a dataset is the average distance between each data point and the mean
