Rates of Return

Question 1

What is the arithmetic rate of return, and how is it calculated?

Answer 1

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.

Question 2

Why is the arithmetic rate of return considered a simple average of returns?

Answer 2

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.

Question 3

What assumptions does the arithmetic rate of return make about the reinvestment of returns over time?

Answer 3

The arithmetic rate assumes that returns are not reinvested and do not compound. It ignores the impact of volatility and time on investment growth.

Question 4

How does the arithmetic rate of return differ from the geometric rate of return in measuring investment performance?

Answer 4

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.

Question 5

In what scenarios might the arithmetic rate of return provide misleading insights into an investment’s long-term performance?

Answer 5

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.

Question 6

Why might the arithmetic rate of return overestimate the average performance of a volatile investment?

Answer 6

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%.

Question 7

How does the absence of compounding in the arithmetic rate of return calculation affect its usefulness for decision-making?

Answer 7

It makes the arithmetic rate less suitable for evaluating long-term growth. Investors might overestimate returns, especially when there is significant volatility.

Question 8

If an investment returns 8%, -4%, and 12% over three years, what is the arithmetic rate of return?

Answer 8

ArithmeticRateofReturn=
{ 8+(−4)+12 }/ 3 = 16/3

=5.33%
The arithmetic rate of return is 5.33%.

Question 9

How would you interpret the arithmetic rate of return in the context of annualised returns for an investor?

Answer 9

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.

Question 10

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?

Answer 10

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.

Question 11

In what situations would a financial analyst use the arithmetic rate of return instead of the geometric rate of return?

Answer 11

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.

Question 12

How can the arithmetic rate of return be used in portfolio optimisation and asset allocation decisions?

Answer 12

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.

Question 13

Calculate the arithmetic rate of return for a portfolio with the following yearly returns: 10%, 5%, -2%, and 15%.

Answer 13

ArithmeticRateofReturn=

10+5+(−2)+15 / 4 = 28/4 =
​
=7%
The arithmetic rate of return is 7%.

Question 14

If the same portfolio had a geometric rate of return of 6%, what conclusions can you draw about the difference between these two measures?

Answer 14

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.

Question 15

What is the geometric rate of return, and how is it calculated?

Answer 15

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||

Question 16

How does the geometric rate of return differ from the arithmetic rate of return?

Answer 16

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.

Question 17

Why is the geometric rate of return considered more accurate for evaluating long-term performance?

Answer 17

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.

Question 18

What are the limitations of the geometric rate of return?

Answer 18

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.

Question 19

How does volatility affect the geometric rate of return?

Answer 19

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.

Question 20

What happens to the geometric rate of return when all periodic returns are identical?

Answer 20

When all returns are identical, the geometric and arithmetic rates of return are equal, as there is no impact from compounding or volatility.

Question 21

How does the geometric rate of return relate to risk-adjusted performance measures?

Answer 21

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.

Question 22

If an investment returns 8%, -4%, and 12% over three years, what is the geometric rate of return?

Answer 22

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%
​

Question 23

How would you interpret the geometric rate of return for a long-term investment?

Answer 23

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.

Question 24

How does the geometric rate of return handle the sequence of returns differently from the arithmetic rate of return?

Answer 24

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.

Question 25

Why is the geometric rate of return always less than or equal to the arithmetic rate of return?

Answer 25

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.

Question 26

If an investment has a geometric return of 5% over 10 years, what does this imply about the overall growth?

Answer 26

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.

Question 27

How is the geometric rate of return used in evaluating fund or portfolio performance?

Answer 27

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.

Question 28

What is the Harmonic mean?

Answer 28

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.

Question 29

Give an example of when you would use the Harmonic Mean

Answer 29

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.

Question 30

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?

Answer 30

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.

Question 31

How does the harmonic mean differ from the arithmetic and geometric means?

Answer 31

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.

Question 32

When is it most appropriate to use the harmonic mean?

Answer 32

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.

Question 33

What are the limitations of the harmonic mean?

Answer 33

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.

Question 34

Why is the harmonic mean commonly used for calculating average price-earnings (P/E) ratios in finance?

Answer 34

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.

Question 35

If a car travels at speeds of 40 km/h and 60 km/h for equal distances, what is the average speed?

Answer 35

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.

Question 36

Why is the harmonic mean preferred over the arithmetic mean in averaging speeds for equal distances?

Answer 36

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.

Question 37

What is the trimmed mean?

Answer 37

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.

Question 38

Calculate the 20% Winsorised mean for the dataset: 3, 7, 8, 12, 15, 18, 24.

Answer 38

• 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.

Question 39

How does the Winsorised mean reduce the impact of outliers compared to the arithmetic mean?

Answer 39

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.

Question 40

What is the Winsorised mean?

Answer 40

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.

Question 41

How to compute the exact time-weighted rate of return on a portfolio?

Answer 41

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.

Question 42

How would you annualise an 18-month holding period return?

Answer 42

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%
.

Question 43

What is the concept continuously compounded return?

Answer 43

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).