Financial assets are primarily defined based on their return-risk characteristics. This definition approach helps when building a portfolio from all the assets available. It’s noteworthy that there are different ways of measuring returns.<\/p>\n
Financial market assets generate two types of returns: Income from cash dividends or interest payments and capital gains or losses from changes in the prices of financial assets.<\/p>\n
Some financial assets give only one stream of return. For instance, headline stock market indices typically only report on price appreciation. They do not include the dividend income unless the index clarifies that it is a \u201ctotal return\u201d series.<\/p>\n
A holding period return is earned from holding an asset for a specified period, such as a day, month, or ten years.<\/p>\n
The general formula of the holding period return is given by:<\/p>\n
$$R=\\frac{\\left(P_1-P_0\\right)+I_1}{P_0}$$<\/p>\n
\\(P_0\\) = Price of an asset at the beginning of the period (t<\/em>=0).<\/p>\n \\(P_1\\) = Price of an asset at the end of the period (t<\/em>=1).<\/p>\n \\(I_1\\) = Income received at the end of the period (t<\/em>=1).<\/p>\n Example: Calculating Holding Period Return<\/strong><\/p>\n An investor purchases 100 shares of a company at a price of USD 50 per share at the beginning of the year. During the year, the investor receives a dividend of USD 2 per share. At the end of the year, the shares are sold for USD 60 per share.<\/p>\n The investor\u2019s one\u2011year holding period return is\u00a0closest to:<\/em><\/p>\n Solution<\/strong><\/p>\n In this case, we have:<\/p>\n $$\\begin{align}P_0&=100 \\text{ shares} \\times \\$50\\ \\text{per share} =\\$5,000\\\\ I_1&=100\\ \\text{shares} \\times\\$2\\ \\text{per share} =\\$200\\\\\u00a0 P_1&=100 \\text{ shares} \\times \\$60\\ \\text{per share} =\\$6,000\\end{align}$$<\/p>\n Therefore,<\/p>\n $$R=\\frac{\\left(P_1-P_0\\right)+I_1}{P_0}=\\frac{6,000-5,000+200}{5,000}=24\\%$$<\/p>\n Holding period returns can also be calculated for periods longer than a year. For instance, if we need to calculate the holding period return for a five-year period, we should compound the five annual returns as follows:<\/p>\n $$R=\\frac{\\left(P_5-P_0\\right)+I_{(1-5)}}{P_0}$$<\/p>\n When we have assets with multiple holding periods, we must aggregate the returns into one overall return.<\/p>\n Denoted by \\({\\bar{R}}_i\\) arithmetic mean for an asset \\(i\\) is a simple process of finding the average holding period returns. It is given by:<\/p>\n $${\\bar{R}}_i=\\frac{R_{i,1}+R_{i,2}+\\ldots+R_{i,T-1}+R_{iT}}{T}=\\frac{1}{T}\\sum_{t=1}^{T}R_{it}$$<\/p>\n Where:<\/p>\n \\(R_{it}\\) = Return of asset \\(i\\) in period \\(t\\).<\/p>\n \\(T\\) = Total number of periods.<\/p>\n For example, if a share has returned 15%, 10%, 12%, and 3% over the last four years, then the arithmetic mean is computed as follows:<\/p>\n $${\\bar{R}}_i=\\frac{1}{T}\\sum_{t=1}^{T}R_{it}=\\frac{1}{4}\\left(15\\%+10\\%+12\\%+3\\%\\right)=10\\%$$<\/p>\n Computing a geometric mean follows a principle similar to the one used to compute compound interest. It involves compounding returns from the previous year to the initial investment’s value at the start of the new period, allowing you to earn returns on your returns.<\/p>\n A geometric return provides a more accurate representation of the portfolio value growth than an arithmetic return.<\/p>\n Denoted by \\({\\bar{R}}_{Gi}\\) the geometric return for asset \\(i\\) is given by:<\/p>\n $$\\begin{align}{\\bar{R}}_{Gi}&=\\sqrt[T]{\\left(1+R_{i,1}\\right)\\times\\left(1+R_{i2}\\right)\\times\\ldots\\times\\left(1+R_{i,T-1}\\right)\\times\\left(1+R_{iT}\\right)}-1\\\\ &=\\sqrt[T]{\\prod_{t=1}^{T}{(1+R_t)}}-1\\end{align}$$<\/p>\n Using the same annual returns of 15%, 10%, 12%, and 3% as shown above, we compute the geometric mean as follows:<\/p>\n $$\\begin{align}\\text{Geometric mean} &=\\ \\left[\\left(1+15\\%\\right)\\times\\ \\left(1+10\\%\\right)\\times\\ \\left(1+12\\%\\right)\\times\\ \\left(1+3\\%\\right)\\right]^\\frac{1}{4}\\ -1\\\\&=\\ 9.9\\%\\end{align}$$<\/p>\n Note that the geometric return is slightly less than the arithmetic return. Arithmetic returns tend to be biased upwards unless the holding period returns are all equal.<\/p>\n The harmonic mean measures central tendency. It’s especially useful for rates or ratios such as P\/E ratios. The harmonic mean’s formula is derived from the harmonic series, a specific mathematical sequence.<\/p>\n $${\\bar{X}}_H=\\frac{n}{\\sum_{i=1}^{n}\\frac{1}{X_i}},\\ X_i>0 \\text{ for all } i=1,2,\\ldots,n$$<\/p>\n The above formula is interpreted as the \u201charmonic mean of observations \\(X_1,\\ X_2,\\ldots,\\ X_n\\).\u201d<\/p>\n The harmonic mean is handy for averaging ratios when those ratios are consistently applied to a fixed quantity, resulting in varying unit numbers. For instance, it’s applied in cost-averaging strategies where you invest a fixed amount of money at regular intervals.<\/p>\n An investor invests in a particular stock over a period of three months, allocating an equal amount of $4,000 each month. The share prices of the stock during these months are $10, $12, and $15, respectively.<\/p>\n The average price paid per share for the three-month period is closest to<\/i>:<\/p>\n Solution<\/strong><\/p>\n Using the harmonic mean formula,<\/p>\n $${\\bar{X}}_H=\\frac{n}{\\sum_{i=1}^{n}\\frac{1}{X_i}}=\\frac{3}{\\frac{1}{10}+\\frac{1}{12}+\\frac{1}{15}}=12$$<\/p>\n Trimmed and Winsorized means seek to lower the effect of outliers in a data set.<\/p>\n The trimmed mean is a measure of central tendency. We calculate it after excluding a small percentage of the lowest and highest values from a dataset.<\/p>\n For example, a data set consists of 10 observations: 12, 15, 18, 20, 22, 25, 27, 30, 35, and 40. We can calculate the trimmed mean after removing the highest and lowest values.<\/p>\n After removing these values, the remaining data set is 15, 18, 20, 22, 25, 27, 30, and 35.<\/p>\n Now, let\u2019s calculate the trimmed mean by taking the average of these remaining values:<\/p>\n $$\\frac{15+18+20+22+25+27+30+35}{8}\\ =\\frac{192}{8}\\ \\ =\\ 24$$<\/p>\n Therefore, the trimmed mean of the given data set is 24.<\/p>\n The Winsorized mean is a central tendency measure. It works by replacing extreme values at both ends of the data with the values of their closest observations. This process is similar to the trimmed mean. Essentially, it helps eliminate outliers in a dataset.<\/p>\n For example, consider a dataset of 12 observations: 8, 12, 15, 18, 20, 22, 25, 27, 30, 35, 40, and 50. We can calculate the Winsorized mean by replacing the lowest and highest values with those closest to the 10th and 90th percentiles, respectively. As such, the new values are 10<\/b>, 12, 15, 18, 20, 22, 25, 27, 30, 35, 37.5<\/b>, and 40, and the winsorized mean is:<\/p>\n $$\\frac{10 +12+15+ 18 + 20 +22 + 25 +27 +30+35+37.5+40}{12} \\approx 24.46$$<\/p>\n What are the arithmetic mean and geometric mean, respectively, of an investment that returns 8%, -2%, and 6% each year for three years?<\/p>\n A. Arithmetic mean = 5.3%; Geometric mean = 5.2%.<\/p>\n B. Arithmetic mean = 4.0%; Geometric mean = 3.6%.<\/p>\n C. Arithmentic mean = 4.0%; Geometric mean = 3.9%.<\/p>\n Solution<\/strong><\/p>\n The correct answer is<\/strong> C<\/strong>.<\/p>\n $$ \\text{Arithmetic mean} = \\frac {8\\% + (-2\\%) + 6\\%} {3} = 4\\% $$<\/p>\n $$ \\text{Geometric mean} = [(1+8\\%) \u00d7 (1+(-2\\%)) \u00d7 (1+6\\%)]^{1\/3} – 1 = 3.9\\% $$<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":" Financial assets are primarily defined based on their return-risk characteristics. This definition approach helps when building a portfolio from all the assets available. It’s noteworthy that there are different ways of measuring returns. Financial market assets generate two types of…<\/p>\n","protected":false},"author":15,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-45204","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"\nArithmetic Return<\/h2>\n
Geometric Return<\/h2>\n
Harmonic Mean<\/h2>\n
Example: Calculating the Harmonic Mean<\/strong><\/h4>\n
Trimmed and Winsorized Means<\/h2>\n
Trimmed Mean<\/strong><\/h3>\n
Winsorized Mean<\/strong><\/h3>\n
\n
Question<\/strong><\/h2>\n