XYZ stock price and divided history are as follows; Year-----------Beginning-of-
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XYZ stock price and divided history are as follows; Year-----------Beginning-of-year-Price 2010----------$100 2011---------$110 2012---------$90 2013----------$95 Dividend paid at Year End are that 2010 is $4, 2011 is $4, 2012 is $4 and 2013 is $4. 1,What are the arithmetric average rate of return and the geometric average rate of return?? these answer should be in perxentage and accurate to the hundredths. 2,Suppose that an investor buys three shares of XYZ at the beginning of 2010, buys another two shares at the beginning of 2011, sells one share at the beginning of 2012, and sells all four remaining shares at the beginning of 2013. What is the dollar-weighted rate of return (IRR)? Your answer should be in percentages and accurate to the hundredthExplanation / Answer
When analyzing investment returns it is important to differentiate between the simple arithmetic return and the geometric return (a.k.a the average annualized rate of return). The geometric return is the more accurate as it is the average compounded return. The arithmetic average is always higher than the geometric average, hence the arithmetic average return is usually the one posted in ads for mutual funds and other investments. The only time when the arithmetic and geometric average will be the same is when the individual returns being averaged are the same for each period being analyzed. Example 1: Mutual Fund XYZ has the following returns for the past 2 years: Year 1: -20% Year 2: +20% This is a rather simplistic example, but its purpose is to illustrate our point. On the surface it would appear that this fund has essentially "broken even" for the past 2 years. The average return for the two years being evaluated is 0%. So; a $10,000 investment at the beginning of year 1 is still worth $10,000 at the end of year 2, correct? Not quite. Even though the arithmetic average return is 0%, the geometric return tells a different and more accurate story. Before we get into the formulas, let's just break down this example: Year 1: $10,000 invested. The fund is down 20% by the end of the first year. The original investment is now worth $8,000 [10000 x (1-.20)] Year 2: You start with $8000. The fund is up 20%. Your investment at the beginning of year 2 has grown to $9600: [8000 x (1+.20)] This does not quite get you back to the original $10,000 invested at the start of Year 1 and is obviously not the break-even scenario it appeared to be initially. At the end of year 2 you are actually down 4%. Example 2: Mutual Fund XYZ has the following returns for the past 3 years: Year 1: -20% Year 2: +20% Year 3: +15% The simple arithmetic avg. would be 15% over the 3 years being evaluated, which averages out to a 5% return per year. As we saw with the previous example, this is not an accurate reflection of the true return on this investment for the past three years. A 5% yield per year would generate the following balances at the end of each year: Year1: $10,000 * 1.05 = 10,500 Year2: $10,500 * 1.05 = 11,025 Year3: $11,025 * 1.05 = 11,576.25 We know from the first example that based on the returns for years one and two, we would be starting year three with only $9600, so the 5% per year return scenario above does not paint a true picture of the resulting balances based on the individual returns for years 1-3. Geometric Average To get the geometric average return, we need to first obtain the total return: Total Return = (1 + percent return) * (1 + percent return) * (1 + percent return) [3 periods in this example] (1-.20) * (1+.20) * (1-.15) = 1.104 Geometric average = ((total return) ^ (1/number of years)) - 1 ((1.104) ^ (1/3))-1 = .03353 You can check this in two ways: Using the total return: Year 1: $10,000 x (1-.2) = 8,000 Year 2: $8,000 x (1+.2) = 9,600 Year 3: $9600 x (1+.15) = 11,040 The result of the Total Return formula just above is 1.104. 10,000 (initial investment) x 1.104 = $11,040 The next way to check this is to use the Geometric average return within the formula for compound interest. P * ((1+r)^n)-1 P= Principal r = geometric return n = number of periods (years in this case) 10000 * ((1+.03353) ^ 3)-1 = 11,040 We have shown here that the actual geometric return is 3.353% over the 3 year period being analyzed, not 5% as indicated by taking the arithmetic return. The key take away from all of this is for you to be able to draw a distinction between the geometric average return and the arithmetic average return, and why it matters. Look beyond the publicized investment returns being touted by the advertisers or investment managers of the products being solicited. Ensure that you fully understand how these investment returns translate in terms of the balances in your personal portfolio. Additionally; take a more critical approach when analyzing potential investments and their historical returns. Knowing the 'average' return of an investment is not very useful unless you are able to differentiate between the arithmetic average return and the geometric average return. Computing the average annual rate of return (ROR) for a bond, stock, fund, or trading strategy should be straightforward, shouldn’t it? For example, to get the average annual ROR for the last five years, don’t you just sum up the 5 individual annual returns, and divide by 5? Well yes, you can do it that way. But what you’ll end up with is the arithmetic average, which is probably not really what you’re after. Why not? Let’s look at an example. Say we’re examining a stock that had the following closing prices on December 31 of the following years: Table 1 Year 2000 2001 2002 2003 2004 2005 Price per Share $111.19 $97.80 $76.12 $97.82 $108.32 $113.49 This translates into the following annual returns: Table 2 Year 2001 2002 2003 2004 2005 Annual ROR -12.0% -22.2% +28.5% +10.7% +4.8% Now we’ll follow our original assumption and compute the arithmetic average: [ (-12.0) + (-22.2) + (28.5) + (10.7) + (4.8) ] / 5 = +2.0% What’s wrong with our computation? Well nothing technically, but the arithmetic mean can mistakenly lead you to assume that if you had invested money 5 years ago, your investment today would be worth the same as if the fund returned that average +2.0% for each of those 5 years. But didn’t it? If we had initially invested $10,000: At the end of year 1 it would be worth: $10,000 x (1-.12) = $8,800 At the end of year 2 it would be worth: $8,800 x (1-.222) = $6,846 At the end of year 3 it would be worth: $6,846 x (1+.285) = $8,798 At the end of year 4 it would be worth: $8,798 x (1+.107) = $9,739 At the end of year 5 it would be worth: $9,739 x (1+.048) = $10,206 Note that this is quite a different return than if we’d made that arithmetic average 2% each year: At the end of year 1 it would be worth: $10,000 x (1+.02) = $10,200 At the end of year 2 it would be worth: $10,200 x (1+.02) = $10,404 At the end of year 3 it would be worth: $10,404 x (1+.02) = $10,612 At the end of year 4 it would be worth: $10,612 x (1+.02) = $10,824 At the end of year 5 it would be worth: $10,824 x (1+.02) = $11,041 Of course the problem with the arithmetic mean is that it doesn’t take into account the compounding effect of each yearly ROR. This would be fine if we invested say $10,000 each year and whether we made money or lost money, we still set our principal back to $10,000 for each subsequent year. But most of us don’t invest that way. If we put money in a fund for a number of years, the amount invested in any particular year is what was leftover from the previous year. Assuming this is the way we’re investing, if we wanted to figure out what equivalent yearly average return would give us what our investment is worth today, we have to use the geometric mean. To compute the geometric mean, multiply each of the yearly returns (rather than summing them) and then take the nth root of the product (rather than dividing by n), where n is the number of years we’re averaging over. For our n=5 years this would work out to be: [(1-.12) x (1-.222) x (1+.285) x (1+.107) x (1+.048)](1/5) = 1.0041 or 0.41% And in fact if we compound our initial principal at this geometric average, we do get what our investment is worth today: At the end of year 1: $10,000 x (1+.0041) = $10,041 At the end of year 2: $10,041 x (1+.0041) = $10,082 At the end of year 3: $10,082 x (1+.0041) = $10,124 At the end of year 4: $10,124 x (1+.0041) = $10,165 At the end of year 5: $10,165 x (1+.0041) = $10,207 Actually you’ll see we’re off by one dollar, due to round-off error. There is a simpler way to compute the geometric mean, and that is to use the beginning and ending prices in Table 1, but annualize the gain, again by taking the nth root where n is the number of years between beginning and end prices. If we do this, we get the same as the geometric mean: (113.49 / 111.19)(1/5) = 1.0041 or 0.41% Apart from the calculation being simpler, the nice thing about annualizing is that you can get a yearly geometric mean, even when you have a non-integer number of years. For example, let’s say that instead of annual prices we’re examining quarterly prices: Table 3 Quarter Q1 2005 Q2 2005 Q3 2005 Q4 2004 Q1 2006 Price per Share $64.08 $67.77 $81.02 $80.85 $83.52 We have 5 quarters, or 1.25 years worth of data, so we annualize to compute the following effective annual ROR: (83.52 / 64.08) (1/1.25) = 1.2361 or 23.6% So by annualizing, when computing average ROR we don’t have to restrict ourselves to periods where we have a complete year’s worth of data. This is helpful because chances are the stock or fund was not created on January 1 of its inception year and usually we’re part of the way into a new year. By annualizing we can therefore include all price data available to us to get an average ROR – including fractional years. Average vs. Geometric Average By JLP | June 6, 2006 Take a look at the total returns for the S&P 500 Index for the last 20 years: S&P 500 Index Average Returns Over the last 20 years, the S&P 500 Index had an average return of 13.17% (.1317 as a decimal) per year. So, if you had invested $10,000 in 1986, it would have been worth $118,748 at the end of 2005. The math looks like this: $10,000 X (1 + .1317)20 $10,000 X 1.131720 $10,000 X 11.88748 $118,748 Is this how much your account would have really been worth? NOPE! Here’s what your account would have looked like through the years: Actual Returns According to those numbers, the account was worth $95,421 at the end of 2005. That’s a far cry from the $118,748 we got from using the average return. So, how do we calculate the Geometric Mean (also called the average annualized rate of return)? If you use Microsoft Excel, it is simple to calculate if you use their function (located in the “Insert” menu). You can also calculate it yourself relatively easily. If you look at the chart above, you will see the year listed, the total return received that year and a column titled “Factor.” To get the factor, you simply add 1 to the return expressed as a decimal. For example: A return of 18.67% is .1876 as a decimal, for a factor of 1.1867 (1 + .1867 = 1.1867) For a year with a negative return, the calculation looks like this: A return of -3.11% is -.0311 as a decimal, for a factor of .9689 (1 + (-.0311) = .9689) So, if you had $10,000 at the start of one year and received a return of 18.67%, you would have $11,867 at the end of the year (10,000 X 1.1867). On the other hand, if you had $10,000 at the beginning of the year and received a return of -3.11%, you would have $9,689 at the end of the year, or only 96.89% of the amount you had at the beginning ($10,000 X .9689 = $9.689). Now, to find your Geometric Average, you simply multiply all the factors together and raise them to the 1/n power (where n is the number of periods—usually years—for which you are making the calculation) . Got that? It’s not as confusing as it sounds. Using the numbers from the table above, it looks like this: (1.1867 X 1.0525 X …X 1.0491)1/20 9.542119.05 1.119392 Subtract 1 from the answer above and you get the geometric average of .119392 or 11.9392%. Now, to check and see if our answer is right, test it out with this equation: $10,000 X (1+.119392)20 $10,000 X (1.119392)20 $10,000 X 9.5421 $95,421 Please visit this as well 1. http://people.stern.nyu.edu/wsilber/Geometric%20Average%20Versus%20Arithmetic%20Average.pdf
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