Risk and return

Risk and return is a complex topic. There are many types of risk, and many ways to evaluate and measure risk. 
In investing, a widely used definition of risk is: "The chance that an investment's actual return will be different than expected."Risk, Investopedia. Retrieved 5 May 2012. 
In this context, different means a variation which can go either positive or negative, and is known as volatility. 
However, risk is a term often used to imply downside risk, meaning the uncertainty of a return and the potential for financial loss.Financial risk, from Wikipedia. Retrieved 5 May 2012. The Merriam-Webster dictionary defines risk as "the chance that an investment (as a stock or commodity) will lose value".Free Merriam-Webster Dictionary, Risk - Definition, viewed January 26, 2015.
This article demonstrates some of the concepts of risk, return, and diversification.
== Risk as the uncertainty of returns ==
Refer to Figure 1 below. The uncertainty inherent in investing is demonstrated by the historical distributions of returns of three major asset classes: cash (typically studied in the form of Treasury bills), bonds, and stocks. A span of 41 years (1970 - 2011) should be a sufficiently long time to show general trends, but 1982 had a significant impact and is highlighted separately.All numerical results presented here are available in the spreadsheet.
Figure 2. Risk vs. Return: 1970 - 2011 Standard deviation is a statistical measure that often is used to quantify the dispersion (variation) of investment returns which provides a standardized way to compare risk for different investments. Standard deviation increases as dispersion of annual returns increases. This is illustrated by the standard deviation of the asset classes shown in the returns history shown in Figure 1.
Figure 1. Range of Returns: 1970 - 2011The effect of removing 1982 from the bonds return was less than 1% (US -0.6%, Cdn -0.9%).Based on data from Libra Investment Management, which is Copyright © Libra Investment Management Inc. 2005-2012. All rights reserved. Used with permission.
As shown in the left/top chart, the left vertical bar represents the range of annual returns for treasury bills. The range (dispersion) of returns is from slightly above 0% to 19%.
The middle vertical bar represents the range of annual returns on bonds over the same time period. Note the larger range (dispersion) of returns; from about -7% to +27% (excluding 1982).
The far right vertical bar in the left chart represents the range of annual returns on Canadian stocks (as represented by stocks in the TSX/Composite index) over the same time period. Note the much larger dispersion of returns; from about -33% to +45%.
The right/bottom chart in Figure 1 represents similar data from the US (3 month treasury bills, 10 year bonds, S&P 500 index) over the same time period. Note that 1982 also had a similar effect on U.S. bonds.
Figure 2 shows this same information, but from a different perspective. These histograms show how many years each return falls within a certain range (5% "bin"). The top chart shows that T-Bills had returns around 5% for 17 years (and 10% for another 17 years) out of the 1970 - 2011 period. The middle chart shows bonds had 10 years around 15% return, and the bottom chart shows stocks had 7 years around 15% return for the same period.
The focus is on the grouping of the vertical bars (dispersion). The T-Bill bars are grouped close together, bonds further apart, and stocks cover the full range.The use of standard deviation implies that returns follow a normal (Gaussian) distribution pattern. This is certainly not the case, which can be further explored in Normal or Log-Normal, by forum member Gummy (gummy stuff ... about Investing (mostly)).
Therefore, the uncertainty of returns (volatility) shows that bonds have more uncertainty than cash, and stocks have more uncertainty than bonds. Said differently; stocks are riskier than bonds, which is riskier than cash (T-Bills).
Another area of interest is the expected return.Expected return is a weighted average of an investment's future ("expected") return; and can be different than the actual return. See: Risk and Return: Variance and Standard Deviation, from Columbia Interactive (Columbia University). Although it is not clear from the data supplied here (but can be seen in the US data), the expected return increases with increasing risk. Investors want to get compensated for taking a chance that their investments may experience a loss. This is a nearly universal concept and is one of the first principles of investing:
Now, let's look at bonds. From Figure 1, 1982 had a significant impact. Figure 2 shows that there was a single occurrence - one year with a return of 45% (details are in the spreadsheet). Will this happen again? It's not possible to predict. This brings about another first principle of investing:
==Portfolio diversification==
Now, combine those same assets into portfolios. Refer to Figure 3.
Table 1 shows the portfolio breakdown. The file used to create this table and all the figures in this article can be downloaded from Google Docs: Risk - Historical Performance of Cdn Bonds and Stocks.xlsx
:
 Table 1. 1970 - 2011 Portfolio Returns and Risk (Hypothetical)
PortfolioAllocation 	Stocks 	20%	30%	40% 	50% 	60% 	70% 	80%
PortfolioAllocation 	T.Bills 	20%	15%	10% 	5% 	5% 	5% 	5%
PortfolioAllocation 	Bonds 	60%	55%	50% 	45% 	35% 	25% 	15%
Expected Return: 	Expected Return: 	10%	10%	10% 	11% 	11% 	11% 	11%
Max predicted loss (95%):	Max predicted loss (95%):	-1%	-2%	-3% 	-3% 	-5% 	-6% 	-7%
2008 return: 	2008 return: 	-4%	-8%	-12%	-15%	-19%	-22%	-26%
* TSX/Composite stock returns include dividends
*Three month Treasury bills
*Canadian Long bonds
*Asset classes are uncorrelated	
* TSX/Composite stock returns include dividends
*Three month Treasury bills
*Canadian Long bonds
*Asset classes are uncorrelated	
* TSX/Composite stock returns include dividends
*Three month Treasury bills
*Canadian Long bonds
*Asset classes are uncorrelated	
* TSX/Composite stock returns include dividends
*Three month Treasury bills
*Canadian Long bonds
*Asset classes are uncorrelated	
* TSX/Composite stock returns include dividends
*Three month Treasury bills
*Canadian Long bonds
*Asset classes are uncorrelated	
* TSX/Composite stock returns include dividends
*Three month Treasury bills
*Canadian Long bonds
*Asset classes are uncorrelated	
* TSX/Composite stock returns include dividends
*Three month Treasury bills
*Canadian Long bonds
*Asset classes are uncorrelated	
* TSX/Composite stock returns include dividends
*Three month Treasury bills
*Canadian Long bonds
*Asset classes are uncorrelated	
* TSX/Composite stock returns include dividends
*Three month Treasury bills
*Canadian Long bonds
*Asset classes are uncorrelated
 Value at risk was used to determine the maximum predicted loss, at 95% confidence. Details are in the Excel file.
The Treasury bonds and bills have been combined into a single category called "bonds." This is done intentionally, as asset classes which contain bonds and treasury bills are commonly grouped together as fixed income.Short bonds have a maturity from 1-5 years (duration about 3 throughout), long bonds have a maturity of 11-30 years (duration of 10-15 years and bounces around). Ref: Forum member Norbert Schlenker.
Going from left to right, the portfolio progresses from (20% stocks / 80% bonds) to (80% stocks / 20% bonds). Similar to Figure 2, the least variation is a portfolio which contains mostly bonds; the highest variation is a portfolio containing mostly stocks. (In many portfolios, the highest risk also has the highest return - but those results are not seen in this data.)
:
Figure 3. Risk vs. Return for Various Portfolios: 1970 - 2011
Caveat: Figure 3 shows a hypothetical portfolio of assets held from 1970 - 2011 and should not be compared with an actual portfolio. Do not assume that the returns vary evenly (linearly) across the portfolio percentages. It is an effect of this specific hypothetical portfolio held for 41 years; one which cannot be reproduced in the real world.
There is a statistical method which attempts to predict the maximum expected loss of a portfolio. Known as Value at risk, there was a 95% confidence that the loss of these portfolios would not exceed (lose more than) the value shown in the table. However, 2008 proved the statisticians wrong. Investors who sold assets during 2008 experienced traumatic losses; but those holding assets for the long term (many years) will recover. 2008 is represented as the lowest return in the Figure 2 histogram, lower chart (1974 is the 2nd lowest return, at -26%).
This demonstrates that one can reduce their portfolio risk by adding bonds, which may result in a lower return (seen in many portfolios, but not here). Alternatively, one can increase their return by adding stocks, at an increased risk of loss.
Diversifying your portfolio is the only way to mitigate risk. It is a compromise of investing horizon (how long until the funds are needed) and acceptable loss. Diversification is determined by setting the ratio of equities (stocks) to bonds, and is known as asset allocation. 
==Notes==
==See also==
*Behavioural pitfalls
*Risk (without charts or algebra)
*Portfolio design and construction
*Canadian asset class returns
*Risk - bonds
==References==
==Further reading==
* 
* 
==External links==
*
*The file used to generate the Figures is available as a free download from Google Docs: Risk - Historical Performance of Cdn Bonds and Stocks.xlsx
:(The US information, including an example of Value at Risk, is also available from Google Docs: Risk - Historical Performance of Bonds and Stocks.xlsx.)
*Stingy Investor: The Stingy Investor Asset Mixer, by forum member NormR. Create your own historical portfolio performance.
*Risk and Return, part of an online course by from Columbia Interactive (Columbia University). This website is no longer maintained, but the material is made available.
===Papers===
Academic Papers	Academic Papers
 	
* Cotton, Benjamin L, (September 16, 1999). "The Uncertain Science of Asset Allocation". Retrieved 14 April 2012.
* Huisman, Ronald, Koedijk, Kees C.G. and Campbell, Rachel A.J. (April 27, 1999). "Asset Allocation in a Value-at-Risk Framework". Retrieved 14 April 2012.
. 
Category:Portfolio design and construction
Category:Investment management