REV: FEBRUARY 20, 2007
ANDRÉ F. PEROLD
Introduction to Portfolio Theory
Portfolio theory is concerned with the risk-reducing role played by individual assets in an investment portfolio of several assets. The benefits of diversification were first formalized in 1952 by Harry Markowitz, who later was awarded the Nobel Prize in economics for this work. Portfolio theory is today a cornerstone of modern financial theory, as well as a widely used tool for managing risk-return tradeoffs in investment portfolios. This note examines the basic building blocks of the theory.
Means and Standard Deviations of Total Return
The return and risk of an asset are commonly measured in terms of the mean and standard deviation of total return, where total return represents income plus capital gains or losses. The mean is the return one expects to obtain on average; standard deviation is a measure of dispersion, in this case total volatility of return. For bell-shaped distributions (Figure 1), the return one actually experiences will fall within one standard deviation to either side of the mean about 68% of the time, within two standard deviations 95% of the time, and within three standard deviations 99.7% of the time.
________________________________________________________________________________________________________________ Professor André F. Perold prepared this note as the basis for class discussion.
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Introduction to Portfolio Theory
The mean and standard deviation of return for a given asset can be computed from historical returns. In that case, however, they are merely summary descriptors of past performance, and may or may not reflect the probability distribution of future returns. Investors, of course, need to estimate the mean and standard deviation of future returns.
Assets can be compared on the basis of their means and standard deviations by drawing a simple diagram:
Asset A is a risk-free asset (i.e., cash or equivalent) since its standard deviation is zero. Asset B has a higher mean and higher standard deviation than A; asset C has a higher mean and standard deviation than B; and D has the highest standard deviation but a lower return than either B or C.
Investing in Only One Asset
If we could invest in only one of these four assets, which would we pick? The top left hand corner of Figure 2 (marked X) is nirvana: much return, and no risk; the bottom right hand corner (marked Y) represents the worst of all worlds: no return, and much risk. As risk-averse investors, we prefer to own assets that are “closest” to X and “furthest away” from Y. For example, we can easily rule out asset D since both B and C have higher expected returns and lower standard deviations. Assets such as D are said to be dominated (by B and/or C). In Figure 2, D is the only dominated asset. Each of the others has either a higher expected return or a lower standard deviation than any other asset.
Dominated assets are relatively easy to remove from consideration. Choosing among
undominated assets (here A, B, & C) is harder and requires knowledge of our risk tolerance. A riskneutral investor would prefer C, and a very risk-averse...