# User:JohnWest/Drafts

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This is space for me to place draft articles. Feel free to contribute, if you desire.--JohnWest 03:30, 6 Oct 2004 (UTC)

# Kuhn-tucker theorem

In mathematics, the Kuhn-Tucker theorem (also known as Kuhn-Tucker conditions) is a result used in non-linear programming. It states that, if certain conditions of the analyzed function and its constraints hold, then an optimum (maximum if the function is convex) for the function can be found.

In other words, given a function ${\displaystyle f({\overline {x}})}$ —where ${\displaystyle {\overline {x}}}$ is a vector of variables— and given a vector of constraints ${\displaystyle {\overline {g}}({\overline {x}})}$

# Capital asset pricing model (merged version)

The Capital asset pricing model (CAPM) is used in finance to determine a theoretically appropriate price of an asset such as a security. The formula takes into account the asset's sensitivity to non-diversifiable risk (also known as systematic risk or market risk), in a number often referred to as beta (β) in the financial industry, as well as the expected return of the market and the expected return of a theoretical risk-free asset.

The model was introduced by William Sharpe, Lintner and Mossin independently, though it is commonly attributed only to the first of them, who published it earliest (in 1964). Sharpe received The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel (jointly with Harry Markowitz and Merton Miller) for his contribution to the field of financial economics.

## The formula

The relationship between a given asset i, and a proxy porfolio m (here, the market portfolio) is described as:

Where:

## Asset pricing

Once the expected return, ${\displaystyle E(r_{i})}$, is calculated using CAPM, the future cash flows of the asset can be discounted to their present value using this rate to establish the correct price for the asset.

In theory, therefore, an asset is correctly priced when its observed price is the same as its value calculated using the CAPM derived discount rate. If the observed price is higher than the valuation, then the asset is overvalued (and undervalued for a too low price).

Alternatively, one can "solve for discount rate" for the observed price given a particular valuation model and compare that discount rate with the CAPM rate. If the discount rate in the model was lower than the CAPM rate then the asset is overvalued (and undervalued for a too high discount rate).

## Asset specific required return

The CAPM returns the asset appropriate required return or discount rate - i.e. the rate at which future cash flows produced by the asset should be discounted given that asset's relative riskiness. Betas exceeding one signify more than average "riskiness"; betas below one indicate lower than average. Thus a more risky stock will have a higher beta and will be discounted at a higher rate; less sensitive stocks will have lower betas and be discounted at a lower rate. The CAPM is consistent with intuition - investors (should) require a higher return for holding a more risky asset.

Since beta reflects asset specific sensitivity to non-diversifiable, i.e. market, risk, the market as a whole, by definition, has a beta of one. Stock market indices are frequently used as local proxies for the market - and in that case (by definition) have a beta of one. An investor in a large, diversisifed portfolio (such as a mutual fund) therefore expects performance in line with the market.

## Risk and diversification

The risk of a portfolio is comprised of systematic risk and specific risk. Systematic risk refers to the risk common to all securities - i.e. market risk. Specific risk is the risk associated with individual assets. Specific risk can be diversified away (specific risks "cancel out"); systematic risk (within one market) cannot. Dependent on market, a portfolio of approximately 15 well selected shares (and more) would be sufficiently diversified to leave the portfolio exposed to systematic risk only.

An investor cannot expect to be rewarded for taking on diversifiable risk, (it is not rational to expose one's wealth to more risk than necessary). Therefore, the required return on an asset, that is, the return that compensates for risk taken, must be linked to its riskiness in a portfolio context - i.e. its contribution to overall portfolio riskiness - as opposed to its "stand alone riskiness." In the CAPM context, portfolio risk is represented by higher variance i.e. less predictability.

## The efficient (Markowitz) frontier

The CAPM assumes that the risk-return profile of a portfolio can be optimized - an optimal portfolio displays the lowest possible level of risk for its level of return. Additionally, since each additional asset introduced into a portfolio further diversifies the portfolio, the optimal portfolio must comprise every asset, (assuming no trading costs) with each asset value-weighted to achieve the above (assuming that any asset is infinitely divisible). All such optimal portfolios, i.e. one for each level of return, comprise the efficient (Markowitz) frontier.

## The market portfolio

An investor might choose to invest a proportion of her wealth in a portfolio of risky assets with the remainder in cash - earning interest at the risk free rate (or indeed may borrow money to fund her purchase of risky assets in which case there is a negative cash weighting). Here, the ratio of risky assets to risk free asset determines overall return - this relationship is clearly linear. It is thus possible to achieve a particular return in one of two ways:

1. By investing all of one’s wealth in a risky portfolio,
2. or by investing a proportion in a second portfolio and the remainder in cash (either borrowed or invested).

For a given level of return, however, only one of these portfolios will be optimal (in the sense of lowest risk). Since the risk free asset is, by definition, uncorrelated with any other asset, option 2) will generally have the lower variance and hence be the more efficient of the two.

This relationship also holds for portfolios along the efficient frontier: a higher return portfolio plus cash is more efficient than a lower return portfolio alone for that lower level of return. For a given risk free rate, there is only one optimal portfolio which can be combined with cash to achieve the lowest level of risk for any possible return. This is the market portfolio.

## Shortcomings of CAPM

• The model does not appear to adequately explain the variation in stock returns. Empirical studies realized in the past 15 years show that low beta stocks may offer higher returns.
• What is the "market portfolio"? Does it include the bond market? Real estate? Commodities? Private placements?
• The market portfolio, and hence its return, are not observable and have to be estimated.
• The model assumes that all investors are risk averse. Some investors (e.g., some day traders), however, can not be considered to be risk averse.
• The model assumes that all investors create mean-variance optimized portfolios. However, there are many investors who don't know what a mean-variance optimized portfolio is.

## References

• Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, Review of Economics and Statistics, 47
• Markowitz, Harry M. (1999). The early history of portfolio theory: 1600-1960, Financial Analysts Journal, 55 (4)
• Sharpe, William F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (3)
• Tobin, James (1958). Liquidity preference as behavior towards risk, The Review of Economic Studies, 25