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EQULIBRIUM, MUTUAL FUNDS AND SHARPE RATIO

Luis A Pons PerezChristian RoblesEconometría y Modelos de FinanzasDr. Balbino García

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Optimal PortfolioThe optimal portfolio concept falls

under the modern portfolio theory. The theory assumes (among other

things) that investors fanatically try to minimize risk while striving for the highest return possible.

The theory states that investors will act rationally, always making decisions aimed at maximizing their return for their acceptable level of risk. 

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Optimal PortfolioThe optimal portfolio was used in 1952

by Harry Markowitz, and it shows us that it is possible for different portfolios to have varying levels of risk and return.

Each investor must decide how much risk they can handle and than allocate (or diversify) their portfolio according to this decision. 

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Optimal Portfolio

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Two-fund Separation  Is a theorem stating that, under certain conditions, any

investor's optimal portfolio can be constructed by holding each of certain mutual funds in appropriate ratios, where the number of mutual funds is smaller than the number of individual assets in the portfolio.

Here a mutual fund refers to any specified benchmark portfolio of the available assets. There are two advantages of having a mutual fund theorem:

a. First, if the relevant conditions are met, it may be easier (or lower in transactions costs) for an investor to purchase a smaller number of mutual funds than to purchase a larger number of assets individually.

b. Second, from a theoretical and empirical standpoint, if it can be assumed that the relevant conditions are indeed satisfied, then implications for the functioning of asset markets can be derived and tested.

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MEASURES OF RETURN

MEASURES OF RETURN◦complicated by addition or

withdrawal of money by the investor◦percentage change is not reliable

when the base amount may be changing

◦timing of additions or withdrawals is important to measurement

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MEASURES OF RETURN

TWO MEASURES OF RETURN◦Dollar-Weighted Returns uses discounted cash flow approach weighted because the period with the

greater number of shares has a greater influence on the overall average

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MEASURES OF RETURN

TWO MEASURES OF RETURN◦Time-Weighted Returns used when cash flows occur between

beginning and ending of investment horizon

ignores number of shares held in each period

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MEASURES OF RETURN

TWO MEASURES OF RETURN◦Comparison of Time-Weighted to

Dollar-Weighted Returns Time-weighted useful in pension fund

management where manager cannot control the deposits or withdrawals to the fund

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MAKING RELEVANT COMPARISONS

PERFORMANCE◦should be evaluated on the basis of

a relative and not an absolute basis this is done by use of a benchmark

portfolio

◦BENCHMARK PORTFOLIO should be relevant and feasible reflects objectives of the fund reflects return as well as risk

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THE USE OF MARKET INDICES

INDICES◦are used to indicate performance but

depend upon the securities used to calculate them the calculation weighting measures

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THE USE OF MARKET INDICES

INDICES◦Three Calculation Weighting

Methods: price weighting

sum prices and divided by a constant to determine average price

EXAMPLE: THE DOW JONES INDICES

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THE USE OF MARKET INDICES

INDICES◦Three Calculation Weighting

Methods: value weighting (capitalization method)

price times number of shares outstanding is summed

divide by beginning value of index EXAMPLE: S&P500 WILSHIRE 5000 RUSSELL 1000

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THE USE OF MARKET INDICES

INDICES◦Three Calculation Weighting

Methods: equal weighting

multiply the level of the index on the previous day by the arithmetic mean of the daily price relatives

EXAMPLE: VALUE LINE COMPOSITE

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ARITHMETIC V. GEOMETRIC AVERAGESGEOMETRIC MEAN FRAMEWORK

GM = ( HPR)1/N - 1where = the summation of the

product of HPR= the holding period

returns n= the number of periods

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ARITHMETIC V. GEOMETRIC AVERAGES

GEOMETRIC MEAN FRAMEWORK◦measures past performance well◦represents exactly the constant rate

of return needed to earn in each year to match some historical performance

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ARITHMETIC V. GEOMETRIC AVERAGES

ARITHMETIC MEAN FRAMEWORK◦provides a good indication of the

expected rate of return for an investment during a future individual year

◦it is biased upward if you attempt to measure an asset’s long-run performance

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RISK-ADJUSTED MEASURES OF PERFORMANCE

THE REWARD TO VOLATILITY RATIO (TREYNOR MEASURE)◦There are two components of risk risk associated with market fluctuations risk associated with the stock

◦Characteristic Line (ex post security line) defines the relationship between

historical portfolio returns and the market portfolio

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TREYNOR MEASURETREYNOR MEASURE

◦Formula

where arp = the average portfolio return

arf = the average risk free rate

bp = the slope of the characteristic

line during the time period

p

fpp

ararRVOL

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TREYNOR MEASURE

THE CHARACTERISTIC LINE

arp

bp

SML

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TREYNOR MEASURE

CHARACTERISTIC LINE◦slope of CL measures the relative volatility of

portfolio returns in relation to returns for the aggregate market, i.e. the portfolio’s beta

the higher the slope, the more sensitive is the portfolio to the market

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TREYNOR MEASURE

THE CHARACTERISTIC LINE

arp

bp

SML

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THE SHARPE RATIO

THE REWARD TO VARIABILITY (SHARPE RATIO)◦measure of risk-adjusted

performance that uses a benchmark based on the ex-post security market line

◦total risk is measured by sp

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THE SHARPE RATIOSHARPE RATIO

◦formula:

where SR = the Sharpe ratio

sp = the total risk

p

fpp

ararSR

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THE SHARPE RATIOSHARPE RATIO

◦indicates the risk premium per unit of total risk

◦uses the Capital Market Line in its analysis

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THE SHARPE RATIO

arp

sp

CML

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THE JENSEN MEASURE OF PORTFOLIO PERFORMANCEBASED ON THE CAPM EQUATION

◦measures the average return on the portfolio over and above that predicted by the CAPM

◦given the portfolio’s beta and the average market return

])([)( RFRrERFRrE mi

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THE JENSEN MEASURE OF PORTFOLIO PERFORMANCETHE JENSEN MEASURE

◦known as the portfolio’s alpha value recall the linear regression equation

y = a + bx + e alpha is the intercept

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THE JENSEN MEASURE OF PORTFOLIO PERFORMANCEDERIVATION OF ALPHA

◦Let the expectations formula in terms of realized rates of return be written

◦subtracting RFR from both sides jttmtjtjt uRFRRRFRR

jttmtjtjt uRFRRRFRR

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THE JENSEN MEASURE OF PORTFOLIO PERFORMANCEDERIVATION OF ALPHA

◦in this form an intercept value for the regression is not expected if all assets are in equilibrium

◦in words, the risk premium earned on the jth portfolio is equal to bj

times a market risk premium plus a random error term

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THE JENSEN MEASURE OF PORTFOLIO PERFORMANCEDERIVATION OF ALPHA

◦to measure superior portfolio performance, you must allow for an intercept a

◦a superior manager has a significant and positive alpha because of constant positive random errors

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COMPARING MEASURES OF PERFORMANCETREYNOR V. SHARPE

◦SR measures uses s as a measure of risk while Treynor uses b

◦SR evaluates the manager on the basis of both rate of return performance as well as diversification

◦for a completely diversified portfolio SR and Treynor give identical rankings

because total risk is really systematic variance

any difference in ranking comes directly from a difference in diversification

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CRITICISM OF RISK-ADJUSTED PERFORMANCE MEASURES

Use of a market surrogate Roll: criticized any measure that

attempted to model the market portfolio with a surrogate such as the S&P500 it is almost impossible to form a portfolio whose

returns replicate those over time making slight changes in the surrogate may

completely change performance rankings

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CRITICISM OF RISK-ADJUSTED PERFORMANCE MEASURESmeasuring the risk free rate

using T-bills gives too low of a return making it easier for a portfolio to show superior performance

borrowing a T-bill rate is unrealistically low and produces too high a rate of return making it more difficult to show superior performance

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¡Gracias y Éxito!