asset pricing models chapter 36 tools & techniques of investment planning copyright 2007, the...

Asset Pricing Models Chapter 36Tools & Techniques of

Investment Planning

Copyright 2007, The National Underwriter Company 1

Asset Pricing Models

• A key factor in valuing securities is a required rate of return.

• Asset pricing models help determine the appropriate required rate of return for equity securities.

• Modern Portfolio Theory provides a means of assessing the rate of return required to compensate investors for the risk involved.


Investment Planning


Asset Pricing Models

• Modern portfolio theory suggests that the rate of return required on a security must compensate investors for the risk involved. – Given the choice between multiple assets (or portfolios) with

equal expected returns, investors prefer the one with the lowest risk.

– Given the choice between multiple assets (or portfolios) of equal perceived risk, investors prefer the one with the highest expected return.


Investment Planning


• Henry Markowitz, the pioneer of Modern Portfolio Theory, believed investors expect returns on an asset approximating that asset’s average (mean) returns.– He defined risk as the variability (standard deviation) of returns.

• Investors can graph all available portfolios of securities to examine the relationship between return (expected return) and risk (standard deviation).

• The efficient frontier is a line that includes those portfolios with the highest return for a particular level of risk.– The area on and under the efficient frontier represents all

available portfolios. – Portfolios below the efficient frontier are considered inefficient.

Markowitz Portfolio Theory


Investment Planning


• An efficient portfolio is preferred over another one depending on the individual investor and their overall tolerance for risk.

• The Markowitz model was expanded into the capital market theory by Sharpe, Lintner, and others by adding the concept of a risk-free asset, which is typically represented by a government fixed income security.

Markowitz Portfolio Theory and the Capital Market Line


Investment Planning


The Capital Market Line

• When the risk-free rate is added to a graph of the efficient frontier, a line can be drawn from the risk-free rate tangent to the efficient frontier (intersecting at Portfolio M). – The capital market line (CML) is this line drawn from the risk free

asset to the highest point of tangency on the efficient frontier.– If the investor can borrow and lend at the risk-free rate (by

purchasing government bonds or by selling them short), a new efficient frontier is created along the CML.

– Portfolio M represents the market-weighted portfolio of all investable assets (the market portfolio).


Investment Planning



• An investor wholly invested in the market portfolio could expect the same risk and return as the market portfolio.

• Investors who put less than 100% in the market portfolio and place the remainder in the risk-free asset (a government security) would have a portfolio with less expected return and risk than the market portfolio.

• Investors willing to accept more risk can borrow at the risk-free rate and invest more than 100% (their own money plus the borrowed money) of their assets in the market portfolio. – Such a portfolio would have a higher expected return, but higher

expected risk than the market portfolio.


Investment Planning



• The expected return for a portfolio is the risk-free rate plus a premium for the risk of the portfolio, based upon the market risk premium (rm-rf) and the risk of the portfolio relative to the risk of the market (as captured by the standard deviations of each).

• A portfolio with the same risk as the market portfolio, would have an expected return equal to that of the market portfolio.


Investment Planning



• The expected return from a portfolio can be measured as:

E(rp) = rf + σp [(rm-rf) / σm], – where E(rp) represents a portfolio’s expected return; rm is the

expected return on the market portfolio; rf is the risk free rate; σp is the standard deviation on the portfolio and σm is the standard deviation on the market.


Investment Planning


Capital Asset Pricing Model (CAPM)

• The capital market line can be applied to an individual equity securities by using Beta as a measure of risk instead of standard deviation (i.e. the Security Market Line).

• The Security Market Line (SML) represents the relationship between expected return and risk measured by Beta.


Investment Planning


Capital Asset Pricing Model (CAPM)

• Beta measures the risk of an individual security relative to the market. – It is determined by running a regression of a security’s return

against returns on a market index (such as the Standard and Poor’s 500).

– A Beta of 1.0 means the security exhibited the same level of risk as the index over the period measured.

– A Beta greater (less) than 1.0 indicates higher (lower) risk

than the index.


Investment Planning


Capital Asset Pricing Model

• The CAPM states that the expected return on an individual asset (Eri) should be the risk free rate plus a premium for risk determined as:

E(ri) = rf + ßi (rm-rf)

– The risk premium on an individual asset is based on the security’s Beta and the overall equity market risk premium, (rm-rf).

– Expected market and risk free returns are a matter of judgment. They are often based on historical market returns combined with current market conditions.


Investment Planning


Example

• Cornerstone Electronics, Inc. (CEI) has a Beta of 1.3. Assuming that the expected return on the market is 12% and the expected risk-free rate is 4%. Under the CAPM, investors should require a return of:

4% + 1.3 (12% - 4%) = 14.4%.

• As one would expect, CEI’s higher risk results in a required return higher than that of the market. This required rate of return could then be used to assess the value of the equity using models such as the dividend discount model presented in Chapter 35. Assuming that CEI has an expected growth rate of 6% per year and a current dividend of $2.00, its value under a constant dividend growth model would be:

V0 = [D0 (1+g)]/r – g = [2.00 (1.06)]/[(0.144 – 0.06)] = $25.24


Investment Planning



• The CAPM assumes that investors own diversified portfolios– It only accounts for the impact of systematic risk; that is the

risk that is common to all market securities and cannot be mitigated by holding diverse securities.

– It does not consider non-systematic risk (also known as unique risk or non-diversifiable risk) because non-systematic risks of individual securities should offset each other in a well-diversified portfolio.

• Beta captures the relative sensitivity of a particular security to overall market risk (systematic risk).


Investment Planning



• The capital asset pricing model can help estimate the appropriate rate of return for valuing a security, but it should be viewed primarily as a starting point. – The CAPM relies on a number of assumptions – Research into the CAPM’s ability to explain

observed returns has been mixed. • An investor should still consider whether the

computed required return, particularly given the level of risk, makes sense.


Investment Planning


Arbitrage Pricing Theory (APT)

• APT, developed by Stephen Ross, introduces multiple risk factors into the assessment of expected returns, as opposed to a single risk factor under CAPM.

• Each of the factors can capture different aspects of risk.

• Individual factors influencing risk and expected return are not pre-specified, unlike CAPM.

E(ri) = a + bi1F1 + bi2F2 + bi3F3 +…+binFn + ,

– where a is some constant term (such as the risk free rate), and bin represents the sensitivity of a security to a particular risk factor, Fn.


Investment Planning



• Investment advisors and analysts can develop their own risk factors for use in an APT model or use those developed in research and practice by others.

• Burmeister, Roll and Ross created a model with five factors: – confidence risk (corporate vs. government bond yields)– time horizon risk – inflation risk – business cycle risk – market-timing risk (market return not explained)

• The constant is represented by the 30 day Treasury bill rate.


Investment Planning



• Each individual factor must be measured periodically, as must an individual security’s sensitivity to each factor.

• APT does not merely focus on changes in these variables, but in unexpected changes in these variables.


Investment Planning


Example

• Dot Murphy, investment advisor, is assessing the expected return for Alpha Enterprises (AE), a large diversified conglomerate. Dot has computed the following coefficients and risk factor components for AE:

• The current Treasury bill rate is 3.0%.

Risk Factor AE’s Coefficient Price of Risk (F)

Confidence Risk 0.60 2.50%

Time Horizon Risk 0.50 -0.70%

Inflation Risk -0.40 -4.00%

Business Cycle Risk

1.50 1.50%

Market-Timing Risk 1.10 3.50%


Investment Planning


Example

• Dot computes the expected return on AE as:

3.0% + 0.60 x 2.50% + 0.50 x (-0.70%) – 0.40 x (-4.0%) + 1.5 x 1.5% + 1.10 x 3.50% = 11.85%


Investment Planning


Option Pricing

• Options are contracts involving the future trade of an underlying asset. In an options contract, only one party is obligated to complete the transaction but must do so only if the other party exercises the option.

• An option gives the holder (buyer of the option) the right, but not the obligation, to buy the underlying asset (a call option) or sell it (a put option) at a specified price for a given period of time.


Investment Planning


Option Pricing

• The writer (seller of the option) collects a fee for selling the option and must fulfill their side of the contract if the option is exercised.

• Options can be negotiated between two parties, though standardized options are traded on organized exchanges.


Investment Planning


Option Pricing

• Rights and obligations to the parties in an options contract

Option Writer (Seller) Option Holder (Buyer)

Call Option

Must sell the underlying security at the agreed upon price if the option exercised

May purchase underlying security at agreed upon price if doing so is advantageous

Put Option

Must buy the underlying security at the agreed upon price if the option exercised

May sell underlying security at agreed upon price if doing so is advantageous


Investment Planning


Option Pricing

• Option values are determined by several important characteristics: – Strike Price or the Exercise Price: Price at which the holder of the

option can buy (or sell) the underlying asset. • Usually, there are standardized option contracts with several strike prices

above and below the current market price.

– Expiration Date: Standardized equity options cease trading on the third Friday of the indicated expiration month and expire the next day (Saturday.)

• Options are generally available for the two near term months plus two additional months, depending upon the security’s designated quarterly cycle

• Most options traded in the U.S. are American style options, which can be exercised any time on or before the expiration date. By contrast, European style options can only be exercised on the expiration date.


Investment Planning


Option Pricing

• Option values are determined by several important characteristics:– Underlying Security: This is the underlying asset on which

the value of the option depends. • For standardized equity options, one contract generally

represents 100 shares of common stock or American Depository Receipts.


Investment Planning


Option Pricing

• The option seller (writer) charges a fee (premium) to the purchaser.– The option premium is determined by the market, based upon

the prospects for the underlying security and the terms of the option.

• Option value can be broken into two parts:– Intrinsic value– Time value


Investment Planning


Option Pricing

• The option’s intrinsic value is the amount by which the current market price exceeds (in the case of a call option) or falls short of (in the case of a put option) the strike price. – An option that has intrinsic value is known as being “in the

money”– The intrinsic value will vary based on the strike price


Investment Planning


Option Pricing

• The option’s time value is represented by the difference between the current option price and the intrinsic value. – The time value of the option reflects the fact that the

underlying price of the security can change before the expiration of the contract.

– The longer to expiration, the higher the time value because the likelihood is greater that the option will confer additional value to the holder prior to expiration.

• Over time, the option’s time value will be reduced to zero (known as decaying), and the option value should equal intrinsic value at expiration.


Investment Planning


Option Pricing

• “Out of the money”– A call option with a strike price above the current stock

price.– The premium is comprised entirely of time value.

• “In the money”– A call option with a strike price below the current market

price of the underlying security.

• “At the money”– A call option with a strike price equal to the current market

price


Investment Planning


Option Pricing

• While determining the intrinsic value is relatively straightforward, it is difficult to determine what the time value, and hence, total value of the option should be. – An investor can observe the current market prices of the

options contracts, but the value could be higher or lower.

• Options-pricing models exist to assist in this task.– The binomial model– The Black-Scholes model


Investment Planning


The Binomial Model

• In a binomial model, it is assumed that in the next period there are only two outcomes: the underlying stock price goes up by x% or goes down by y%. The x & y percentages are the investor’s expectations for a particular security.

• First, the intrinsic value at maturity is determined at the far right of the binomial tree, which is a diagram of the call option values. Then each sub-period option value is determined working right to left until the current option value is determined.


Investment Planning


The Binomial Model

• The current value of an option is the present value of the weighted average of the future call values.

• The more periods that are used, the better the estimate of value.– Note: The computation is similar for European and

American style options, except that when using an American option, the computed option value must be compared to the intrinsic value at each node, and if the intrinsic value is greater than the computed value, the intrinsic value is used instead.


Investment Planning


The Black-Scholes Model

• Fischer Black, Myron Scholes, and Robert Merton developed a formula for pricing call options. – While the binomial model assumes two outcomes and

discrete time periods, the Black-Scholes Model provides a mathematical extension to the binomial model to a continuous time period with a wider distribution of outcomes.

– While mathematically complex, the Back-Scholes Model is useful to examine what factors influence option value.


Investment Planning


The Black-Scholes Model

• The following conclusions can be derived from the model:– As the value of the underlying stock increases, so does the

value of the call option.– As the time to expiration increases, so does the value of the

option.– As the volatility of the underlying stock increases, so does the

the value of the option.– As the strike price increases, the value of the call option

decreases. – As the risk free rate of return increases, the value of the call

option increases. • Note: The Black-Scholes Model is based on European style

options and is best used in valuing those types of options.


Investment Planning


Put Option Pricing

• The holder of a put option has the right to sell the underlying security at a certain price.

• Intrinsic value of a put option only exists where the strike price exceeds the current market price, since the holder of the option would not want to exercise his right to sell something at a price lower than the current market price.– The intrinsic value is measured as the strike price less the

current market price

• Put options with a strike price below the current market price are referred to as “out of the money” and have no intrinsic value.


Investment Planning


Put Option Pricing

• “Out of the money”– A put option with a strike price below the current market

price– Has no intrinsic value

• “In the money”– A put option with a strike price above the current market

price of the underlying security

• “At the money”– A put option with a strike price equal to the current market

price


Investment Planning


Put Option Pricing

• The time value equals the difference between the current option price (premium) and the intrinsic value.– As with call options, a longer time to maturity equates to

greater time value.

• To determine the value of a put option, there are several methods:– Variation of the Binomial Model– Variation of the Black-Scholes Model– Put-Call Parity Model


Investment Planning


Put Option Pricing

• Variation of the Binomial Method– Applied in the same manner as in the case of call options.– The intrinsic value is determined at maturity based on the

excess of the strike price over the value of the underlying security.

– The put value at each node is computed based on the weighted average of the values for the put option in the upside and downside cases.

• Variation of the Black-Scholes Model – Represents the excess of strike price over current stock price

adjusting for time value of money and normal distribution outcomes.


Investment Planning


Put Option Pricing

• Put-Call Parity Model– Derived from the assumption that puts and calls should be

priced relative to the underlying security such that no arbitrage opportunity exists.

– The formula assumes that purchasing a stock and a put option should be an equivalent position to holding cash equal to the exercise price and purchasing a call option (where the put and call options have the same exercise price)

– This is not always the case for real data due to bid-ask spreads and the timing of the quotations.


Investment Planning


Put Option Pricing

• The equation is:S + P = (K / ert) + C

– where S is the security’s current market price, P is the price of a put option, K is the exercise price discounted at the risk free rate, and C is the call price.

• Rearranging the previous equation leads to:P = (K / ert) + C - S,

– so that holding a put option is equivalent to investing the present value of the strike price, purchasing a call option, and selling the stock short.

asset pricing models chapter 36 tools & techniques of investment planning copyright 2007, the...

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