Not everything that counts can be counted,
and not everything that can be counted counts.
Albert Einstein, 1879 – 1955
The Arbitrage Pricing Theory (APT) was developed in the 1970’s and, like the CAPM, relates the expected return on securities or portfolios to risk. Where CAPM is a single-factor model that relates investment returns to the return on the overall market, APT is a multifactor model. It describes expected returns as a function of their sensitivities to a set of macroeconomic risk factors, such as changes in inflation, GDP, and interest rates. While APT usually takes the form of a multifactor model, it could also be formulated as a single risk factor model using only one economic risk factor.
APT and CAPM both assume market efficiency; assets are properly priced in relationship to their riskiness and if they are not, arbitrage activity will restore their equilibrium pricing. However, APT assumptions are generally less stringent than the CAPM in that it only assumes:
Asset returns can be described as a set of factors.
Investors form well-diversified portfolios that eliminate asset-specific risk.
Asset prices will be set such that no profitable arbitrage will be possible between well-diversified portfolios.
While multifactor models are used in explaining the return of assets or portfolios during some specific time period as a function of surprises in certain factors, APT is used to explain the long-run excess return above the risk-free rate for exposure to various risk factors. In a sense, multifactor models explain past returns while APT projects future returns.
APT models take the form of the following equation:
E(RP) = RF + βP,iFi + βP,jFj
where:
|
E(RP) |
Is the expected return on the portfolio or asset P |
|
RF |
Is the risk-free rate. If no risk-free asset exists, an asset with zero sensitivity to all the factors is substituted for RF. |
|
Fi |
Is the factor risk premium (factor price) for accepting risk exposure to factor i. |
|
βP,i |
Is the sensitivity of the portfolio or asset to risk factor i. |
|
Fj |
Is the factor risk premium (factor price) for accepting risk exposure to factor j. |
|
βP,j |
Is the sensitivity of the portfolio or asset to risk factor j. |
APT can be used to project expected returns on an asset or a portfolio.
APT Exercise 1
Suppose there are two factors that determine the returns on stocks; the economy as measured by the growth rate of real GDP and interest rates. Assume the risk-free rate is 5%. The following parameters have been determined using statistical regression:
|
FGDP |
= |
.03 |
The market requires an extra 3% return premium for every 1% uncertainty with respect to predicting real GDP. |
|
Fr |
= |
.02 |
The market requires an extra 2% return premium for every 1% uncertainty with respect to predicting interest rates. |
|
βGDPx |
= |
1.2 |
The response of stock X to unanticipated changes in the rate of growth of GDP is 1.2 times the amount of the change. |
|
βrx |
= |
0.1 |
The response of stock X to unanticipated changes in interest rates is 0.1 times the amount of the change. |
Project the expected long run return on stock X.
Arbitrage
Underlying the portfolio models is the assumption that if assets or portfolios with the same expected risk do not have the same expected return, investors will form arbitrage portfolios to earn riskless profits. This activity of arbitragers buying undervalued assets (raising prices) and selling overvalued assets (lowering prices) will eliminate the mispricing, thus restoring market equilibrium.
An arbitrage portfolio is a portfolio with zero factor risk, that is, all the factor sensitivities are equal to zero and the portfolio requires no capital investment. Furthermore, if the market is efficient, the return on an arbitrage portfolio will be zero.
If the expected return on an arbitrage portfolio is not zero, it will be possible to earn a riskless profit on that portfolio. An arbitrage opportunity can be captured in three steps.
Identify and purchase the undervalued asset or portfolio.
Fund the purchase by short selling the overvalued assets. This will also eliminate (hedge) the risk of the assets purchased.
Close out the initial transactions when asset prices return to equilibrium.
These steps are illustrated in the following example.
Assume the following portfolios are available and have been analyzed for their risk and expected returns over one year using a single-factor model. Further assume that the risk-free rate is 1.5% and factor value is 10.
|
Portfolio |
1-year Quoted E(R) |
Factor Sensitivity βP,i |
|
U |
0.0350 |
0.200 |
|
V |
0.0375 |
0.225 |
|
S |
0.0425 |
0.250 |
|
T |
0.0450 |
0.300 |
|
.50 U + .50 T |
0.0400 |
0.250 |
Question: What should the relationship be between factor sensitivity and return?
Answer: Factor sensitivity measures the riskiness of the portfolio with respect to a non-diversifiable (priced) risk. There should be a linear relationship between that risk and expected return where portfolios with greater risk should offer higher returns. The relationship is described by the following model:
E(RP) = RF + βP,iFi
Question: Are the portfolios’ quoted prices consistent with the assumptions of modern portfolio theory and, if not, how could an arbitrage portfolio be constructed using the portfolio of 50% U and 50% T?
Answer: The portfolios’ prices are not consistent with the assumptions because they do not exhibit the linear relationship of return to risk. In particular, portfolio S has the same sensitivity (.250) as the combined portfolio of U and T, but their returns are different.
For a given factor model (Fi), if the expected return calculated by the model does not equal the expected return quoted in the market, an arbitrage opportunity exists. Using the factor value of 10 and the risk-free rate of 1.5%, the model would indicate the following values for the portfolios.
|
Portfolio |
1-year Quoted E(R) |
Factor Sensitivity βP,i |
Model E(R) |
|
U |
0.0350 |
0.200 |
3.50% |
|
V |
0.0375 |
0.225 |
3.75% |
|
S |
0.0425 |
0.250 |
4.00% |
|
T |
0.0450 |
0.300 |
4.50% |
|
.50 U + .50 T |
0.0400 |
0.250 |
4.00% |
Notice that the combined portfolio consisting of 50% portfolio U and 50% portfolio T has the same sensitivity (risk) as portfolio S but a different quoted return. This violates the law of one price (portfolios with the same risk should also have the same returns) and is indicative of an arbitrage opportunity. According to the factor model, portfolio S is mispriced (4.25% quoted versus 4.00% calculated by the model). Since the quoted return on portfolio S is higher than the model’s expected return, the portfolio is undervalued.
To exploit this mispricing one should create an arbitrage portfolio by:
Buying the undervalued asset by investing $100,000 in portfolio S.
Fund the purchase and hedge the risk by short selling portfolio U + T.
At the end of one year, close out the initial transactions and take a risk-free profit of $250.
APT Summary
A multi-factor model.
Attempts to explain individual stock returns by identifying several relevant variables (one of which might be the market return).
Should include industry group, economic indices, and other explanatory factors.
Increases in complexity and computation and input requirements.
APT Exercise 2
A portfolio manager has an investment horizon of one year. She assumes that the expected returns on Techmar stock and Riexa stock are 10% and 9%, respectively. She also makes the following projections and calculates the factor betas (factor sensitivities) for the two stocks.
|
|
|
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Write the multifactor equations for each stock. If the actual levels of inflation and GDP growth are 2.5% and 2.7% respectively, what does the model predict will be the return on the stocks and on a portfolio consisting of 50% Techmar and 50% Riexa?
An analyst has determined that the return on XYZ stock can be explained by the following model:
RXYZ = 2.0% + 0.5(IP) + 2.0(INFL) + 1.0(EXCH)
where:
IP = % change in industrial production
INFL = Inflation Rate
EXCH = % change in the $ exchange rate
For 20X8, IP is expected to rise by 3%, inflation is expected to be 5%, and the dollar is expected to fall 2% against a broad basket of currencies. The market is expected to earn a 10% return in 20X8 and the risk-free rate is 6%.
Based on the information provided, calculate the expected return for XYZ in 20X8.