Arbitrage Pricing Model (APM) assumes the residual to be normally distributed. This article empirically
checks this assumption in APM. In this paper, an Arbitrage Pricing Model is built on returns from stocks
traded in National Stock Exchange (NSE). The APM of returns from shares has four explanatory
variables—Market Trend (Market Index), Sector-specific trend in the Market (IT Index), Size of the
company (Daily Turnover) and Location factor of the company (Index of Industrial Production). The
normal distribution is compared with lognormal and exponential distribution. It has been observed that
the exponential distribution performs better than lognormal and normal distributions. Univariate kernel
smoothing method is also undertaken for univariate model based on returns dependent on IT Index. It
has been observed that Exponential distribution performs better than Kernel Smoothing and Normal
distributions in an univariate model.
One of the most common models for asset returns is the temporally Independently and
Identically Distributed (IID) normal model, in which returns are assumed to be
independent over time and normally distributed. The original formulation of Capital
Asset Pricing Model (CAPM) employed this assumption of normality, although returns
were implicitly assumed to be temporally IID.
While the temporally IID normal model may be tractable, it suffers from at least two
important drawbacks. First, most financial assets exhibit limited liability, so that the
largest loss an investor can realize is his total investment and no more. This implies that
the smallest net return achievable is –1 or –100%. But since normal distribution supports
the entire real line, its lower bound of –1 is clearly violated by normality assumption. Of
course, it may be argued that by choosing the mean and variance appropriately the
probability of realization below –1 can be made arbitrarily small; however it will never be
zero, as limited liability requires.
Second issue is that the normal distribution has skewness equal to zero and kurtosis equal
to 3. Fat-tailed distributions with extra probability mass in the tail areas have higher or
infinite kurtosis. Sample estimates of excess kurtosis for daily US stock returns are large and
positive for both indexes and individual stocks, indicating that returns have more mass in
the tail areas than would be predicted by a normal distribution. For many studies in this area,
lognormal distributions are also considered in place of normal distribution. |
