The most familiar probability distribution is, no doubt, the normal. Can we say that the distribution of return from a stock share is normal? Empirical investigation tends to support a different view, as the normal distribution is symmetric and features thin tails, i.e., it tends to underestimate the probability of extreme events. In any case, the worst stock return we may experience is \(100 \%\), or ( 1), i.e., we lose all of our investment (this is related to the limited liability property of stock shares, discussed later). In other words, the worst gain is 0 , and a stock share price can never be negative. Since the support of a normal random variable is unbounded, \((+)\), according to this uncertainty model there is always a nonzero probability of observing an impossible price.

However, let us discuss the matter from a very limited viewpoint, namely, convenience. One nice feature of a normal distribution is that if we add normal variables, we get another normal (to be precise, we should be considering jointly normal variables). This is nice when we add returns from different stock shares over the same time period. If we have invested \(30 \%\) of our wealth in stock share \(a\) and \(70 \%\) in stock share \(b\), the holding period return for the portfolio is

\[\begin{equation*}R_{p}=03 R_{a}+07 R_{b} \tag{1.3}\end{equation*}\]

where we denote the return of stock shares \(a\) and \(b\) by \(R_{a}\) and \(R_{b}\), respectively, and \(R_{p}\) is the portfolio return. To justify Eq. (1.3), let us consider:

- Initial stock prices \(S_{a}(0)\) and \(S_{b}(0)\)

- Initial wealth \(W(0)\)

- Stock prices \(S_{a}(T)\) and \(S_{b}(T)\) at the end of the holding period

- Wealth \(W(T)\) at the end of the holding period Then, if initial wealth is split as we have assumed, we may write

\[\begin{aligned}W(0) & =\frac{03 \quad W(0)}{S_{a}(0)} S_{a}(0)+\frac{07 \quad W(0)}{S_{b}(0)} S_{b}(0) \\& =N_{a} \quad S_{a}(0)+N_{b} \quad S_{b}(0)\end{aligned}\]

where \(N_{a}\) and \(N_{b}\) are the number of stock shares \(a\) and \(b\), respectively, that we buy. At the end of the holding period, we have

\[\begin{aligned}W(T) & =N_{a} \quad S_{a}(T)+N_{b} \quad S_{b}(T) \\& =N_{a} \quad\left(1+R_{a}\right) \quad S_{a}(0)+N_{b} \quad\left(1+R_{b}\right) \quad S_{b}(0)\end{aligned}\]

\[\begin{aligned}& =W(0) \quad 1+\frac{N_{a} R_{a} S_{a}(0)+N_{b} \quad R_{b} \quad S_{b}(0)}{W(0)} \\& =W(0)\left(1+03 \quad R_{a}+07 \quad R_{b}\right) \\& =W(0)\quad\left(1+R_{p}\right)\end{aligned}\]
which gives Eq. (1.3).
If \(R_{a}\) and \(R_{b}\) are jointly normal, then \(R_{p}\) is a nice normal, too, which is quite convenient. Furthermore, if the holding period return \(R\) is normal, so is the corresponding stock price \(S(T)=S(0) \quad(1+R)\). However, imagine that we take a different perspective. Rather than considering two stock shares over one time period, let us consider one stock share over two consecutive time periods. In other words, we take a longitudinal view (a single variable over multiple time periods) rather than a cross-sectional view (multiple variables over a single time period). Let us denote by \(R(1)\) and \(R(2)\) the two holding period returns of that single stock share, over the two consecutive time periods. As we have mentioned, in this case we should not add returns, but rather multiply gains \(G(1)\) and \(G(2)\) to find the holding period gain \[\begin{aligned}
G & =G(1) \quad G(2) \\
& =1+R(1) \quad 1+R(2) \\
& =1+R(1)+R(2)+R(1) \quad R(2)
\end{aligned}\]
The last expression involves a product of returns. Unfortunately, if \(R(1)\) and \(R(2)\) are normal, their product is not. Hence, the holding period gain \(G\) is not normal, and the same applies to the holding period return \(R=G \quad 1\). We may only say that the holding period return is approximately normal if the single-period returns are small enough to warrant neglecting their product.
One way out is to consider the logarithmic return, or log-return for short, \[r=\log (1+R) \quad \log G\]
where we use log rather than ln to denote natural logarithm. It is interesting to note that, given the well-known Taylor expansion (Maclaurin series, if you prefer)
\[\log (1+x) \quad x \quad \frac{x^{2}}{2}+\frac{x^{3}}{3} \quad \frac{x^{4}}{4}+\]

for a small \(x\), the log-return can be approximated by the return. Since \[\begin{aligned}
\log 1+R(1) \quad 1+R(2) & =\log 1+R(1)+\log 1+R(2) \\
& =r(1)+r(2)
\end{aligned}\]
we see that log-returns are additive, and if we assume that they are normal, we preserve normality over time.
Since \[S(T)=S(0) \quad G=S(0) \quad e^{r}\]
the normality of the log-return \(r\) implies that the gain and the stock prices are lognormally distributed, i.e., they may be expressed as the exponential of a normal random variable. On the one hand, this is nice, as it is consistent with the fact that we cannot observe negative stock prices. Furthermore, the product of lognormals is lognormal, which is nice in the longitudinal sense. Unfortunately, this is not nice in the cross-sectional sense, since the sum of lognormals is not lognormal, and we get in trouble when we consider the return of a portfolio of different stock shares.
To summarize, whatever modeling choice we make, some complication will arise. On the one hand, normal returns/gains (and stock prices) simplify the analysis of a portfolio over a single holding period, but they are empirically questionable and complicate the analysis over multiple time periods. On the other hand, lognormal gains (and stock prices) are fine for dynamic modeling of a single stock share, but they complicate the analysis of a portfolio. We may conclude that, whatever we choose, we have to accept some degree of approximation somewhere. The alternative is to tackle complicated distributions by numerical methods.