Let's say the return on a hypothetical investment can take on three possibilities, all equally likely: $30\%$, $10\%$, and $-10\%$. The expected value of the investment after n trials (assuming we started with $1) is:
$$ E[(1+i_1)(1+i_2)(1+i_3)\cdots(1+i_n)] $$ Luckily, since the returns are all independent and identically distributed I can use the fact that the returns have zero covariance (and therefore $E[XY] = E[X]E[Y]$), so that the expectation can be written like so:
$$ E[1+i_1]\cdot E[1+i_2]\cdot E[1+i_3]\cdots E[1+i_n] $$
$$ =(E[1+i])^n $$
$$= (1.1)^n $$
(I used the fact that $E[1+i] = 1/3\cdot(1.3 + 1.1 + 0.9) = 1.1$.)
Nowhere did we use any approximation; the result is exact.
Now let’s do it another way. Starting from back here:
$$ E[(1+i_1)(1+i_2)(1+i_3)\cdots(1+i_n)] $$
We could change variables, where $1+i = e^δ$, (where $δ$ would be $\ln(1.3)$, $\ln(1.1)$, $\ln(0.9)$, each with $1/3$ probability). Then we have:
$$ E[e^{δ_1}\cdot e^{δ_2}\cdot e^{δ_3}\cdots e^{δ_n}] $$
$$ = E[e^{δ_1+δ_2+δ_3+\cdots+δ_n}] $$
Now let us use the Central Limit Theorem. If n is sufficiently large, then it would be reasonable to replace the exponent (i.e. $δ_1+δ_2+δ_3+\cdots+δ_n \approx n\mu $) with a normally distributed random variable, $N(n\mu,n\sigma^2)$ with
$µ = 1/3\cdot [\ln(1.3) + \ln(1.1) + \ln(0.9)] \approx 0.0841046$, and $σ^2 = 1/3 \cdot [(\ln(1.3)-µ)^2 + (\ln(1.1)-µ)^2 + (\ln(0.9)-µ)^2] \approx 0.0225997$.
Then we have:
$E[e^{N(nµ,nσ^2)}]$ (The expression inside the brackets is a lognormal random variable.)
$= e^{nµ + nσ^2/2}$ (This is the expectation of a lognormal random variable.)
$$ = (e^{µ + σ^2/2})^n $$
If I plug in the values for $µ$ and $σ^2$, I get this:
$$ = (1.10010375)^n $$
This is close to the actual answer of $(1.1)^n$ that we arrived at earlier, but here’s my problem: I would think that as n gets large, the true Expectation and the approximate expectation using the CLT should converge – after all, doesn’t the Central Limit Theorem say that the estimate only gets better the greater $n$ is? However, you can clearly see that even though $1.1$ and $1.10010375$ are close, as they are raised to the nth power, and as $n$ gets larger and larger, their values diverge!
So what’s going on here?? I’ve either made a mistake or I am inappropriately using the CLT.
Any thoughts??
The central limit theorem here implies that as $n\to\infty$, $$ \frac1{\sqrt{n}}\left(\sum_{i=1}^n(\delta_i-\mu)\right)\to \mathcal N(0,\sigma^2)\ \text{in distribution}. $$ It does not say anything about the convergence of $\sum_{i=1}^n\delta_i$ because it does not converge to anything. In particular, you cannot use it to say $$ \sum_{i=1}^n\delta_i\to\mathcal N(n\mu,n\sigma^2)\ \text{in distribution}.$$