When Log Returns Imply a Lognormal Asset Value

Summary

The question asks whether normally distributed log returns imply that an asset's value is lognormally distributed. The answer expresses the terminal price as its starting price multiplied by the exponential of the sum of period log returns. Taking the logarithm of the price ratio turns that product into a sum, so if the cumulative log return is normally distributed, the price ratio is lognormal; with a fixed starting value, the terminal price is lognormal as well.

This is a distributional implication under stated assumptions, not evidence that real asset prices follow the model. In particular, normality of each period's log return alone does not guarantee normality of their sum unless the joint behavior supports it. The answer's closing sentence reverses the distributional terminology: log returns are normal under the assumption, while simple gross returns and price ratios are lognormal. The argument concerns a fixed initial value and a specified horizon.

Key ideas

• A price evolves as its initial value multiplied by the exponentials of period log returns.
• The logarithm of the terminal-to-initial price ratio equals the sum of period log returns.
• A normally distributed cumulative log return implies a lognormally distributed price ratio.
• A fixed initial price preserves the lognormal form for the terminal price.
• Normality of individual period returns requires suitable dependence assumptions for their sum to be normal.

Tags

# Is the value also log-normally distributed?


# Is the value also log-normally distributed?












My book assumes many times that $log(1+R)$ is normally distributed, so R is log-normal. But does this also mean that the value process is log-normal? Since $V=V_0(1+R)\rightarrow V/V_0=1+R$, and since $1+R$ is log-normal, $V/V_0$ is log-normal, so $log(V/V_0)=log(V)-log(V_0)$ is normal, and hence V is also log-normally distributed?

## Answer by SolitonK (score 2)

https://quant.stackexchange.com/a/15580

There are many ways answering this, here is one:

We assume the asset price at $t=T$, $S_T = S_{T-1} \times (S_T / S_{T-1})$.

Assuming continuous compounding, we can write, $S_T = S_{T-1} \times \exp(R_{T-1})$.

Working the same way for the previous period, we get $S_{T} = S_{T-2} \times \exp(R_{T-1}+R_T)$.

Working all the way back to the initial value of the asset price, $S_0$, we have that: $S_T = S_0 \times \exp(R_1 + R_2 + ... + R_T)$.

Dividing by $S_0$ and taking $\log$s yields: $\ln(S_T/S_0) = R_1 + R_2 + ... + R_T$.

Here, $(S_T/S_0)$, which is the return of the asset over the entire period, is lognormally distributed if $R_1 + R_2 + ... + R_T$ is normally distributed. To arrive to the appropriate format of the lognormal distribution,and since $R$ is normally distributed, by $\exp$ing: $(S_T/S_0) = \exp(R_1 + R_2 + ... + R_T)$

Hence, to pose on another way, if we assume that prices are lognormally distributed (which could not always be the case), then $\log(1+R_t)$ is lognormally distributed.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.