Time Diversification

Time Diversification#

Notation#

notation	description
$r_t$	return rate, ($t-1$ to $t$)
$r_{t,t+h}$	$\left(\prod_{\tau=1}^h R_{t+\tau}\right)-1$
$r^{\log}_t$	$\log\left(1+r_t\right)$
$r^{\log}_{t,t+h}$	$\log\left(1+r_{t,t+h}\right)$

Cumulative Return Risk and Autocorrelation#

Log cumulative returns are simply a portfolio of single-period returns.

Define $\mu$ and $\sigma^2$ as $$\mathbb{E}\left[r^{\log}_{t,t+1}\right] = \mu, \quad \text{Var}\left[r^{\log}_{t,t+1}\right] = \sigma^2$$
For the $h$-period log return, $$\mathbb{E}\left[r^{\log}_{t,t+h}\right] = h\mu$$\text{Var}\left[r^{\log}_{t,t+h}\right] = \sum_{j=1}^h\sum_{i=1}^h \text{Cov}\left[r^{\log}_{t+i},r^{\log}_{t+j}\right]$$

Like the case of diversifying one-period returns across $k$ assets, the covariances determine the overall risk but do not affect the mean return.

Autocorrelation Models#

As seen in the previous section,

The variance of cumulative returns, $r^{\log}_{t,t+h}$, depends critically on the auto-covariance of the return series, denoted as $$\text{Cov}\left[r^{\log}_{t},r^{\log}_{t+i}\right], \quad \text{or} \quad \sigma_{t,t+i}$$
Specifying a form for the autocorrelations, $\text{Corr}\left[r^{\log}_{t},r^{\log}_{t+i}\right]$, is equivalent.
A model of autocorrelations uniquely specifies a linear time series model.

AR(1) Models#

Autoregressive (AR) models are among the most popular in time-series statistics.

Consider the AR(1) model, $$\text{Cov}\left[r^{\log}_{t},r^{\log}_{t+i}\right] = \rho^i \sigma^2$$\text{Corr}\left[r^{\log}_{t},r^{\log}_{t+i}\right] = \rho^i$$

With AR models, covariances are easy to scale over time.

Diversification and Cumulative Returns#

Mean returns scale linearly in horizon $h$, $$\mathbb{E}\left[r^{\log}_{t,t+h}\right] = h\mu$$

But scaling of variance depends on correlation:

$\rho=1$: Std.Dev. is linear in cumulation: $\text{Std}\left[r^{\log}_{t,t+h}\right] = h\sigma$
$\rho=0$: Variance is linear in cumulation: $\text{Var}\left[r^{\log}_{t,t+h}\right] = h\sigma^2$
$\rho=-1$: The return is riskless: $\text{Var}\left[r^{\log}_{t,t+h}\right] = 0$

Example: Riskless Bond#

At time $t=0$, consider a bond with riskless payout at $t=10$. The yield of the bond at $t=0$ is 5%.

The 10-year cumulative return, $r^{\log}_{0,10}$ is riskless, and equals $5\% \times 10$.
At any intermediate time, ($t$, such that $0< t < 10$,) the bond price is uncertain.
Thus, the intermediate returns, $r^{\log}_{0,t}$ and $r^{\log}_{t,10}$ are uncertain.
However, $r^{\log}_{0,10} = r^{\log}_{0,t} + r^{\log}_{t,10}$.
So if $r^{\log}_{0,t}$ is unexpectedly high, then $r^{\log}_{t,10}$ must be relatively low, such that the riskless return $r^{\log}_{0,10}$ ends up at $5\%\times 10$.

Example: Bond Mean Reversion#

A ten-year risk-free bond has a 10-year return with perfect mean reversion. (Source: Cochrane, 2011)

Bond mean reversion demo

Negative Serial Correlation in Bonds#

Thus, riskless bonds should have negatively autocorrelated returns: $$\text{Corr}\left(r^{\log}_t,r^{\log}_{t+1}\right)<0$$

And if the bond matures at $T$, then for any $0<h<T-t$, $$\text{Corr}\left(r^{\log}_{t,t+h},r^{\log}_{t+h,T}\right) = -1$$

Default-free bonds are safer in the long-run, with $\text{Var}\left[r^{\log}_{t,T}\right]=0$.

Cumulative Sharpe Ratios in AR(1) Model#

Consider again the cumulative return, $r^{\log}_{t,t+h}$.

For $\rho=1$: $$\text{SR}\left(r^{\log}_{t,t+h}\right) = \text{SR}\left(r^{\log}_t\right)$$
For $|\rho|<1$: $$\text{SR}\left(r^{\log}_{t,t+h}\right) > \text{SR}\left(r^{\log}_t\right)$$
For $\rho=0$: $$\text{SR}\left(r^{\log}_{t,t+h}\right) = \sqrt{h}~ \text{SR}\left(r^{\log}_t\right)$$

Mean Annualized Returns#

The annualized mean (log) return on the $h$-period investment, $r^{\log}_{t,t+h}$ is

\[\frac{r^{\log}_{t,t+h}}{h} = \frac{\sum_{i=1}^h r^{\log}_{t+i}}{h}\]

This is just the usual sample estimate of the mean of one-period returns, $\bar{r}$, based on a sample-size of $h$!

For any $\rho$	For $\rho=0$
$\mathbb{E}\left[\bar{r}\right] = \mu$	$\text{Var}\left[\bar{r}\right] = \frac{\sigma^2}{h}$

So as the investment horizons gets large, the mean annualized return, $\bar{r}$, converges to the true annual mean return, $\mu$.
Even if $\rho\ne 0$ we can still conclude that $\bar{r}\to \mu$. (Law of large numbers still holds.)