3. 1-Page Draft#

3.1. Inflation and the need for investing#

In contemporary economic systems, institutions targets a low non-zero inflation, usually \(2\%\). Thus, the protection fo real valu of money is the bare minimum goal of investing.

Thus, need for investing

  • starting from personal goals and constraints (saving rate, risk tolerance,…)

  • knowing the available assets and their properties, e.g. 1-period — usually 1-year — return probability distribution

  • knowing the fundamentals of composition of returns, and math facts about multi-period performances

First, analysis of 1-asset over 1-period, and eventually multi-asset over many periods of time.

3.2. Return, risk, and risk-adjusted return#

3.2.1. Single asset, single period#

The single period return of an asset \(X\) over period \(i\) from time \(i-1\) and \(i\) is defined as

\[r_t := \dfrac{X_{t} - X_{t-1}}{X_{t-1}} = \frac{X_t}{X_{t-1}} - 1 \ ,\]

being \(X_t\) the value of the asset \(X\) at time \(t\). A 1-period return of an asset \(X\) can be usually modelled as a random variable with probability distribution \(p_t(r)\).

3.2.2. Single asset, multiple periods#

The return compounds over many periods,

\[X_t = ( 1 + r_t ) X_{t-1} = ( 1 + r_t ) \dots ( 1 + r_1 ) X_0 = \prod_{i=1}^{t} ( 1 + r_i ) X_0 = \prod_{i=1}^t \frac{X_{i}}{X_{i-1}} \ .\]

The compund average return (usually called Compound Annual Growth Rate, if the reference period is a year) can be defined as the constant return \(r_i = r\) for all \(i = 1:t\) producing the actual result, and thus

\[X_t = \left( 1 + r \right)^t X_0 \ ,\]

or

\[r = \left(\frac{X_t}{X_0}\right)^{\frac{1}{t}} - 1 \ .\]
Log-return.
\[\begin{split}\begin{aligned} \log \frac{X_t}{X_0} & = \sum_{i=1}^t \log \frac{X_{i}}{X_{i-1}} = \sum_i \log \left( 1 + r_i \right) \\ & = t \log ( 1 + r ) \ , \end{aligned}\end{split}\]

so that

\[\log ( 1 + r ) = \frac{1}{t} \sum_i \log \left( 1 + r_i \right) \ .\]
Probability distribution, expected value and variance

Skewness of the probability density of the result \(\frac{X_t}{X_0}\), typically producing median value \(lt\) expected value. Under some assumptions (…) the expected value of the log-return has expected value that is smaller than the 1-period expected return due to dispersion (volatility drag)

\[\mathbb{E}[ r ] \sim \mathbb{E}[ r_i ] - \frac{\text{Var}[r_i]}{2} \ ,\]

and variance

\[\text{Var}[ r ] \sim \frac{\text{Var}[r_i]}{t} \ .\]

3.2.3. Multiple assets, single period#

Given a portfolio with many assets of value \(X_{t-1}\) at time \(t-1\), with weights

\[w_{i,t-1} := \frac{X_{i,t-1}}{X_{t-1}} = \frac{X_{i,t-1}}{\sum_{i} X_{i,t-1}} \ ,\]

and constant composition in time until \(t\), the value of the final portfolio becomes

\[X_{t} = \sum_{i} X_{i,t} = \sum_i ( 1 + r_{i,t} ) X_{i,t-1} = \sum_i r_{i,t} X_{i,t-1} + X_{t-1}, \]

and the return

\[r_t := \frac{X_t}{X_{t-1}} - 1 = \dfrac{\sum_i r_{i,t} X_{i,t-1}}{X_{t-1}} = \sum_i r_{i,t} w_{i,t-1} = \mathbf{w}_{t-1}^T \mathbf{r}_t \ .\]

The return of the assets \(\mathbf{r}\) can be modelled as a (vector) random variable. The expected value and the variance of the 1-period return of the portfolio with weights \(\mathbf{w}\) read

\[\mu_r := \mathbb{E}[ r ] = \mathbb{E}\left[ \mathbf{w}^T \mathbf{r} \right] = \mathbf{w}^T \mathbb{E}\left[ \mathbf{r} \right] = \mathbf{w}^T \boldsymbol\mu_{\mathbf{r}} \ ,\]

and

\[\begin{split}\begin{aligned} \sigma^2_r := \text{Var}[ r ] & = \mathbb{E}\left[ ( r - \mathbb{E}[r] )^2 \right] = \\ & = \mathbb{E}\left[ \mathbf{w}^T ( \mathbf{r} - \boldsymbol\mu ) ( \mathbf{r} - \boldsymbol\mu )^T \mathbf{w} \right] = \\ & = \mathbf{w}^T \boldsymbol\sigma^2_{\mathbf{r}} \mathbf{w} \ . \end{aligned}\end{split}\]

3.2.4. Multiple assets, multiple periods#

Without rebalancing. No action is taken. Every individual asset of the portfolio evolves indipendently. At time \(0\)

\[X_0 = \sum_i X_{i,0} = \sum_i w_{i,0} X_0 \ .\]

At successive times,

\[X_t = \sum_i X_{i,t} = \sum_i X_{i,0} \prod_{\tau=1}^{t} ( 1 + r_{i,\tau} ) = X_0 \sum_i w_{i,0} \prod_{\tau=1}^{t} ( 1 + r_{i,\tau} ) \ .\]

With rebalancing (assuming no cost…). At the end of each period, weights of individual assets are set back to the original desired value. Following this strategy, under the assumption of stationariety, the 1-period performance of the portfolio is constant, with 1-period return

\[r_1 = \mathbf{w}^T \mathbf{r} \ ,\]

and expected value and variance

\[ \mu_{(1)} = \mathbf{w}^T \boldsymbol{\mu}_{\mathbf{r}} \quad , \quad \sigma_{(1)}^2 = \mathbf{w}^T \boldsymbol\sigma^2_{\mathbf{r}} \mathbf{w} \ . \]

Under the assumption of “small” 1-period returns, the geometric average return reads

\[\overline{r} \sim \mathscr{N} \left( \mu_{(1)} - \dfrac{\sigma_{(1)}^2}{2}, \dfrac{\sigma_{(1)}^2}{N} \right) \ .\]
Shannon demon and the rebalancing premium: 2-asset portfolio

A 2-asset portfolio is determined by the weights \(\mathbf{w} = ( w_1, w_2 )\) of the two assets. One-period return of the 2 assets is modeleld as a multidimensional random variable \(\mathbf{r}\), whose expected value and variance read

\[\begin{split}\boldsymbol\mu_{(1)} = \mathbb{E}[ \mathbf{r} ] = \begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}\end{split}\]

and

\[\begin{split}\boldsymbol\sigma^2_{(1)} = \mathbb{E}[ \Delta \mathbf{r} \Delta \mathbf{r}^T ] = \begin{bmatrix} \sigma^2_{11} & \sigma^2_{12} \\ \sigma^2_{12} & \sigma^2_{22} \end{bmatrix} = \begin{bmatrix} \sigma^2_1 & \rho_{12} \sigma_1 \sigma_2 \\ \rho_{12} \sigma_1 \sigma_2 & \sigma_2^2 \end{bmatrix} \ .\end{split}\]

Under certain conditions, the expected value of the multi-period return of the diversified and rebalanced portfolio may exceed the largest expected value of multi-period returns from individual assets, i.e.

\[\mu - \dfrac{\sigma^2}{2} > \mu_i - \dfrac{\sigma_i^2}{2} \ .\]
Some algebra and analytic geometry

Let’s evaluate the conditions for \(i = 1\)

\[\begin{split}\begin{aligned} 0 & < w_1 \mu_1 + w_2 \mu_2 - \dfrac{w_1^2 \sigma_1^2 + 2 \rho_{12} \sigma_1 \sigma_2 w_1 w_2 + w_2^2 \sigma_2^2}{2} - \mu_1 + \dfrac{\sigma_1^2}{2} = \\ & = c(w_1, w_2) \end{aligned}\end{split}\]

Given the properties \(\mu_i\), \(\sigma_i\), \(\rho_{12}\), the equation \(c(w_1, w_2) = 0\) is the equation of a conic section in the plane \(w_1-w_2\). With some high-school math, see general expression of conic sections, the reduced discriminant \(\frac{\Delta}{4} = \frac{B^2}{4} - AC\) determines the nature of the curve. As,

\[\dfrac{\Delta}{4} = \dfrac{B^2}{4} - 4 A C = \dfrac{\rho_{12}^2 \sigma_1^2 \sigma_2^2 - \sigma_1^2 \sigma_2^2}{4} = \dfrac{1}{4} \sigma_1^2 \sigma_2^2 ( \rho_{12}^2 - 1 ) \le 0 \ ,\]

the equation \(c(w_1, w_2) = 0\) is the equation of an ellipse for \(\rho_{12} \ne 1\), and a parabola for \(\rho_{12}^2 = 1\) (perfectly correlated assets).

Non-perfectly correlated assets, \(\ |\rho_{12}| < 1\)

The general expression of the quadratic curve in the \(\mathbf{w}\)-plane reads

\[0 = \frac{1}{2} \mathbf{w}^T \boldsymbol\sigma^2 \mathbf{w} - \boldsymbol\mu^T \mathbf{w} + \mu_1^{multi}\]

In order to find the center of the ellipse, a translation \(\mathbf{w} = \mathbf{w}_C + \mathbf{w}_1\) should produce zero 1-degree terms, i.e.

\[\begin{split}\begin{aligned} 0 & = \dfrac{1}{2} \left( \mathbf{w}_1 + \mathbf{w}_C \right)^T \boldsymbol\sigma^2 \left( \mathbf{w}_C + \mathbf{w}_1 \right) - \boldsymbol\mu^T \left( \mathbf{w}_C + \mathbf{w}_1 \right) + \mu_1^{multi} = \\ & = \dfrac{1}{2} \mathbf{w}_1^T \boldsymbol\sigma^2 \mathbf{w}_1 + \left( 2 \mathbf{w}_C^T \boldsymbol\sigma^2 - \boldsymbol\mu^T \right) \mathbf{w}_1 + \dfrac{1}{2} \mathbf{w}_C^T \boldsymbol\sigma^2 \mathbf{w}_C - \boldsymbol\mu^T \mathbf{w}_C + \mu_1^{multi} \ , \end{aligned}\end{split}\]

i.e.

\[\mathbf{w}_C = \dfrac{1}{2} \left( \boldsymbol\sigma^2 \right)^{-1} \boldsymbol\mu \ .\]

Using coordinates \(\mathbf{x}_1\), the expression of the ellipse becomes

\[0 = \dfrac{1}{2} \mathbf{w}_1^T \boldsymbol\sigma^2 \mathbf{w}_1 - \boldsymbol\mu^T \left( \boldsymbol\sigma^2 \right)^{-1} \boldsymbol\mu + \mu_1^{multi} \ .\]

The angle of the semi-axes of the ellipse w.r.t. the coordinate axes and their lengths appears after applying the rotation transformation, \(\mathbf{w}_1 = \mathbf{R} \mathbf{w}_2\), that makes the quadratic form diagonal, i.e. as the solution of the spectral decomposition of the covariance matrix,

\[\mathbf{R}^T \boldsymbol\sigma^2 \mathbf{R} =: \boldsymbol\sigma^2_d \ .\]

Using the results obtained for general expression of conic sections, if \(\rho_1 \ne \rho_2\), the rotation angle \(\theta\) satisfies

\[\text{tan} 2 \theta = \frac{2 \rho_{12} \sigma_1 \sigma_2}{\sigma_1^2 - \sigma_2^2} \ ,\]

and the length of the semi-axes, given \(\boldsymbol\mu^T \left( \boldsymbol\sigma^2 \right)^{-1} \boldsymbol\mu - \mu_1^{multi} > 0\)1, is

\[a_{1,2} = \sqrt{\frac{\boldsymbol\mu^T \left( \boldsymbol\sigma^2 \right)^{-1} \boldsymbol\mu - \mu_1^{multi}}{s_{1,2}}} \ ,\]

with the eigenvalue of the covariance matrix

\[s_{1,2} = \dfrac{\sigma_1^2 + \sigma_2^2}{2} \mp \dfrac{\sqrt{ \left( \sigma_1^2 - \sigma_2^2 \right)^2 + 4 \rho_{12}^2 \sigma_1^2 \sigma_2^2}}{2}\]
Perfectly correlated assets, \(\ |\rho_{12}| = 1\)

The general expression of the quadratic curve in the \(\mathbf{w}\)-plane reads

\[0 = - c(\mathbf{w}) = \frac{1}{2} \left( w_1 \sigma_1 + w_2 \sigma_2 \right)^2 - w_1 \mu_1 - w_2 \mu_2 + \mu_1 - \frac{\sigma_1^2}{2} \ .\]

At the end of any period, weights are set back to the desired fractions. At time \(0\)

\[X_0 = \sum_i X_{i,0} = \sum_{i} w_{i} X_0 \ ,\]

At time \(1\) before rebalancing

\[X_1 = \sum_i X_{i,1}^- = \sum_i X_{i,0} ( 1 + r_{i,0} ) = X_0 \left( 1 + \sum_i w_i r_{i,0} \right) \ .\]

At time \(1\) after rebalancing

\[X_1 = \sum_i X_{i,1}^+ = \sum_i w_i X_1 \]

At time \(2\) before rebalancing

\[X_2 = \sum_i X_{i,1}^+ ( 1 + r_{i,1} ) = X_1 \sum_i w_i ( 1 + r_{i,1} ) \ . \]

and thus

\[\frac{X_t}{X_0} = \prod_{\tau=1}^{t} \left( 1 + \mathbf{w}^T \mathbf{r}_{\tau} \right) \ .\]

The log-return comes from

\[\begin{split}\begin{aligned} \log \frac{X_t}{X_0} & = \sum_{\tau=1}^{t} \log ( 1 + \mathbf{w}^T \mathbf{r}_{\tau} ) = \\ &\sim \sum_{\tau=1}^{t} \left( 1 + \mathbf{w}^T \mathbf{r}_{\tau} + \dfrac{1}{2} \mathbf{w}^T \mathbf{r}_{\tau} \mathbf{r}_{\tau}^T \mathbf{w} \right) \end{aligned}\end{split}\]

The expected value and the variance read…

1

Otherwise the general expression doesn’t represent any curve in the real plane, as that equation is never satisfied by any values of \(\mathbf{w}\).