(fin-edu:draft)=
# 1-Page Draft

## 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.

## Return, risk, and risk-adjusted return

### 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 $p_t(r)$.

### 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 

$$r = \left(\frac{X_t}{X_0}\right)^{\frac{1}{t}} - 1 \ .$$

```{dropdown} Log-return.

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

so that 

$$\log ( 1 + r ) = \frac{1}{t} \sum_i \log \left( 1 + r_i \right) \ .$$

```

```{dropdown} 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

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


```

### 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

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

### 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

$$\overline{r} \sim \mathscr{N} \left( \mu_{(1)} - \dfrac{\sigma_{(1)}^2}{2}, \dfrac{\sigma_{(1)}^2}{N} \right) \ .$$

```{dropdown} Shannon demon and the rebalancing premium: 2-asset portfolio
:open:

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

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

and

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

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.

$$\mu - \dfrac{\sigma^2}{2} > \mu_i - \dfrac{\sigma_i^2}{2} \ .$$

```

```{dropdown} Some algebra and analytic geometry
:open:

Let's evaluate the conditions for $i = 1$

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

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](https://basics2022.github.io/bbooks-math-miscellanea-hs/intro.html), see [general expression of conic sections](https://basics2022.github.io/bbooks-math-miscellanea-hs/ch/analytic_geometry/analytic_geometry_2d/conics-general.html), 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).

```

```{dropdown} Non-perfectly correlated assets, $\ |\rho_{12}| < 1$
:open:

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.

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

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](https://basics2022.github.io/bbooks-math-miscellanea-hs/ch/analytic_geometry/analytic_geometry_2d/conics-general.html), 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 $\boldsymbol\mu^T \left( \boldsymbol\sigma^2 \right)^{-1} \boldsymbol\mu - \mu_1^{multi} > 0$[^degenerate-conics], is

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

[^degenerate-conics]: Otherwise the general expression doesn't represent any curve in the real plane, as that equation is never satisfied by any values of $\mathbf{w}$.

with the eigenvalue of the covariance matrix

$$s_{1,2}  = \dfrac{\sigma_1^2 + \sigma_2^2}{2} \mp \dfrac{\sqrt{ \left( \sigma_1^2 - \sigma_2^2 \right)^2 + 4 \rho_{12}^2 \sigma_1^2 \sigma_2^2}}{2}$$


<!--
By direct computation, 

$$\begin{aligned}
  0 = | \boldsymbol\sigma^2 - s \mathbf{I} |
  & = \left|\begin{bmatrix} \sigma_1^2 - s & \rho_{12} \sigma_1 \sigma_2 \\ \rho_{12} \sigma_1 \sigma_2 & \sigma_2^2 - s  \end{bmatrix}\right| = \\
  & = ( \sigma_1^2 - s )( \sigma_2^2 - s ) - \rho_{12}^2 \sigma_1^2 \sigma_2^2 = \\
  & = s^2 - \left( \sigma_1^2 + \sigma_2^2 \right) s + \left( 1 - \rho^2_{12} \right) \sigma_1^2 \sigma_2^2 \ ,
\end{aligned}$$

the eigenvalues read

$$\begin{aligned}
  s_{1,2}
  & = \dfrac{\sigma_1^2 + \sigma_2^2}{2} \mp \dfrac{\sqrt{\sigma_1^4 + 2 \sigma_1^2 \sigma_2^2 + \sigma_2^4 - 4 \sigma_1^2 \sigma_2^2 + 4 \rho_{12}^2 \sigma_1^2 \sigma_2^2}}{2} = \\
  & = \dfrac{\sigma_1^2 + \sigma_2^2}{2} \mp \dfrac{\sqrt{ \left( \sigma_1^2 - \sigma_2^2 \right)^2 + 4 \rho_{12}^2 \sigma_1^2 \sigma_2^2}}{2} \ ,
\end{aligned}$$

and the eigenvectors are evaluated as a result of

$$\begin{aligned}
  0 
  & = \begin{bmatrix} \frac{\sigma_1^2 - \sigma_2^2}{2} \pm \frac{\sqrt{(\sigma_1^2 - \sigma_2^2)^2 + 4 \rho_{12}^2\sigma_1^2 \sigma_2^2}}{2} & \rho_{12} \sigma_1 \sigma_2 \end{bmatrix} \begin{bmatrix} \hat{w}_{1,12} \\ \hat{w}_{2,12} \end{bmatrix} \ ,
\end{aligned}$$

i.e.

$$\begin{bmatrix} \widetilde{w}_{1,12} \\ \widetilde{w}_{2,12} \end{bmatrix} \propto \begin{bmatrix} \rho_{12} \sigma_1 \sigma_2 \\ \frac{\sigma_2^2-\sigma_1^2}{2} \mp \frac{\sqrt{(\sigma_2^2-\sigma_1^2)^2 + 4 \rho_{12}\sigma_1^2 \sigma_2^2}}{2} \end{bmatrix} \ .$$

or, after normalization,

$$\begin{aligned}
|\widetilde{\mathbf{w}}_{12}|^2
  & = \rho_{12}^2 \sigma_1^2 \sigma_2^2 + \dfrac{(\sigma_2^2 - \sigma_1^2)^2}{4} + \dfrac{(\sigma_2^2-\sigma_1^2)^2+4\rho_{12} \sigma_1^2 \sigma_2^2}{4} \mp \dfrac{(\sigma_2^2-\sigma_1^2) \sqrt{...}}{2} = \\
  & = \dfrac{(\sigma_2^2 - \sigma_1^2)^2}{2} + 2 \rho_{12} \sigma_1^2 \sigma_2^2 \mp \dfrac{(\sigma_2^2 - \sigma_1^2) \sqrt{...}}{2}
\end{aligned}$$
-->


```

```{dropdown} Perfectly correlated assets, $\ |\rho_{12}| = 1$
:open:

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

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

The expected value and the variance read...