# Chapter 10: Association between random variables

STAT 1010 - Fall 2022

# Learning outcomes

By the end of this lesson you should:

• Understand the Sharpe ratio and know how to compute it to compare stocks that are not independent
• Understand and use a joint probability distribution function
• Know the 3 rules of independence
• Understand how covariance, correlation, and independence intertwine
• Explain how independent and identically distributed random variables impact upon expectation, variance, and standard deviation
• Compute expectation and variance of weighted sums

# The Sharpe Ratio

$S(X) = \frac{\mu_X - r_f}{\sigma_X}$

• $r_f$ is the return on a risk-free investment
• $\mu$ is the mean of a random variable $X$ that measures the performance of an investment
• $\sigma$ is the standard deviation of a random variable $X$ that measures the performance of an investment
• all inputs must be measured over the same time period (ie. yearly, monthly, …)

## Compute the Sharpe ratio

Company Random Variable Mean per month SD
Apple A 2.45% 13.3%
McDonalds M 1.14% 6.2%

assume $r_f$ is the risk-free rate of interest is $0.1\%$

## Compute the Sharpe ratio

• \begin{aligned} S(A) &= \frac{\mu_A - r_f}{\sigma_A}\\ &= \frac{2.45 - 0.1}{13.3}\\ &= 0.177 \end{aligned}

• \begin{aligned} S(M) &= \frac{\mu_M - r_f}{\sigma_M}\\ &= \frac{1.14 - 0.1}{6.2}\\ &= 0.168 \end{aligned}

We prefer Apple because it has a higher Sharpe ratio $0.177 > 0.168$

## Revision 2

Instead of knowing $\mu$ and $\sigma$ we have a pdf

IBM stock IBM stock Microsoft Microsoft
$x$ $P(X = x)$ $y$ $P(Y = y)$
Increases $5 0.11$4 0.18
No change 0 0.80 0 0.67
Decreases -$5 0.09 -$4 0.15

## Sharpe ratio by hand

IBM IBM IBM IBM IBM IBM IBM
$x$ $P(X = x)$ $\mu_X$ $(x-\mu_X)^2 \cdot p_X(x)$ $Var(X)$ $sd(X)$ Sharpe ratio
Increases $5 0.11 No change 0 0.80 Decreases -$5 0.09

## Sharpe ratio by hand

IBM IBM IBM IBM IBM IBM IBM
$x$ $P(X = x)$ $\mu_X$ $(x-\mu_X)^2 \cdot p_X(x)$ $Var(X)$ $sd(X)$ Sharpe ratio
Increases $5 0.11 0.1 2.6411 4.99 2.23 0.03805123 No change 0 0.80 0.1 0.0080 4.99 2.23 0.03805123 Decreases -$5 0.09 0.1 2.3409 4.99 2.23 0.03805123

## Sharpe ratio by hand

MCSFT MCSFT MCSFT MCSFT MCSFT MCSFT MCSFT
$y$ $P(Y =y)$ $\mu_Y$ $(y-\mu_Y)^2 \cdot p_Y(y)$ $Var(Y)$ $sd(Y)$ Sharpe ratio
Increases $4 0.18 No change 0 0.67 Decreases -$4 0.15

## Sharpe ratio by hand

MCSFT MCSFT MCSFT MCSFT MCSFT MCSFT MCSFT
$y$ $P(Y =y)$ $\mu_Y$ $(y-\mu_Y)^2 \cdot p_Y(y)$ $Var(Y)$ $sd(Y)$ Sharpe ratio
Increases $4 0.18 0.12 2.709792 5.2656 2.29469 0.04575782 No change 0 0.67 0.12 0.009648 5.2656 2.29469 0.04575782 Decreases -$4 0.15 0.12 2.546160 5.2656 2.29469 0.04575782

## The Sharpe ratio - in R

# Input the data and the pdf
stock <- tibble(x = c(5, 0, -5),
p_x = c(0.11, 0.8, 0.09),
y = c(4, 0, -4),
p_y = c(.18, .67, .15))

Sharpe_ratio <- # name this the Sharpe ratio
stock %>% # use the data from above
mutate(part_mean_x = x*p_x, # find mean for each part of x
part_mean_y = y*p_y, # now for y
mean_x = sum(part_mean_x), # find mean x
mean_y = sum(part_mean_y),# now for y
part_var_x = p_x * (mean_x - x)^2, # find var for each part of x
part_var_y = p_y * (mean_y - y)^2,# now for y
var_x = sum(part_var_x), # find var of x
var_y = sum(part_var_y),# now for y
sd_x = sqrt(var_x), # find sd of x
sd_y = sqrt(var_y)) %>% # now for y
summarise(S_x = (mean_x - 0.015)/sd_x, # find Sharpe ratio of x with rf = 0.015
S_y = (mean_y - 0.015)/sd_y) %>% # now for y
slice(1)

# Joint probability distributions

$X$ $X$ $X$
$x = -5$ $x = 0$ $x = 5$ $p(y)$
$Y$ $y=4$ $0.00$ $0.11$ $0.07$ $0.18$
$Y$ $y=0$ $0.03$ $0.62$ $0.02$ $0.67$
$Y$ $y=-4$ $0.06$ $0.07$ $0.02$ $0.15$
$p(x)$ $0.09$ $0.80$ $0.11$ $1$
• What is the probability that $x = 0$ and $y =4$? - $11\%$

• What is the probability that $x = 5$ and $y =4$? - $7\%$

• What is the probability that $x = 0$ and $y =0$? - $62\%$

• What outcome will never occur? - $x = -5$ and $y =4$

## Expected value of $X + Y$

$X$ $X$ $X$
$x = -5$ $x = 0$ $x = 5$ $p(y)$
$Y$ $y=4$ $0.00$ $0.11$ $0.07$ $0.18$
$Y$ $y=0$ $0.03$ $0.62$ $0.02$ $0.67$
$Y$ $y=-4$ $0.06$ $0.07$ $0.02$ $0.15$
$p(x)$ $0.09$ $0.80$ $0.11$ $1$
• $E(X+Y) = E(X) + E(Y)$
• $E(X+Y) = E(X) + E(Y) = 0.1 + 0.12 = 0.22$

# 3 Rules of independence

If the probability of one event occuring has no impact on another event occuring, they are independent.

• If $X$ and $Y$ are independent then $p(x,y) = p(x) \cdot p(y)$ for all pairs $(x,y)$

• If $p(x,y) = p(x) \cdot p(y)$ for all $(x, y)$ pairs, then $X$ and $Y$ are independent.

• It follows that if $X$ and $Y$ are independent, then $E(XY) = E(X) \cdot E(Y)$

## Are $X$ and $Y$ independent?

$X$ $X$ $X$
$x = -5$ $x = 0$ $x = 5$ $p(y)$
$Y$ $y=4$ $0.00$ $0.11$ $0.07$ $0.18$
$Y$ $y=0$ $0.03$ $0.62$ $0.02$ $0.67$
$Y$ $y=-4$ $0.06$ $0.07$ $0.02$ $0.15$
$p(x)$ $0.09$ $0.80$ $0.11$ $1$
• $p_x(-5) \cdot p_y(4) = 0.18 \cdot 0.09 \neq 0.00$

• $p_x(-5) \cdot p_y(-4) = 0.09 \cdot 0.15 = 0.0135 \neq 0.06$

• NO! NO! NO!

# Covariance of Random Variables

• $cov(x, y) = \frac{(x_1 - \bar{x})(y_1 - \bar{y}) + (x_2 - \bar{x})(y_2 - \bar{y}) + \ldots + (x_n - \bar{x})(y_n - \bar{y})}{n-1}$

• assumes each $(x, y)$ pair appears once

• $Cov(X, Y) = E((X - \mu_X)(Y - \mu_Y))$

## Covariance of sum

• $(a+b)^2 = a^2 + 2ab +b^2$
• it follows that
• $Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y)$

## Correlation

• $corr(x, y) = \frac{cov(x, y)}{s_x \cdot s_y}$

• $\rho = Corr(X, Y) = \frac{Cov(X, Y)}{\sigma_X \cdot \sigma_Y}$

As $X$ increases, what happens to $Y$?

$X$ $X$ $X$
$x = -5$ $x = 0$ $x = 5$ $p(y)$
$Y$ $y=4$ $0.00$ $0.11$ $0.07$ $0.18$
$Y$ $y=0$ $0.03$ $0.62$ $0.02$ $0.67$
$Y$ $y=-4$ $0.06$ $0.07$ $0.02$ $0.15$
$p(x)$ $0.09$ $0.80$ $0.11$ $1$

## Independence & Cov

How does independence impact upon correlation and covariance?

$E((X-\mu_X)(Y-\mu_Y))$

• \begin{aligned} Cov(X,Y) &= E((X-\mu_X)(Y-\mu_Y))\\ &= E(X - \mu_X)E(Y-\mu_Y)\\ &= 0 \end{aligned}

• If $X$,$Y$ are independent, then $Cov(X, Y) = 0$

• The opposite is not true (if $Cov(X, Y) = 0$, then $X$,$Y$ are independent)

## Independence & Var

$Var(X) + Var(Y) = Var(X) + Var(Y) + 2Cov(X, Y)$

• If $X$, $Y$ are independent, $Cov(X, Y) = 0$, so

• $Var(X+ Y) = Var(X) + Var(Y)$

## Sharpe ratio of a sum

$S(X + Y) = \frac{(\mu_X + \mu_Y) - 2r_f}{\sqrt{(Var(X+Y))}}$

## Sharpe ratio of sum - R

# input x & y
x <- c(-5, 0, 5)
y <- c(4, 0, -4)

# input data
data <- bind_cols(expand_grid(x, y),
probs = c(0, .03, .06,
.11, .62, .07,
.07, .02, .02),
mu_x = rep(0.1, 9),
mu_y = rep(0.12, 9),
var_x = rep(4.99, 9),
var_y = rep(5.2656, 9))

data %>%
mutate(cov_xy_part = (x - mu_x)*
(y - mu_y)*probs, # multiply together
cov_xy = sum(cov_xy_part), # sum them
var_xy = var_x + var_y + 2*cov_xy) %>% # using the rule
summarise(S_r = (mu_x + mu_y - 2*0.015)/
(sqrt(var_xy))) %>% # find the Sharpe value
slice(1) # only the first
# A tibble: 1 × 1
S_r
<dbl>
1 0.0497
• The Sharpe ratio for $X$ is $0.038$, for $Y$ it’s $0.046$, investing in both gives a better return $0.050$

# Double the investment

• Invest twice as much for one day
• Invest in one stock for two subsequent days
• Which is better? Why?

## Double for one day

$S(2X) = \frac{2\mu_X - 2r_f}{\sqrt{Var(2X)}}$

• $\mu_X = 0.1$

• $Var(X) = 4.99$

• $r_f = .015$

• \begin{aligned} &= \frac{2 \cdot 0.1 - 2 \cdot 0.015}{\sqrt{4 \cdot 4.99}}\\ &= \frac{.2 - 0.03}{\sqrt{19.96}}\\ &= 0.038 \end{aligned}

• This is the same as before, how do we compute if we invest for two subsequent days?

## IID

If we invested in a stock on 2 subsequent days instead of investing in 2 stocks on one day the return on those two days are:

independent and identically distributed (IID)

• identically distributed - the outcomes on each day are likely to be different, but the probability of the outcomes is the same

• independent - very common assumption for stocks (part of the reason for the 2008 financial crises)

## Addition rules for IID variables

Important

If $n$ random variables $(X_1, X_2, ..., X_n)$ are iid with mean $\mu_X$ and standard deviation $\sigma_X$, then

$E(X_1 + X_2 + ... + X_n) = n \cdot \mu_X$ $Var(X_1 + X_2 + ... + X_n) = n \cdot \sigma^2_x$ $SD(X_1 + X_2 + ... + X_n) = \sqrt{n} \sigma_X$

## Sharpe ratio - two days

$S(X_1 + X_2) = \frac{2\mu_X - 2r_f}{\sqrt{2 \sigma^2_x}}$

• \begin{aligned} &= \frac{0.20 - 0.03}{\sqrt{9.98}}\\ &= 0.054 \end{aligned}
• you expect more variance if you have two stock for one day b/c anything that happens that day is magnified
• if you have one stock for two days you reduce the variance

# Weighted sums

If we decide to leave money in the bank, there will be interest that accrues. How can we include this in our model?

Important

$E(aX + bY + c) = aE(X) + bE(Y) + c$

$Var(aX + bY + c) = a^2Var(X) + b^2Var(Y) + 2abCov(X, Y)$

## Expectation of weighted sum

$E(2X + 4Y + 0.06)$

• \begin{aligned} &= 2E(X) + 4E(Y) + 0.06\\ &= 2(0.10) + 4(0.12) + 0.06\\ &= \0.74 \end{aligned}

## Variance of weighted sum

$Var(2X + 4Y + 0.06)$

• \begin{aligned} &= 2^2Var(X) + 4^2Var(Y) + 2 \cdot(2 \cdot 4) \cdot Cov(X,Y)\\ &= 4(4.99) + 16(5.27) + 16(2.19)\\ &= 139.32 \end{aligned}