Portfolios

BUSI 721: Data-Driven Finance I

Kerry Back, Rice University

Portfolio returns

• Two assets, prices $p_1$ and $p_2$
• Own shares $x_1$ and $x_2$
• Portfolio value is $p_1x_1 + p_2x_2$
• Fraction of value in asset $i$ (weight of asset) is

$$w_i = \frac{p_ix_i}{p_1x_1+p_2x_2}$$

• Future prices + dividends $\Rightarrow$ returns $r_1$ and $r_2$
• Portfolio return is $w_1r_1 + w_2r_2$.

Example

• $\text{\$}200$ in asset 1 and $\text{\$}300$ in asset 2
• asset 1 goes up 10% and asset 2 goes up 5%
• asset 1 $\mapsto \text{\$}220$
• asset 2 $\mapsto \text{\$}315$
• portfolio value is $\text{\$}535$
• This is a 7% gain and

$$\frac{2}{5} \times 0.1 + \frac{3}{5} \times 0.05 = 0.07$$

Expected return

• Returns are random variables
• Expected = mean
• mean portfolio return $w_1r_1 + w_2r_2$ is

$$w_1 \times \text{mean of $r_1$} + w_2 \times \text{mean of $r_2$}$$

Variance

• The variance of the portfolio return is

$$w_1^2 \sigma_1^2 + w_2 \sigma_2^2 + 2 w_1w_2\rho\sigma_1\sigma_2$$

• where $\sigma_i=$ std dev of $r_i$ and $\rho$ is the correlation of $r_1$ and $r_2$.

• Lower correlation implies lower portfolio risk!

Proof of variance formula

• Variance is expected squared deviation from mean
• Set $r_p = w_1r_1+w_2r_2$ and use overbars to denote means
• Variance of $r_p$ is

$$ \text{mean of } (r_p - \bar{r}_p)^2$$

• And

$$r_p - \bar{r}_p = w_1r_1 + w_2r_2 - (w_1\bar{r}_1 + w_2\bar{r}_2)$$

• So

$$r_p - \bar{r}_p = w_1(r_1 -\bar{r}_1) + w_2(r_2 - \bar{r}_2)$$

• To square this, use $(a+b)^2 = a^2 + b^2 + 2ab$

• So portfolio variance is

$$w_1^2 \times \text{mean of $(r_1-\bar{r}_1)^2$}$$$$ \quad + w_2^2 \times \text{mean of $(r_2-\bar{r}_2)^2$}$$$$ \qquad \qquad \qquad + 2w_1w_2 \times \text{mean of $(r_1-\bar{r_1})(r_2-\bar{r}_2)$}$$

• This is

$$w_1^2 \sigma_1^2 + w_2 \sigma_2^2 + 2 w_1w_2\rho\sigma_1\sigma_2$$

Example

mu1, mu2 = 0.06, 0.1
sigma1, sigma2 = 0.2, 0.3
rho = 0.3
w1, w2 = 0.4, 0.6

import numpy as np

mn = w1*mu1 + w2*mu2
var = w1**2*sigma1**2 + w2**2*sigma2**2 + 2*w1*w2*rho*sigma1*sigma2

print(f"mean portfolio return is {mn:.2%}")
print(f"std dev of portfolio return is {np.sqrt(var):.2%}")

mean portfolio return is 8.40%
std dev of portfolio return is 21.78%

Simulation

cov = [
    [sigma1**2, rho*sigma1*sigma2],
    [rho*sigma1*sigma2, sigma2**2]
]

from scipy.stats import multivariate_normal as multinorm
rets = multinorm.rvs(
    mean=[mu1, mu2], 
    cov=cov, 
    size=1000000
)
rp = w1*rets[:,0] + w2*rets[:,1]

print(f"simulated mean is {np.mean(rp):.2%}")
print(f"simulated std dev is {np.std(rp):.2%}")

simulated mean is 8.39%
simulated std dev is 21.77%

Cash

• Adding cash (money market investment) to a portfolio has a simple effect on expected return and risk.
• Let asset 2 be cash. Its return has negligible risk. Call its return $r_{mm}$.
• Portfolio mean is $w_1 \mu_1 + w_2 r_{mm}$
• Portfolio variance is $w_1^2\sigma_1^2$
• Portfolio std dev is $w_1\sigma_1$.

Margin loans

• You can have negative cash by borrowing from your broker.
• Example: put $\text{\$1,000}$ in an account, borrow $\text{\$200}$ and buy $\text{\$1,200}$ of a stock.
• Continue to call asset 2 cash.
• Your portfolio weights are $w_1 = 1.2$ and $w_2=-0.2$.
• Your expected return is $1.2\times\mu_1 - 0.2\times r_{ml}$ where $r_{ml}$ is the margin loan rate.
• Your std dev is $1.2 \sigma_1$.

Short selling

• You can have a negative weight on a stock by selling short.
• To sell short, your broker borrows shares on your behalf and sells them.
• You eventually have to buy the shares back in the market and return them.
• Profit by selling high and buying low.
• Example: short sell 100 shares $\text{\$}$100 stock.
  • Stock falls to $\text{\$}$90 and you cover the short (buy and return shares).
  • Paid $\text{\$}$90 and sold at $\text{\$}$100 $\Rightarrow$ profit $\text{\$}$10 per share on 100 shares.

Long and short returns (simplified version)

• Assume we invest $\text{\$1,000}$, short sell $\text{\$400}$ of stock 2 and buy $\text{\$1,400}$ of stock 1.
• The weights are 1.4 in stock 1 and -0.4 in stock 2.
• Suppose stock 1 goes up 10% and stock 2 goes up 5%.
  • Stock 1 $\rightarrow$ $\text{\$1,540}$.
  • Stock 2 position $\rightarrow$ $-\text{\$420}$.
  • Portfolio value $\rightarrow$ $\text{\$1,120} = 12$% return
  • $w_1r_1 + w_2r_2 = 1.4 \times 0.10 - 0.4 \times 0.05 = 0.12$

Long and short returns (practical version)

• Proceeds from short sales are retained as collateral
• Investor may get some interest on the proceeds while they are held (called short interest rebate)
• We can invest $\text{\$1,000}$, short sell $\text{\$400}$ of stock 2 and buy $\text{\$1,400}$ of stock 1 only if we take out a margin loan for $\text{\$400}$.
• Actual return in example is

$$1.4 \times 0.10 - 0.4 \times 0.05 $$$$- 0.4 \times r_{ml} + 0.4 \times r_{sir}- 0.4 \text{$\times$ short borrowing fee}$$

• where $r_{sir}$ is the short interest rebate rate.

Enhanced index return example

• Invest $\text{\$1,000}$. Borrow $\text{\$1,000}$ on margin loan.
• Buy $\text{\$1,000}$ of SPY and buy $\text{\$1,000}$ of CVX.
• Short sell $\text{\$1,000}$ of COP.
• Return is SPY return + CVX return - COP return minus margin loan/short interest rebate/short borrowing fee drag.
• If CVX beats COP enough, you will beat SPY.

More assets

• $n$ stocks
• weights $w_1, \ldots, w_n$
• expected returns $\mu_1, \ldots, \mu_n$
• covariance matrix $\Sigma$
  • diagonal elements of covariance matrix are variances
  • off-diagonal elements are correlation $\times$ std dev $\times$ std dev.

Portfolio risk

• Portfolio variance is $w'\Sigma w$
• Portfolio std dev is $\sqrt{w'\Sigma w}$

# example

w = np.array([0.2, 0.2, 0.4])
sigma1, sigma2, sigma3 = 0.2, 0.3, 0.1
rho12, rho13, rho23 = 0.3, 0.5, 0.4

cov = np.array([
    [sigma1**2, rho12*sigma1*sigma2, rho13*sigma1*sigma3],
    [rho12*sigma1*sigma2, sigma2**2, rho23*sigma2*sigma3],
    [rho13*sigma1*sigma3, rho23*sigma2*sigma3, sigma3**2]
])
stdev = np.sqrt(w @ cov @ w)

print(f"portfolio std dev is {stdev:.2%}")

portfolio std dev is 10.84%

Portfolio expected return

• If $w_i \ge 0$ and sum to 1, then portfolio mean is $w'\mu$.
• If $w_i \ge 0$ and sum to less than 1, then portfolio mean is

$$w'\mu + \left(1-\sum w_i\right)r_{mm}$$

• If $w_i \ge 0$ and sum to more than 1, then portfolio mean is

$$w'\mu + \left(1-\sum w_i\right)r_{ml}$$

• So we can say that if $w_i \ge 0$, then portfolio mean is

$$w'\mu + \left(1-\sum w_i\right)r_f$$

• where $r_f = r_{mm}$ if cash > 0 and $r_m = r_{ml}$ if cash < 0.
• With short sales, portfolio mean is also

$$w'\mu + \left(1-\sum w_i\right)r_f$$

• minus the drag from difference between margin loan and short interest rebate rates and minus short borrowing fees.

# continuing prior example

rf = 0.04
mu1, mu2, mu3 = 0.1, 0.12, 0.08
mu = np.array([mu1, mu2, mu3])

port_mean = w @ mu + (1-np.sum(w))*rf
print(f"portfolio mean is {port_mean:.2%}")

portfolio mean is 8.40%

Estimating from historical returns

• Example:
  • SPY = S&P 500
  • IEF = Treasury bonds
  • GLD = gold
• Get adjusted closing prices from Yahoo
• Compute returns as percent changes

import yfinance as yf

tickers = ["SPY", "IEF", "GLD"]
prices = yf.download(tickers, start="1970-01-01")["Adj Close"]
rets = prices.pct_change().dropna()
rets.head(3)

[*********************100%%**********************]  3 of 3 completed

	GLD	IEF	SPY
Date
2004-11-19	0.009013	-0.005480	-0.011117
2004-11-22	0.003796	0.000704	0.004769
2004-11-23	-0.004449	-0.000938	0.001526

rets.corr()

	GLD	IEF	SPY
GLD	1.000000	0.208371	0.049425
IEF	0.208371	1.000000	-0.320228
SPY	0.049425	-0.320228	1.000000

Annualizing

• May be easier to interpret means and std devs when expressed in annual terms
• Annualize daily means by multiplying by 252 (# of trading days in a year)
• Annualize daily variances by multiplying by 252
• Annualize daily std devs by multiplying by square root of 252

print(f"annualized means are \n{252*rets.mean()}")
print(f"\nannualized std devs are \n{np.sqrt(252)*rets.std()}")

annualized means are 
GLD    0.090313
IEF    0.031442
SPY    0.106698
dtype: float64

annualized std devs are 
GLD    0.176744
IEF    0.068225
SPY    0.193171
dtype: float64

Portfolio returns

w = np.array([0.2, 0.3, 0.5])
mean = w @ rets.mean()
var = w @ rets.cov() @ w
stdev = np.sqrt(var)

print(f"annualized mean portfolio return is {252*mean:.2%}")
print(f"annualized std dev is {np.sqrt(252)*stdev:.2%}")

annualized mean portfolio return is 8.08%
annualized std dev is 10.18%