13. Monte Carlo and portfolios#

We saw how to simulate the price of an asset back when we looked at the price of Bitcoin and using the Nasdaq API. We are going to take a more complete look here at simulating correlated assets within a portfolio. These methods can be used for measuring portfolio risk, for simulating client portfolios in a financial planning setting, or pricing complex options like Asian options.

I am basing a lot of my code and discussion on this blog post.

This material is related to the Heston model for simulating the prices of correlated assets. The volatility of the assets are linked together. In the Heston model, the volatility of an asset today is also related to past volatility. We’ll do more on this when we get to GARCH models.

We’ll also use yfinance to bring in some stock prices from Yahoo! finance. You’ll need to run pip install yfinance in the terminal. You can read more about it here.

13.1. How correlation affects our simulations#

However, when we start to form portfolios, we are going to care about the correlations across assets. We’ve already seen this when looking at portfolio math. When simulating the Bitcoin prices, we took random numbers from the standard normal distribution and then scaled them by our estimate for the volatility of BTC. This allowed us to capture the randomness of the returns.

However, when dealing with more than one asset, we can’t do that, since the assets are not independent from one another. If Apple goes up, Google is more likely to have gone up as well. The two stocks are correlated. How can we account for this in our simulation? We will create correlated random shocks (\(W\)), where the random movement of one asset depends on another assets movement.

Let’s try a simulation to see what’s going on. I’ll draw two random variables from the standard normal distribution, N(0,1). The random part of the return for Asset 1, \(W_1\), is simply the random draw, \(x_1\). However, the random part of the return for Asset 2, \(W_2\), is correlated with whatever \(x_1\) ends up being. Mathematically, this dependent random movement is given as:

(13.1)#\[\begin{align} W_2 = \rho x_1 + (1 - \sqrt{\rho^2}) x_2 \end{align}\]

I’ll set the correlation, \(\rho\), to be -0.80. These two assets are negatively correlated, so they should move in opposite directions. We’ll set each asset to have the same mean return and same volatility. I’ll set the starting price of each of these assets to 1.

We’ll bring in our usual packages and try this out. This code should look a lot like the Bitcoin code, except that I now have two assets. I am only doing one simulation for each asset, so two total simulations.

How do we simulative prices? Let me remind you of a formula:

(13.2)#\[\begin{align} S(t) = S(0) \exp \left(\left(\mu - \frac{1}{2}\sigma^2\right)t + \sigma W(t)\right) \end{align}\]

Take a starting price \(S_0\). We use exp for continuous compounding. Then, the price today, \(S_t\) is a function of two return sources. The first source is called drift: \(\mu - \frac{1}{2}\sigma^2\). This is what an asset returns on average over some period, like a day. This is the deterministic piece of the return and would be related to risk. \(\mu\) is the arithmetic average and \(\sigma\) is the volatility (standard deviation). They should be in the same units (e.g. daily).

The second piece of the return is the diffusion: \(\sigma W(t)\). \(W(t)\) is called a Brownian motion and is, essentially, cumulative random wiggles. You multiply by \(\sigma\) to scale it: more volatility, more wiggles.

\(dW\) denotes the random wiggle in just one period.

In this example, our random wiggles are going to be correlated across assets.

import datetime
import pandas as pd
import yfinance as yf
import numpy as np
import matplotlib.pyplot as plt

# Include this to have plots show up in your Jupyter notebook.
%matplotlib inline 

T = 251 # How long is our simulation? Let's do 252 days (0 to 251)
N = 251 # number of time points in the prediction time horizon, making this the same as T means that we will simulate daily returns 

rho = -0.80 # Correlation between our two assets. Change me to get different patterns!
mu = 0.05/252 # Mean return (log). Using daily. 
sigma = 0.12/np.sqrt(252)  # Volatility of returns (standard deviation). Using daily. 
drift = mu - 0.5 * sigma ** 2 # Drift is the same each day.

S_0 = 1 # Starting price

dt = T/N # One day price movements
dW = np.zeros([2, N]) # Set up random shocks with zeroes for now

# Sample two random variables from the standard normal distribution standard normal distribution. Two rows of 251 values from N(0,1).
X = np.random.normal(scale = np.sqrt(dt), size=(2, N))

# Generate correlated random shock, W. 
dW[0] = X[0] # Same as first row in X
dW[1] = rho * X[0] + (1 - np.sqrt(rho**2)) * X[1] # Dependent on both sets of random values. How dependent? Correlation determines.

#An array containing the sum of random shocks for our assets.
W = np.cumsum(dW, axis=1) 

diffusion = sigma * W # This is CUMULATIVE diffusion returns over time, rather than daily diffusion returns. 

time_step = np.linspace(dt, T, N) # An array of 1, 2, 3, 4, 5..., keeping track of what time period we are in.
time_steps = np.broadcast_to(time_step, (2, N)) # An array of 1, 2, 3, 4, 5..., but for each asset. 

# This is taking the drift (mu - 0.5 * sigma ** 2), which is the same each day, and multiplying it by 1 on Day 1, 2 on Day 2, 3 on Day 3, etc. Then, that term has sigma (volatility) * the cumulative shock added to it. 
# Then, Staring Price * e^(r) = Price Today, since r is really a cumulative return.
S_t = S_0 * np.exp((drift) * time_steps + diffusion) 

S_t = np.insert(S_t, 0, S_0, axis=1)

Let’s plot the two sets of cumulative returns to see if they make sense.

sims = pd.DataFrame(np.transpose(S_t))

# plotting
ax = sims.plot(alpha=0.2, legend=False)

ax.set_title('Simulated Asset Returns with Correlation of -0.80', fontsize=12);
../_images/13_monte_carlo_5_0.png

You can really see the negative correlation!

Let’s move to some real data now. I am using yfinance to bring in price and volume data. I then keep just the adjusted closing prices for five stocks. Finally, I can calculate discrete returns and check my descriptives.

start_date = datetime.datetime(2019, 1, 1)
end_date = datetime.datetime(2020, 1, 1)
ticker_list = ["AAPL", "F", "CAG", "AMZN", "XOM"]

stock_data = yf.download(ticker_list, start_date, end_date)

prices = stock_data['Adj Close']

rets = prices.pct_change().dropna()

[                       0%                       ]

[*******************   40%                       ]  2 of 5 completed

[**********************60%****                   ]  3 of 5 completed

[**********************80%*************          ]  4 of 5 completed

[*********************100%***********************]  5 of 5 completed



stock_data
Adj Close Close ... Open Volume
AAPL AMZN CAG F XOM AAPL AMZN CAG F XOM ... AAPL AMZN CAG F XOM AAPL AMZN CAG F XOM
Date
2019-01-02 37.893333 76.956497 17.998070 6.351992 53.962502 39.480000 76.956497 21.330000 7.90 69.690002 ... 38.722500 73.260002 21.139999 7.53 67.349998 148158800 159662000 9529800 47494400 16727200
2019-01-03 34.118870 75.014000 18.192142 6.255506 53.133980 35.547501 75.014000 21.559999 7.78 68.620003 ... 35.994999 76.000504 21.270000 7.97 70.000000 365248800 139512000 8526000 39172400 13866100
2019-01-04 35.575394 78.769501 18.445276 6.496722 55.093025 37.064999 78.769501 21.860001 8.08 71.150002 ... 36.132500 76.500000 21.690001 7.91 69.639999 234428400 183652000 10307800 43039800 16043600
2019-01-07 35.496201 81.475502 18.824984 6.665572 55.379517 36.982498 81.475502 22.309999 8.29 71.519997 ... 37.174999 80.115501 21.850000 8.11 71.250000 219111200 159864000 7688400 40729400 10844200
2019-01-08 36.172874 82.829002 18.090887 6.729896 55.782162 37.687500 82.829002 21.440001 8.37 72.040001 ... 37.389999 83.234497 22.120001 8.42 72.540001 164101200 177628000 12699700 45644000 11439000
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
2019-12-24 69.236237 89.460503 29.664688 8.119878 56.814789 71.067497 89.460503 34.049999 9.47 70.019997 ... 71.172501 89.690498 34.060001 9.44 70.349998 48478800 17626000 2459600 11881600 3979400
2019-12-26 70.609909 93.438499 29.351055 8.102728 56.904045 72.477501 93.438499 33.689999 9.45 70.129997 ... 71.205002 90.050499 33.959999 9.47 70.190002 93121200 120108000 4672700 28961300 8840200
2019-12-27 70.583099 93.489998 29.777946 8.025558 56.709309 72.449997 93.489998 34.180000 9.36 69.889999 ... 72.779999 94.146004 33.770000 9.45 70.199997 146266000 123732000 2574400 28272800 10516100
2019-12-30 71.002022 92.344498 29.655973 7.931242 56.376633 72.879997 92.344498 34.040001 9.25 69.480003 ... 72.364998 93.699997 34.119999 9.34 70.089996 144114400 73494000 1813200 36074900 12689400
2019-12-31 71.520813 92.391998 29.830221 7.974115 56.620045 73.412498 92.391998 34.240002 9.30 69.779999 ... 72.482498 92.099998 34.119999 9.25 69.019997 100805600 50130000 2558500 32342100 13151800

252 rows × 30 columns

rets
AAPL AMZN CAG F XOM
Date
2019-01-03 -0.099608 -0.025241 0.010783 -0.015190 -0.015354
2019-01-04 0.042690 0.050064 0.013914 0.038561 0.036870
2019-01-07 -0.002226 0.034353 0.020586 0.025990 0.005200
2019-01-08 0.019063 0.016612 -0.038996 0.009650 0.007271
2019-01-09 0.016982 0.001714 -0.002799 0.041816 0.005275
... ... ... ... ... ...
2019-12-24 0.000951 -0.002114 0.000882 0.003178 -0.003842
2019-12-26 0.019840 0.044467 -0.010573 -0.002112 0.001571
2019-12-27 -0.000380 0.000551 0.014544 -0.009524 -0.003422
2019-12-30 0.005935 -0.012253 -0.004096 -0.011752 -0.005866
2019-12-31 0.007307 0.000514 0.005876 0.005406 0.004318

251 rows × 5 columns

rets.describe()
AAPL AMZN CAG F XOM
count 251.000000 251.000000 251.000000 251.000000 251.000000
mean 0.002671 0.000832 0.002271 0.001053 0.000257
std 0.016498 0.014382 0.022887 0.017158 0.011499
min -0.099608 -0.053819 -0.120982 -0.074540 -0.040290
25% -0.004820 -0.006649 -0.008724 -0.006961 -0.006711
50% 0.002768 0.001163 0.002338 0.000964 0.000694
75% 0.011560 0.008535 0.013240 0.010034 0.007803
max 0.068335 0.050064 0.158692 0.107447 0.036870

Seems legit! The average daily returns are all positive, but around zero. The worst daily return is an Apple return of -9.96%. Each stock has a count of 251 daily returns over the course of the year.

OK, now it’s time to set up our simulation. We will do 50 different future portfolio paths. We’ll simulate 252 days worth of prices. We’ll use these prices to find returns.

We are simulating random numbers. But, computers cant really generate truly random numbers. We can set a seed value that will always generate the same set of random values.

mu has our arithmetic average returns.

We then find our usual variance-covariance matrix using .cov().

np.random.seed(1986)

T = 252 # How long is our simulation? Let's do 252 days.
N = 252 # number of time points in the prediction time horizon, making this the same as T means that we will simulate daily returns 
N_SIM = 50  # How many simulations to run?
dt = T/N # daily steps
noa = 5 # Number of assets

weights = np.array(noa * [1. / noa,])  # EW portfolio based on number of assets. You can change this array to have any weights you want.

mu = rets.mean()
cov = rets.cov()
sigma = rets.std()
corr = rets.corr()

# initial matrix
port_returns_all = np.full((T-1, N_SIM), 0.) # One less return than price

We can look at the actual correlations among our assets to get a sense of what we are using to form portfolios.

corr
AAPL AMZN CAG F XOM
AAPL 1.000000 0.592888 0.152909 0.305895 0.416556
AMZN 0.592888 1.000000 0.116090 0.364216 0.370256
CAG 0.152909 0.116090 1.000000 0.066560 0.160013
F 0.305895 0.364216 0.066560 1.000000 0.375397
XOM 0.416556 0.370256 0.160013 0.375397 1.000000

Let’s start with a simple case. We’ll simulate one series of returns for each of our assets, but include correlated shocks. We’ll get returns by actually simulating prices and then calculating returns.

The first line picks out he latest prices for each stock. This is where we will start. You could start each stock at $1 and get the same return result, since it’s the change that matters.

The second line is our math magic. This takes our variance-covariance matrix and does something called the Cholesky Decomposition. The variance-covariance matrix contains the information for how our assets move together (i.e. covariance). This method will allow us to use the correlation among our assets when simulating the random price paths. We want correlated assets to move together. You can read more about these methods here.

A is a \(5\times5\) matrix, just like the variance-covariance matrix, but the upper triangular part is all zeroes, while the lower triangular part still contains the dependencies among our assets.

S is 252 days of “empty” prices for our 5 assets. Why 252? We want our starting price plus 251 days of simulated prices. We’ll fill these in with simulated prices. We make the first price in the array equal to the latest price.

Then, we have a for loop that simulates prices for each asset across all 252 days. Ranges in Python start at and include the first, but stop one before the last number. Why do we start at 1? S[:, 0] are our initial prices. See how the fourth line has i-1. So, the first time through, it will use the starting price S[:,1-1], or S[:,0], to find the first simulated price S[:,1]. The loop will then go to 252 and stop one before, or 251. This gives us a starting price plus 251 daily simulated prices. Counting isn’t always easy!

We first calculate the drift, or deterministic component, of the return. This is based on the mean and standard deviation of the returns. dt is just 1 day in this case.

We then pull 5 random variables, one for each asset, from the standard normal distribution. We’ll call these 5 numbers Z.

If you multiply A and Z, then you get correlated standard normal variables, instead of five sets of random variables with no relationships among them.

The last line in the for loop is again how we simulate prices. I have done the drift (deterministic) and diffusion (random) terms separately to make it easier to see what’s going on. With this type of simulation, the stock price today is the stock price yesterday times \(\exp(\text{drift} + \text{diffusion})\). This is continuous compounding.

I take these prices and calculate returns again. But, these are now one set of simulated returns.

Finally, I take the returns and multiply them by our chosen weights to get a single set of portfolio returns.

S_0 = prices.iloc[-1]
A = np.linalg.cholesky(cov)
S = np.zeros([noa, N]) # Why N+1? We want our starting price + 252 days of prices.
S[:, 0] = S_0

for i in range(1, N):    
    drift = (mu - 0.5 * sigma**2) * dt # dt = 1. This is the deterministic part of the daily return. It's the same every day.
    Z = np.random.normal(0., 1., noa) # Putting as period after a number in Python makes division work correctly when dealing with integers. Not sure we even need it here.
    diffusion = np.matmul(A, Z) * np.sqrt(dt) # dt = 1. This is the random part. 
    S[:, i] = S[:, i-1]*np.exp(drift + diffusion) # S_t = S_t-1 * e^(r). Continuous compounding, where r is just that day's return, rather than a cumulative return. 

R = pd.DataFrame(S.T).pct_change().dropna() # Create returns from those simulated prices.

port_rets = np.cumprod(np.inner(weights, R) + 1) # Weights x returns, cumulative product to get cumulative portfolio returns.

This method is coded up a bit differently from the two-asset and BTC example. There are no timesteps. Instead, dt = 1. Each previous stock return S_t-1 is being multiplied by a single daily return (drift+diffusion), rather than a cumulative return.

Hint

I recommend running each line separately and then looking at the resulting variable/array. This is how you can figure out what’s going on in the simulation.

Let’s check the correlations and descriptives for these simulated returns.

R.corr()
0 1 2 3 4
0 1.000000 0.595471 0.266348 0.304451 0.501392
1 0.595471 1.000000 0.213023 0.387220 0.443719
2 0.266348 0.213023 1.000000 0.156464 0.257570
3 0.304451 0.387220 0.156464 1.000000 0.349659
4 0.501392 0.443719 0.257570 0.349659 1.000000 
R.describe()
0 1 2 3 4
count 251.000000 251.000000 251.000000 251.000000 251.000000
mean 0.001904 0.000292 0.000791 0.000038 -0.000043
std 0.016625 0.014635 0.023678 0.017270 0.011896
min -0.045377 -0.038192 -0.067844 -0.049063 -0.034151
25% -0.010133 -0.008758 -0.014598 -0.011360 -0.008491
50% 0.001480 -0.001245 -0.000619 0.000610 -0.000171
75% 0.010484 0.010548 0.018418 0.012514 0.007760
max 0.045815 0.040962 0.061667 0.044706 0.032788

They seem to make sense! The simulated correlations, in particular, are close to the empirical correlations.

Finally, let’s add one more element to this loop. We can find 50 different portfolio returns by wrapping the code above in another for loop.

The first loop is counting a bit differently. Our first simulation is indexed at 0 in Python. We will go up to, but not include the number of simulations we are doing. So, 0 through 49. This gives us 50 total simulations.

Remember, the indentation matters! It tells us how the loops are nested.

S_0 = prices.iloc[-1]
A = np.linalg.cholesky(cov)
S = np.zeros([noa, N])
S[:, 0] = S_0

for t in range(0, N_SIM):
    for i in range(1, N):    
        drift = (mu - 0.5 * sigma**2) * dt 
        Z = np.random.normal(0., 1., noa) 
        diffusion = np.matmul(A, Z) * np.sqrt(dt) 
        S[:, i] = S[:, i-1]*np.exp(drift + diffusion) 

        R = pd.DataFrame(S.T).pct_change().dropna()

        port_rets = np.cumprod(np.inner(weights, R) + 1)
    port_returns_all[:, t] = port_rets

I don’t claim that this is elegant code. I’m sure there’s a better way to do this via vectorization, without loops.

Let’s graph these 50 different portfolio paths to see what our future may hold.

port_returns_all = pd.DataFrame(port_returns_all)

plt.plot(port_returns_all)
plt.ylabel('Portfolio Returns')
plt.xlabel('Days')
plt.title('Simulated Portfolio Returns For 252 days');
../_images/13_monte_carlo_24_0.png

That’s a large spread of possible cumulative portfolio returns over the year! This type of simulation is potentially useful for financial planners, and let’s you “answer” questions like, “What’s the probability that my portfolio falls below a particular level over the next decade?”. You can also use price paths like this to price certain options, where the value of the option depends on the paths that a basket of stocks took.