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Compound Interest Part 1 6
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IntroductionLecture1.1
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Basics 30 minLecture1.2
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Plotting Compound InterestLecture1.3
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Plotting Compound Interest Part 2Lecture1.4
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Comparing rLecture1.5
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More PlottingLecture1.6
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Compound Interest Part 2 3
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Changing rLecture2.1
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Interest with WithdrawalsLecture2.2
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Effective Interest RateLecture2.3
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Present Value 4
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Plotting CashflowsLecture3.1
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Present ValueLecture3.2
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Present Value Part 2Lecture3.3
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Multiple CashflowsLecture3.4
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Annuities 6
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IntroductionLecture4.1
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Annuity EquationLecture4.2
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Annuity DiscountingLecture4.3
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Annuity DueLecture4.4
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Semi-annual RateLecture4.5
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Annuities ConclusionLecture4.6
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Perpetuities 2
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IntroductionLecture5.1
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PerpetuitiesLecture5.2
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Bonds 6
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IntroductionLecture6.1
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Bond ValuationLecture6.2
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Discount BondLecture6.3
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Par BondLecture6.4
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Premium BondLecture6.5
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Finding rLecture6.6
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Dividend Discount Model 3
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IntroductionLecture7.1
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Dividend Discount ModelLecture7.2
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Dividend Discount Model Part 2Lecture7.3
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Risk 8
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IntroductionLecture8.1
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Market ReturnsLecture8.2
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Market RiskLecture8.3
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CovarianceLecture8.4
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Graphing Market ReturnsLecture8.5
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Correlation and BetaLecture8.6
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Creating PortfoliosLecture8.7
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DiversificationLecture8.8
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Capital Asset Pricing Model 6
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IntroductionLecture9.1
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CAPM InputsLecture9.2
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Comparing CAPMLecture9.3
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CAPM PortfoliosLecture9.4
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Shorting a StockLecture9.5
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AlphaLecture9.6
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Graphing Market Returns
Covariance¶
Covariance measures the way in which two things move together or opposite one and other. These values need to be paired up, so when we look at stock returns over time, for example, we compare each dates stock returns! Or in our case the market and the stock returns. The formulas are:
Population Covariance:
$ COV(X, Y) = \frac{\Sigma (X_{i} – \bar{X})(Y_{i} – \bar{Y})}{N} $
Sample Covariance:
$ COV(X, Y) = \frac{\Sigma (X_{i} – \bar{X})(Y_{i} – \bar{Y})}{N-1} $
where
$ X_{i} = \text{The ith value of X} $
$ \bar{X} = \text{Average value of X} $
$ Y_{i} = \text{The ith value of Y} $
$ \bar{Y} = \text{Average value of Y} $
$ N = \text{The number of pairs of observations} $
First, we could find the covariance manually.
#Let's find covariance manually
#Find the deviations
deviations_market = [r - mu_market for r in market_returns]
deviations_stock = [r - mu_stock for r in stock_returns]
#Multiply each deviation by eachother and sum then divide by N or N-2
cov_sample = sum([x*y for x,y in zip(deviations_market, deviations_stock)]) / (len(market_returns) - 1)
cov_population = sum([x*y for x,y in zip(deviations_market, deviations_stock)]) / (len(market_returns))
print(cov_sample)
print(cov_population)
0.00030720450305014174
0.00030659009404404147
If you use numpy's covariance function you will get back a covariance matrix between the two assets. The i=0, j=0 cell will be the variance for the first asset, i=1, j=1 will be the variance for the second asset, and then the other two cells are going to be the covariance between the two assets.
#Or find it with numpy, it returns a covariance matrix
print(np.cov(market_returns,stock_returns))
[[0.00039156 0.0003072 ]
[0.0003072 0.00034763]]
If you index in with [0][1] we can get that covariance between the two assets.
#Find the covariance
print(np.cov(market_returns,stock_returns)[0][1])
0.00030720450305014163
The default version returns the sample covariance but we could change it to be the population covariance with ddof=0.
#Find the population covariance
print(np.cov(market_returns,stock_returns, ddof=0)[0][1])
0.00030659009404404136