Update black_litterman.md

Author: longye-tian

lectures/black_litterman.md

@@ -88,7 +88,7 @@ import matplotlib.pyplot as plt
 from numba import jit
 ```
 
-## Mean-Variance Portfolio Choice
+## Mean-variance portfolio choice
 
 A risk-free security earns one-period net return $r_f$.
 
@@ -145,7 +145,7 @@ which implies the following design of a risky portfolio:
 w = (\delta \Sigma)^{-1} \mu
 ```
 
-## Estimating  Mean and Variance
+## Estimating mean and variance
 
 The key inputs into the portfolio choice model {eq}`risky-portfolio` are
 
@@ -158,7 +158,7 @@ squares; that amounts to estimating $\mu$ by a sample mean of
 excess returns and estimating $\Sigma$ by a sample covariance
 matrix.
 
-## Black-Litterman Starting Point
+## Black-Litterman starting point
 
 When estimates of $\mu$ and $\Sigma$ from historical
 sample means and covariances have been combined with **plausible** values
@@ -329,7 +329,7 @@ plt.legend(numpoints=1)
 plt.show()
 ```
 
-## Adding Views
+## Adding views
 
 Black and Litterman start with a baseline customer who asserts that he
 or she shares the **market's views**, which means that he or she
@@ -443,7 +443,7 @@ def BL_plot(τ):
 BL_plot(τ)
 ```
 
-## Bayesian Interpretation
+## Bayesian interpretation
 
 Consider the following Bayesian interpretation of the Black-Litterman
 recommendation.
@@ -487,7 +487,7 @@ Hence, the Black-Litterman recommendation is consistent with the Bayes
 update of the prior over the mean excess returns in light of the
 realized average excess returns on the market.
 
-## Curve Decolletage
+## Curve decolletage
 
 Consider two independent "competing" views on the excess market returns
 
@@ -734,7 +734,7 @@ def decolletage(λ):
 decolletage(λ)
 ```
 
-## Black-Litterman Recommendation as Regularization
+## Black-Litterman recommendation as regularization
 
 First, consider the OLS regression
 
@@ -883,7 +883,7 @@ So the Black-Litterman procedure results in a recommendation that is a
 compromise between the conservative market portfolio and the more
 extreme portfolio that is implied by estimated "personal" views.
 
-## A Robust Control Operator
+## A robust control operator
 
 The Black-Litterman approach is partly inspired by the econometric
 insight that it is easier to estimate covariances of excess returns than
@@ -1016,7 +1016,7 @@ $$
 \frac{v'v}{2} = \frac{1}{2\theta^2}  w' C C' w
 $$
 
-## A Robust Mean-Variance Portfolio Model
+## A robust mean-variance portfolio model
 
 According to criterion {eq}`choice-problem`, the mean-variance portfolio choice problem
 chooses $w$ to maximize
@@ -1150,7 +1150,7 @@ variances".
 That is, we need significantly more data to obtain a given
 precision of the mean estimate than for our variance estimate.
 
-## Special Case -- IID Sample
+## Special case -- IID sample
 
 We start our analysis with the benchmark case of IID data.
 
@@ -1187,7 +1187,7 @@ $$
 We are interested in how this (asymptotic) relative rate of convergence
 changes as increasing sampling frequency puts dependence into the data.
 
-## Dependence and Sampling Frequency
+## Dependence and sampling frequency
 
 To investigate how sampling frequency affects relative rates of
 convergence, we assume that the data are generated by a mean-reverting
@@ -1270,7 +1270,7 @@ ax.set(title=r'Autocorrelation functions, $\Gamma_h(n)$',
 plt.show()
 ```
 
-## Frequency and the Mean Estimator
+## Frequency and the mean estimator
 
 Consider again the AR(1) process generated by discrete sampling with
 frequency $h$. Assume that we have a sample of size $N$ and