misc

Author: jstac

lectures/pv.md

@@ -1,7 +1,19 @@
-<!-- #region -->
+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+    format_version: 0.13
+    jupytext_version: 1.14.5
+kernelspec:
+  display_name: Python 3 (ipykernel)
+  language: python
+  name: python3
+---
+
 # Present Values
 
-## Overview for present value calculations
+## Overview 
 
 This lecture describes the  **present value model** that is a starting point
 of much asset pricing theory.
@@ -14,77 +26,223 @@ Let's dive in.
 
 Let 
 
- * $\{d_t\}_{t=0}^T $ be a sequence of dividends or ``payouts''
- 
+ * $\{d_t\}_{t=0}^T $ be a sequence of dividends or "payouts"
  * $\{p_t\}_{t=0}^T $ be a sequence of prices of a claim on the continuation of
-    the asset stream from date $t$ on, namely, $\{p_s\}_{s=t}^T $ 
-    
- * $ \delta  \in (0,1) $ be a one-period ``discount rate'' 
- 
- * $p_{T+1}^*$ a terminal price of the asset at time $T+1$
+    the asset stream from date $t$ on, namely, $\{d_s\}_{s=t}^T $ 
+ * $ \delta  \in (0,1) $ be a one-period "discount rate" 
+ * $p_{T+1}^*$ be a terminal price of the asset at time $T+1$
 
 We  assume that the dividend stream $\{d_t\}_{t=0}^T $ and the terminal price 
-$p_{T+1}$ are both exogenous.
+$p_{T+1}^*$ are both exogenous.
 
 Assume the sequence of asset pricing equations
 
 $$
-p_t = d_t + \delta p_{t+1}, \quad t = 0, 1, \ldots , T
+    p_t = d_t + \delta p_{t+1}, \quad t = 0, 1, \ldots , T
 $$ (eq:Euler1)
 
+This is a "cost equals benefits" formula.
+
+It says that the cost of buying the asset today equals the reward for holding it
+for one period (which is the dividend $d_t$)and then selling it, at $t+1$.
+
+The future value $p_{t+1}$ is discounted using $\delta$ to shift it to a present value, so it is comparable with $d_t$ and $p_t$.
 
 We want to solve for the asset price sequence  $\{p_t\}_{t=0}^T $ as a function
 of $\{d_t\}_{t=0}^T $ and $p_{T+1}^*$.
 
+In this lecture we show how this can be done using matrix algebra.
+
+We will use the following imports
+
++++
+
+```{code-cell} ipython3
+import numpy as np
+import matplotlib.pyplot as plt
+```
 
++++
 
-Write the system {eq}`eq:Euler1` of $T+1$ asset pricing  equations as the single matrix equation
+## Present value calculations
+
+The equations in [](eq:Euler1) can be stacked, as in
 
 $$
-\begin{bmatrix} 1 & -\delta & 0 & 0 & \cdots & 0 & 0 \cr
-                0 & 1 & -\delta & 0 & \cdots & 0 & 0 \cr
-                0 & 0 & 1 & -\delta & \cdots & 0 & 0 \cr
-                \vdots & \vdots & \vdots & \vdots & \vdots & 0 & 0 \cr
-                0 & 0 & 0 & 0 & \cdots & 1 & -\delta \cr
-                0 & 0 & 0 & 0 & \cdots & 0 & 1 \end{bmatrix}
-\begin{bmatrix} p_0 \cr p_1 \cr p_2 \cr \vdots \cr p_{T-1} \cr p_T 
-\end{bmatrix} 
-=  \begin{bmatrix}  
-d_0 \cr d_1 \cr d_2 \cr \vdots \cr d_{T-1} \cr d_T
-\end{bmatrix}
-+ \begin{bmatrix} 
-0 \cr 0 \cr 0 \cr \vdots \cr 0 \cr \delta p_{T+1}^*
-\end{bmatrix}
+\begin{aligned}
+    p_0 & = d_0 + \delta p_1 \\
+    p_1 & = d_1 + \delta p_2 \\
+    \vdots \\
+    p_{T-1} & = d_{T-1} + \delta p_T \\
+    p_T & = d_T + \delta p^*_{T+1}
+\end{aligned}
+$$ (eq:Euler_stack)
+
+Write the system {eq}`eq:Euler_stack` of $T+1$ asset pricing  equations as the single matrix equation
+
+$$
+    \begin{bmatrix} 1 & -\delta & 0 & 0 & \cdots & 0 & 0 \cr
+                    0 & 1 & -\delta & 0 & \cdots & 0 & 0 \cr
+                    0 & 0 & 1 & -\delta & \cdots & 0 & 0 \cr
+                    \vdots & \vdots & \vdots & \vdots & \vdots & 0 & 0 \cr
+                    0 & 0 & 0 & 0 & \cdots & 1 & -\delta \cr
+                    0 & 0 & 0 & 0 & \cdots & 0 & 1 \end{bmatrix}
+    \begin{bmatrix} p_0 \cr p_1 \cr p_2 \cr \vdots \cr p_{T-1} \cr p_T 
+    \end{bmatrix} 
+    =  \begin{bmatrix}  
+    d_0 \cr d_1 \cr d_2 \cr \vdots \cr d_{T-1} \cr d_T
+    \end{bmatrix}
+    + \begin{bmatrix} 
+    0 \cr 0 \cr 0 \cr \vdots \cr 0 \cr \delta p_{T+1}^*
+    \end{bmatrix}
 $$ (eq:pieq)
 
-Call the matrix on the left side of equation {eq}`eq:pieq` $A$.
++++
 
+```{exercise-start} 
+:label: pv_ex_1
+```
 
-It is easy to verify that the  inverse of the matrix on the left side of equation
-{eq}`eq:pieq` is
+Carry out the matrix multiplication in [](eq:pieq) by hand and confirm that you
+recover the equations in [](eq:Euler_stack).
 
+```{exercise-end}
+```
 
-$$ A^{-1} = 
-\begin{bmatrix}
-1 & \delta & \delta^2 & \cdots & \delta^{T-1} & \delta^T \cr
-0 & 1 & \delta & \cdots & \delta^{T-2} & \delta^{T-1} \cr
-\vdots & \vdots & \vdots & \cdots & \vdots & \vdots \cr
-0 & 0 & 0 & \cdots & 1  & \delta \cr
-0 & 0 & 0 & \cdots & 0 & 1 \cr
-\end{bmatrix}
-$$ (eq:Ainv)
+In vector-matrix notation, we can write the system [](eq:pieq) as 
+
+$$
+    A p = d + b
+$$ (eq:apdb)
 
-By multiplying both sides of equation {eq}`eq:pieq` by the inverse of the matrix on the left side, we can calculate
+Here $A$ is the matrix on the left side of equation {eq}`eq:pieq`, while
 
 $$
-\vec p \equiv \begin{bmatrix} p_0 \cr p_1 \cr p_2 \cr \vdots \cr p_{T-1} \cr p_T 
-\end{bmatrix} 
+    p = 
+    \begin{bmatrix}
+        p_0 \\
+        p_1 \\
+        \vdots \\
+        p_T
+    \end{bmatrix},
+    \quad
+    d = 
+    \begin{bmatrix}
+        d_0 \\
+        d_1 \\
+        \vdots \\
+        d_T
+    \end{bmatrix},
+    \quad \text{and} \quad
+    b = 
+    \begin{bmatrix}
+        0 \\
+        0 \\
+        \vdots \\
+        p^*_{T+1}
+    \end{bmatrix}
 $$
 
-If we perform the indicated matrix multiplication, we shall find  that
+The solution for prices is given by 
 
 $$
-p_t =  \sum_{s=t}^T \delta^{s-t} d_s +  \delta^{T+1-t} p_{T+1}^*
+    p = A^{-1}(d + b)
+$$ (eq:apdb_sol)
+
+
+Here is a small example, where the dividend stream is given by
+
++++
+
+```{code-cell} ipython3
+T = 6
+current_d = 1.0
+d = []
+for t in range(T+1):
+    d.append(current_d)
+    current_d = current_d * 1.05 
+
+fig, ax = plt.subplots()
+ax.plot(d, 'o', label='dividends')
+ax.legend()
+ax.set_xlabel('time')
+plt.show()
+```
+
+We set $\delta$ and $p_{T+1}^*$ to
+
+```{code-cell} ipython3
+δ = 0.99
+p_star = 10.0
+```
+
+Let's build the matrix $A$
+
+```{code-cell} ipython3
+A = np.zeros((T+1, T+1))
+for i in range(T+1):
+    for j in range(T+1):
+        if i == j:
+            A[i, j] = 1
+            if j < T:
+                A[i, j+1] = -δ
+
+```
+
+Let's inspect $A$
+
+```{code-cell} ipython3
+A
+```
+
+Now let's solve for prices using [](eq:apdb_sol).
+
+```{code-cell} ipython3
+b = np.zeros(T+1)
+b[-1] = δ * p_star
+p = np.linalg.solve(A, d + b)
+fig, ax = plt.subplots()
+ax.plot(p)
+plt.show()
+```
+
++++
+
+
+## Analytical Expressions
+
+It can be verified that the  inverse of the matrix $A$ in {eq}`eq:pieq` is
+
++++
+
+$$ A^{-1} = 
+    \begin{bmatrix}
+        1 & \delta & \delta^2 & \cdots & \delta^{T-1} & \delta^T \cr
+        0 & 1 & \delta & \cdots & \delta^{T-2} & \delta^{T-1} \cr
+        \vdots & \vdots & \vdots & \cdots & \vdots & \vdots \cr
+        0 & 0 & 0 & \cdots & 1  & \delta \cr
+        0 & 0 & 0 & \cdots & 0 & 1 \cr
+    \end{bmatrix}
+$$ (eq:Ainv)
+
+
+
+```{exercise-start} 
+:label: pv_ex_2
+```
+
+Check this by showing that $A A^{-1}$ is equal to the identity matrix.
+
+(By the [inverse matrix theorem](https://en.wikipedia.org/wiki/Invertible_matrix), a matrix $B$ is the inverse of $A$ whenever $A B$ is the identity.)
+
+```{exercise-end}
+```
+
+
+If we use the expression [](eq:Ainv) in [](eq:apdb_sol) and perform the indicated matrix multiplication, we shall find  that
+
+$$
+    p_t =  \sum_{s=t}^T \delta^{s-t} d_s +  \delta^{T+1-t} p_{T+1}^*
 $$ (eq:fisctheory1)
 
 Pricing formula {eq}`eq:fisctheory1` asserts that  two components sum to the asset price 
@@ -94,6 +252,12 @@ $p_t$:
   
   * a **bubble component** $\delta^{T+1-t} p_{T+1}^*$
   
+The fundamental component is pinned down by the discount rate $\delta$ and the
+"fundaments" of the asset (in this case, the dividends).
+
+The bubble component is the part of the price that is not pinned down by
+fundaments.
+
 It is sometimes convenient to rewrite the bubble component as
 
 $$ 
@@ -106,6 +270,7 @@ $$
 c \equiv \delta^{T+1}p_{T+1}^*
 $$
 
++++
 
 ## More about bubbles
 
@@ -120,6 +285,7 @@ d_0 \cr d_1 \cr d_2 \cr \vdots \cr d_{T-1} \cr d_T
 \end{bmatrix}
 $$
 
++++
 
 In this case  system {eq}`eq:Euler1` of our $T+1$ asset pricing  equations takes the
 form of the single matrix equation
@@ -157,6 +323,7 @@ $$
 p_t = c \delta^{-t}
 $$ (eq:bubble)
 
++++
 
 ## Gross rate of return
 
@@ -174,7 +341,7 @@ $$
 R_t = \delta^{-1} > 1 .
 $$
 
-
++++
 
 <!-- #endregion -->
 
@@ -188,4 +355,4 @@ We'll try various settings for $\vec d, p_{T+1}^*$:
   
   * $p_{T+1}^* = 0, d_t = 0$ to get a worthless stock
   
-  * $p_{T+1}^* = c \delta^{-(T+1)}, d_t = 0$ to get a bubble stock 
+  * $p_{T+1}^* = c \delta^{-(T+1)}, d_t = 0$ to get a bubble stock