two fund rf fix

portfolio-theory.qmd

@@ -144,36 +144,49 @@ and a variance
 
 $$ \sigma_\mathrm{min}^2 = \vec{w}_\mathrm{min}^\intercal \, V \, \vec{w}_\mathrm{min} = \left( \frac{\vec{1}^\intercal \, V^{-1}}{a} \right) V \left( \frac{V^{-1} \, \vec{1}}{a} \right) = \frac{\vec{1}^\intercal \, V^{-1} \, \vec{1}}{a^2} = \frac{1}{a} $$
 
-The *tangent portfolio* is
+The *tangent portfolio* is the maximum-Sharpe portfolio, and so it depends on the
+risk-free rate of return, $r_\mathrm{f}$, as well as on $\vec{\mu}$ and $V$.
+It is derived in the section on the [One-fund theorem](#one-fund-theorem) below;
+quoting the result here because it is one of the two funds we want:
 
-$$ \vec{w}_\mathrm{tan} = \frac{V^{-1} \, \vec{\mu}}{b} = \frac{V^{-1} \, \vec{\mu}}{\vec{1}^\intercal \, V^{-1} \, \vec{\mu}} $$
+$$ \vec{w}_\mathrm{tan}(r_\mathrm{f}) = \frac{V^{-1} \, (\vec{\mu} - r_\mathrm{f} \, \vec{1})}{\vec{1}^\intercal \, V^{-1} \, (\vec{\mu} - r_\mathrm{f} \, \vec{1})} = \frac{V^{-1} \, (\vec{\mu} - r_\mathrm{f} \, \vec{1})}{b - a \, r_\mathrm{f}} $$
 
 It has a return
 
-$$ r_\mathrm{tan} = \vec{w}_\mathrm{tan} \cdot \vec{\mu} = \frac{\vec{\mu}^\intercal \, V^{-1} \, \vec{\mu}}{b} = \frac{c}{b} $$
+$$ r_\mathrm{tan} = \vec{w}_\mathrm{tan} \cdot \vec{\mu} = \frac{c - b \, r_\mathrm{f}}{b - a \, r_\mathrm{f}} $$
 
 and a variance
 
-$$ \sigma_\mathrm{tan}^2 = \vec{w}_\mathrm{tan}^\intercal \, V \, \vec{w}_\mathrm{tan} = \left( \frac{\vec{\mu}^\intercal \, V^{-1}}{b} \right) V \left( \frac{V^{-1} \, \vec{\mu}}{b} \right) = \frac{\vec{\mu}^\intercal \, V^{-1} \, \vec{\mu}}{b^2} = \frac{c}{b^2} $$
+$$ \sigma_\mathrm{tan}^2 = \vec{w}_\mathrm{tan}^\intercal \, V \, \vec{w}_\mathrm{tan} = \frac{(\vec{\mu} - r_\mathrm{f} \, \vec{1})^\intercal \, V^{-1} \, (\vec{\mu} - r_\mathrm{f} \, \vec{1})}{(b - a \, r_\mathrm{f})^2} = \frac{a \, r_\mathrm{f}^2 - 2 \, b \, r_\mathrm{f} + c}{(b - a \, r_\mathrm{f})^2} $$
 
 The efficient frontier can be written as a linear combination of any two efficient portfolios.
 This is discussed in more detail in the section on [Fund theorems](#fund-theorems).
 Written as a combination of the minimum variance and the tangent portfolios gives
 
 $$ \vec{w}_{\ast} = \psi \, \vec{w}_\mathrm{min} + (1-\psi) \, \vec{w}_\mathrm{tan} $$
 
-where
+where, solving $r_{\ast} = \psi \, r_\mathrm{min} + (1-\psi) \, r_\mathrm{tan}$ for $\psi$
+and using $r_\mathrm{tan} - r_\mathrm{min} = d \, / \, [\,a \, (b - a \, r_\mathrm{f})\,]$,
 
-$$ \psi = (c - b \, r_{\ast}) \, a \, / \, d $$
+$$ \psi = \frac{a \, \left[ (c - b \, r_{\ast}) + r_\mathrm{f} \, (a \, r_{\ast} - b) \right]}{d}, \qquad 1 - \psi = \frac{(a \, r_{\ast} - b) \, (b - a \, r_\mathrm{f})}{d} $$
+
+Both weights carry $r_\mathrm{f}$, because which frontier portfolio the tangent
+portfolio is depends on $r_\mathrm{f}$.
 
 The efficient frontier portfolio can be equivalently written
 
 \begin{align}
 \vec{w}_{\ast} &= \psi \, \vec{w}_\mathrm{min} + (1-\psi) \, \vec{w}_\mathrm{tan} \\
-      &= \left( \frac{c - b \, r_{\ast}}{d} \right) a \, \vec{w}_\mathrm{min} + \left( \frac{a \, r_{\ast} - b}{d} \right) b \, \vec{w}_\mathrm{tan} \\
+      &= \left( \frac{(c - b \, r_{\ast}) + r_\mathrm{f} \, (a \, r_{\ast} - b)}{d} \right) V^{-1} \, \vec{1} + \left( \frac{a \, r_{\ast} - b}{d} \right) V^{-1} \, (\vec{\mu} - r_\mathrm{f} \, \vec{1}) \\
       &= \left( \frac{c - b \, r_{\ast}}{d} \right) V^{-1} \, \vec{1} + \left( \frac{a \, r_{\ast} - b}{d} \right) V^{-1} \, \vec{\mu}
 \end{align}
 
+using $a \, \vec{w}_\mathrm{min} = V^{-1} \vec{1}$ and
+$(b - a \, r_\mathrm{f}) \, \vec{w}_\mathrm{tan} = V^{-1} (\vec{\mu} - r_\mathrm{f} \vec{1})$
+in the second line. The $r_\mathrm{f}$ terms cancel in the third, as they must: the
+frontier is a property of $\vec{\mu}$ and $V$ alone. Only the *decomposition* into two
+funds depends on $r_\mathrm{f}$.
+
 Along the frontier, the return is
 
 $$ r_{\ast} = \psi \, r_\mathrm{min} + (1-\psi) \, r_\mathrm{tan} $$
@@ -182,9 +195,34 @@ The variance is
 
 $$ \sigma^2_{\ast} = \frac{a}{d} \, r_{\ast}^{2} - \frac{2 \, b}{d} \, r_{\ast} + \frac{c}{d} $$
 
-TODO: Note calculation order of $\vec{w}_\mathrm{min}(\mu, V)$
-and $\vec{w}_\mathrm{tan}(\mu, V, r_\mathrm{f})$, then calculate $r_{\ast}(\sigma_{\ast})$,
-scanning from $\sigma_\mathrm{min}$ to $\sigma_\mathrm{max}$.
+which has no $r_\mathrm{f}$ in it either, consistent with the frontier being independent
+of the risk-free rate.
+
+::: {.callout-warning title="The last line is the one to implement"}
+$\psi$ and $\vec{w}_\mathrm{tan}$ both depend on $r_\mathrm{f}$ and only their combination
+does not, so the two have to be taken at the same $r_\mathrm{f}$. Mixing a
+$\vec{w}_\mathrm{tan}(r_\mathrm{f})$ with coefficients derived at some other rate
+$r_\mathrm{f}'$ --- the $\psi = (c - b \, r_{\ast}) \, a \, / \, d$ of the
+$r_\mathrm{f}' = 0$ case is the one usually quoted --- returns weights that still sum to
+one and still lie on the frontier, and so survive the obvious checks, but they are the
+frontier portfolio for the return
+
+$$ r_{\ast} + \left( r_{\ast} - r_\mathrm{min} \right) \frac{r_\mathrm{f} - r_\mathrm{f}'}{r_\mathrm{min} - r_\mathrm{f}} $$
+
+rather than for $r_{\ast}$. The error vanishes at $r_{\ast} = r_\mathrm{min}$, so a spot
+check at the minimum-variance point will not see it, and it diverges as
+$r_\mathrm{f} \rightarrow r_\mathrm{min}$. Only $\vec{w}_{\ast} \cdot \vec{\mu} = r_{\ast}$
+catches it.
+
+The last line above avoids the issue: it contains no $r_\mathrm{f}$, and needs only
+$V^{-1} \vec{1}$ and $V^{-1} \vec{\mu}$.
+:::
+
+To scan out the frontier: compute $a$, $b$, $c$, $d$ from $\vec{\mu}$ and $V$, then step
+$r_{\ast}$ over the range of interest, taking $\vec{w}_{\ast}$ from the last line above
+and $\sigma_{\ast}$ from $\sigma^2_{\ast}$. Neither step needs $\vec{w}_\mathrm{min}$ or
+$\vec{w}_\mathrm{tan}$, so neither inherits any rounding or clipping applied to those two
+for reporting.
 
 ![The "Markowitz Bullet", the efficient frontier shown in
     @Markowitz_1959_Portfolio_Selection_Efficient_Diversification, p. 152.
@@ -397,7 +435,8 @@ introducing or varying the risk-free rate of return, which portfolio
 along the frontier that is the tangent portfolio will depend on the
 risk-free rate of return.
 
-The *tangent portfolio with a risk-free asset* is
+The *tangent portfolio with a risk-free asset*, quoted above in the
+[two-fund](#two-fund-theorem) discussion, is
 
 $$ \vec{w}_\mathrm{tan}  = \frac{V^{-1} \, (\vec{\mu} - r_\mathrm{f} \, \vec{1})}{\vec{1}^\intercal \, V^{-1} \, (\vec{\mu} - r_\mathrm{f} \, \vec{1})} $$
 
@@ -407,7 +446,7 @@ $$ r_\mathrm{tan} = \vec{\mu} \cdot \vec{w}_\mathrm{tan} = \frac{c - b \, r_\mat
 
 and a variance
 
-$$ \sigma_\mathrm{tan}^{2} = \frac{\left|\vec{\mu} - r_\mathrm{f} \, \vec{1}\right|^2}{ (\vec{\mu} - r_\mathrm{f} \, \vec{1})^\intercal \, V^{-1} \, (\vec{\mu} - r_\mathrm{f} \, \vec{1})}  = \frac{a \, r_\mathrm{f}^2 - 2 \, b \, r_\mathrm{f} + c}{(b - a \, r_\mathrm{f})^2} $$
+$$ \sigma_\mathrm{tan}^{2} = \frac{ (\vec{\mu} - r_\mathrm{f} \, \vec{1})^\intercal \, V^{-1} \, (\vec{\mu} - r_\mathrm{f} \, \vec{1}) }{ \left( \vec{1}^\intercal \, V^{-1} \, (\vec{\mu} - r_\mathrm{f} \, \vec{1}) \right)^2 }  = \frac{a \, r_\mathrm{f}^2 - 2 \, b \, r_\mathrm{f} + c}{(b - a \, r_\mathrm{f})^2} $$
 
 The tangent portfolio is the portfolio with the maximum
 [Sharpe ratio](https://en.wikipedia.org/wiki/Sharpe_ratio), $S_i$.