Financial Risk Forecasting - Errata

Figure 1.3

2022-10-18T00:00:00+00:00

My FM442 student Jingxuan Chen pointed out that the reference in the first line of the second paragraph on page 10 is not quite correct. It says Figure 1.1 but refers to Figure 1.3 on the same page.

Tail index in dependent data

2019-03-28T00:00:00+00:00

Antonella Altamura and Marco Bee spotted that the language of the discussion on tail index for ARCH type data was not correct. It said that

\begin{equation*} \Gamma(\iota/2+1/2)=\sqrt{\pi}(2\alpha)^{-\iota/2} \end{equation*} was the unconditional distribution of $ Y $ which of course does not make sense.

Instead it should say that the value of $\iota$ can be found by solving for $\iota$ in the equation.

Arch kurtosis

2018-01-28T00:00:00+00:00

Stefano Soccorsi pointed out that the one of the Kurtosis equations on page 37 is wrong. The rest of of it is correct, as is the final Kurtosis value. The typo was in the second equation below:

\begin{align*} E(Y^4) &= 3 E\left(\left(\omega+\alpha Y_{t-1}^2\right)^2\right) \ &= 3 \omega^2+ 6\alpha \omega E(Y^2)+3 \alpha^2 E(Y^4)\ &= 3 \omega^2+ 6\alpha \omega \frac{\omega}{1-\alpha}+3 \alpha^2 E(Y^4)\ \end{align*}

Sequential moments

2018-01-22T00:00:00+00:00

Stefano Soccorsi pointed out that the sequential moments equation on page 19 is wrong. It should be:

\begin{equation*} \frac{1}{t}\sum\limits_{i=1}^{t}x_i^m. \end{equation*}

Testing the independence of violations, Section 8.3.2 p. 155-6

2017-12-27T00:00:00+00:00

My FM320 students Hongshen Chen, Yida Li and Yanfei Zhou pointed out that the discussion in Section 8.3.2 could be more clear, so I repeat the relevant parts of the section here with more clarifications.

We need to calculate the probabilities of two consecutive violations, $p_{11}$, as well the probability of a violation, if there was no violation on the previous day, i.e. $p_{01}$. More generally, where $i$ and $j$ are either 0 or 1: \begin{equation*} p_{ij}=\Pr \left( \eta_{t}=j|\eta_{t-1}=i\right). \end{equation*} The violation process can be represented as a Markov chain with two states, so the first order transition probability matrix is defined as: \begin{equation*} \Pi_1=\left( \begin{array}{cc} 1-p_{01} & p_{01} \ 1-p_{11} & p_{11} \end{array} \right) . \end{equation*} The likelihood function is: \begin{equation} L_1(\Pi_1) =\left( 1-p_{01}\right) ^{v_{00}}p_{01}^{v_{01}}\left( 1-p_{11}\right) ^{v_{10}}p_{11}^{v_{11}} \tag{8.5}\label{eq:risk2:lik:bt:int} \end{equation} where $v_{ij}$ is the number of observations where $j$ follows $i$.

The maximum likelihood (ML) estimates are obtained by maximizing the likelihood function which is simple since the parameters are the ratios of the counts of the outcomes: \begin{gather*} \hat{\Pi}_{1}= \begin{pmatrix} \frac{v_{00}}{v_{00}+v_{01}} & \frac{v_{01}}{v_{00}+v_{01}} \\\\[-2mm] \frac{v_{10}}{v_{10}+v_{11}} & \frac{v_{11}}{v_{10}+v_{11}} \\ \end{pmatrix} . \end{gather*} Under the null hypothesis of no clustering, the probability of a violation tomorrow does not depend on today being a violation, then \$p_{01}=p_{11}=p\$ and the transition matrix is simply: \begin{align*} \Pi_{2} & =\left( \begin{array}{cc} 1-p & p \\ 1-p & p \end{array} \right) \end{align*} and the ML estimate is: \[ \hat{p} =\frac{v_{01}+v_{11}}{v_{00}+v_{10}+v_{01}+v_{11}}. \] so \begin{align*} \hat{\Pi}_{2} & =\left( \begin{array}{cc} 1-\hat{p} & \hat{p} \\ 1-\hat{p} & \hat{p} \end{array} \right) \end{align*}

The likelihood function then is \begin{equation} L_2(\Pi_2) =\left( 1-p\right) ^{v_{00}+v_{10}} p^{v_{01}+v_{11}} .\tag{8.6} \label{eq:risk2:lik:bt:int2} \end{equation}

Note in \eqref{eq:risk2:lik:bt:int2} we impose independence but do not in \eqref{eq:risk2:lik:bt:int}. Replace the \$\Pi\$ by the estimated numbers, \$\hat{\Pi}\$. The LR test is then: \begin{equation*} LR=2\left( \log L_1\left( \hat{\Pi}_{1}\right) -\log L_2\left( \hat{\Pi}_{2}\right) \right) \overset{\rm asymptotic}{\sim}\chi _{\left( 1\right) }^{2}. \end{equation*}

Example 4.4

2017-05-16T00:00:00+00:00

Example 4.4 could be more clear, it is not strictly wrong, but could be better, i.e. have weights on the right side of the inequality, i.e.

$$ VaR^{5\%}(0.5 X+ 0.5 Y) \approx 50 > 0.5 VaR^{5\%}(X) + 0.5 VaR^{5\%}(Y) = 0+0. $$

Example 4.5

2016-10-30T00:00:00+00:00

My FM320 student Emily Wong spotted a typo in line 3, Example 4.5. The VaRs in the equation are missing a minus, and should be $$ 0 > - VaR_1 > -VaR_0$$

E and ES and Q

2015-06-17T00:00:00+00:00

My FM320 student and summer intern Yiying Zhong spotted a typo in Chapter 4 page 86. The ES equation at the bottom of the page says $$ ES = - [Q|Q \le -VaR(p)]$$ but is missing the expectation $$ ES = - E[Q|Q \le -VaR(p)]$$

Listings 3.3, 3.4

2015-03-19T00:00:00+00:00

It had been pointed out that Listings 3.3, 3.4 might be better if they had y[i-1] inside the loop. It is not wrong as it is, but this is better

few issues

2015-02-05T00:00:00+00:00

My FM447 student Gevorg Saakyan pointed out a few typos

The tail index on page 1 uses the letter $t$ where it should use $\iota$.

Chapter 1, page 8. The date in the code comment does not correspond to the date in the code.

Chapter 8.2 Backtesting the S&P 500, pages 147-148. The dates in the text do not correspond with the dates in the code, the code should use February 2, this means that the number of observations is not 4000.

Table 2.2

2014-05-19T00:00:00+00:00

My Fm320 student, Chetan Varsani, noted that it might be better to reverse the first line of the table, i.e. ARCH(1) and ARCH(4)

Listings 8.9 to 8.12

2013-05-12T00:00:00+00:00

the code in Listings 8.9 to 8.12 is correct, but it can have numerical problems when sample size is large. It is better to do the code in logs. Here is Listing 8.9 with that alternative, and it would be straightforward to do same adjustment to the other 3.

bern_test</span>=function</span>(</span>p</span>,</span>v</span>){</span></span>
# calculation with numerical problems when sample size is large</span></span>
	a</span>=</span>p</span>^</span>(</span>sum</span>(</span>v</span>))</span>*</span>(</span>1</span>-</span>p</span>)</span>^</span>(</span>length</span>(</span>v</span>)</span>-</span>sum</span>(</span>v</span>))</span></span>
	b</span>=</span>(</span>sum</span>(</span>v</span>)</span>/</span>length</span>(</span>v</span>))</span>^</span>(</span>sum</span>(</span>v</span>))</span>*</span>(</span>1</span>-</span>(</span>sum</span>(</span>v</span>)</span>/</span>length</span>(</span>v</span>)))</span>^</span>(</span>length</span>(</span>v</span>)</span>-</span>sum</span>(</span>v</span>))</span></span>
#	return(-2*log(a/b))</span></span>
</span>
# more stable calculation when sample size is large</span></span>
	lv</span>=</span>length</span>(</span>v</span>)</span></span>
	sv</span>=</span>sum</span>(</span>v</span>)</span></span>
</span>
	al</span>=</span>log</span>(</span>p</span>)</span>*</span>sv</span>+</span>log</span>(</span>1</span>-</span>p</span>)</span>*</span>(</span>lv</span>-</span>sv</span>)</span></span>
	bl</span>=</span>log</span>(</span>sv</span>/</span>lv</span>)</span>*</span>sv</span> +</span>log</span>(</span>1</span>-</span>sv</span>/</span>lv</span>)</span>*</span>(</span>lv</span>-</span>sv</span>)</span></span>
	return</span>(</span>-</span>2</span>*</span>(</span>al</span>-</span>bl</span>))</span></span>
}</span></span></code></pre>



Section 8.3.2
2013-05-06T00:00:00+00:00
Gian Giacomo pointed our that the likelihoods in the independence discussion are mislabeled, so restricted is unrestricted, and vice versa.</p>


Figure 8.1
2013-04-20T00:00:00+00:00
My FM320 student Chi Li spotted yet another problem in the cursed  Figure 8.1, where MW should be MA.</p>


More on ES for the normal
2013-04-20T00:00:00+00:00
My FM320 student Richard Dunn spotted that second equation under section 5.3.4 is missing a minus in front.</p>


page 189, endogenous price section
2013-04-20T00:00:00+00:00
My FM320 student Richard Dunn spotted that below the equation the \(\Delta P\) was incorrectly defined. It should be:
$$\Delta P_2=P_2-P_1$$</p>


page 48.
2013-04-20T00:00:00+00:00
My FM320 student Chi Li spotted that  section 2.6.4 second paragraph, last sentence: This is consistent with the residual analysis in Table 2.4. This should instead be Table 2.5.</p>


Table 8.3 and 8.4
2013-04-20T00:00:00+00:00
My FM320 students Jocelyn Tete and Chi Li independently spotted incorrect references where the reference to the four VaR models reported in Table 8.2/8.3 should have been 8.1 and 8.2 instead.</p>


page 90
2012-11-08T00:00:00+00:00
my FM320 student Richard Dunn  spotted a typo on page 90. The 10th word on the 5th line should be ‘variance’ instead of ’variable</p>


page 44, 149
2012-06-14T00:00:00+00:00
My FM320 student Alexander Stampfer found a number of small typos</p>
The last  paragraph ion page 44 “ The restricted log-likelihood minus the unrestricted log-likelihood” which should be reversed.</p>
Page 149.  There is a bracket missing in the Matlab code in the very last line of the page for EWMA</p>


4 and 20 on page 96
2012-06-09T00:00:00+00:00
My FM320 student Akash Jhunjhunwala points out that on  page 96, the last line on the first paragraph, the 1% and 5% levels corresponds to the 4th and 20th values respectively, not the 5th and 25th as suggested in the brackets.</p>


LR ratio in (8.4) page 154
2012-06-05T00:00:00+00:00
My FM320 student, Bide Liu, spotted that the LR test in (8.4) has the wrong order in the first line. Should be unrestricted - the restricted, i.e. something like
$$ LR=2(\log L_U (\hat{p})- \log L_R (p)) $$</p>


\(p_{ij}\) and \(p_{ji}\) bottom of page 155
2012-06-05T00:00:00+00:00
My FM320 student, Bide Liu, spotted that I should have \(p_{ji}\) and not \(p_{ij}\) in the penultimate equation on page 155. It should read like
$$p_{ji}= \Pr (\eta_t=i | \eta_{t-1} = j)$$</p>


standard deviation not variance on page 44
2012-06-05T00:00:00+00:00
My FM320 student Amith Bhattacharyya spotted that on the penultimate line on page 42 standard deviation should replace variance.</p>


Var(WE+1) and not Var(WT+1) on page 144
2012-06-05T00:00:00+00:00
My class teacher Marcela Valenzuela and Ehsan Ramezanifar, University of Tehran, both spotted a typo in the middle of page 144, where the subscript T should be E.</p>


example 4.3
2011-11-16T00:00:00+00:00
My FM320 student Daniel Payne pointed out that in example 4.3 in third line from the bottom the subscript on the weight is wrong, its right in the preceding line, so in both cases it should be:$$ (w_X \sigma_X + w_Y \sigma_Y)^2$$</p>


page 85 ( and not [
2011-11-16T00:00:00+00:00
My MF320 student Ken Starling pointed out that in the penultimate line on page 85 the left side of the interval should be ( since infinity is not included, so $$ (- \infty ,-VaR(p)] $$</p>


vol and mean numbers page 104
2011-11-16T00:00:00+00:00
My MF320 student Ken Starling pointed out that I refer to Table 1.2 on page 104, but cite different numbers, so instead of  0.021% and 1.1%  for mean and vol, use 0.019% and 1.16%.</p>


Wrong word order on page 44
2011-10-26T00:00:00+00:00
My FM320 student Daniel Payne pointed out that  the words restricted and unrestricted are reversed on the bottom of page 44. It should read:
“The unrestricted log-likelihood minus the restricted log-likelihood”</p>


3rd equation from the bottom on page 38
2011-10-22T00:00:00+00:00
My FM320 student Ken Starling pointed out a missing +  in the 3rd equation from the bottom on page 38, it should be
$$\sigma^2= E(\omega+\alpha Y_{t-1}^2 +\beta \sigma_{t-1}^2) =\omega+\alpha \sigma^2 +\beta \sigma^2.
$$</p>


Equation on top of page 37
2011-10-22T00:00:00+00:00
My FM320 student Yong Bin Ng pointed out a typo in the equation on top of page 37. It should be:$$
E(Y^4)=3E\left[(\omega+\alpha Y_{t-1}^2)^2\right]=3(\omega^2+2\alpha \omega \sigma^2 + \alpha^2 E(Y^4))$$</p>
Also, in case you were wondering how to derive the equation we use previous results on page 36, independence of Y’s and Z and properties of the normal distribution, and it’s done as follows.
\begin{aligned}
E(Y^4)&=E(Y_t^4)\
&=E(\sigma_t^4 Z_t^4)\
&=E(\sigma_t^4)E(Z_t^4)\
&=E(\sigma_t^4)3(E(Z_t^2))^2\
&=3E((\sigma_t^2)^2)\
&=3E\left[(\omega+\alpha Y_{t-1}^2)^2\right]\
&=3(\omega^2+2\alpha \omega \sigma^2 + \alpha^2 E(Y_{t-1}^4))\
&=3\omega^2+6\alpha \omega \sigma^2 + 3\alpha^2 E(Y^4)\
&=3\omega^2+6\alpha \omega \frac{\omega}{1-\alpha} + 3\alpha^2 E(Y^4)
\end{aligned}
then,</p>
$$
E(Y^4)(1-3\alpha^2)(1-\alpha) =3\omega^2(1-\alpha)+6\alpha \omega^2
$$</p>
and
$$
E(Y^4)=\frac{3\omega^2(1+\alpha)}{(1-3\alpha^2)(1-\alpha)}
$$</p>


Multiperiod volatility
2011-10-22T00:00:00+00:00
My FM320 student Dominic Clark pointed out that I could have been more clear on page 39 in the last text paragraph. Its not wrong, but a better way is:</p>
where \( E_t(Y^2_{t+1})\) indicates volatility…</p>


Table 1.5
2011-10-22T00:00:00+00:00
My FM320 student Han Wang pointed out that Table 1.5 is not right. It is supposed to have a two tailed probability of outcomes, but the %1 number is 1-the one tailed prob, and the rest are one tailed. So, here are the correct numbers. Set the volatility to 1.16 as per Table 1.2, and get</p>



1%</td>
0.3886496</td>
</tr>

2%</td>
0.08468295</td>
</tr>

3%</td>
0.009703866</td>
</tr>

5%</td>
1.630002e-05</td>
</tr>

15%</td>
3.007448e-38</td>
</tr>

23%</td>
1.721029e-87</td>
</tr>
</tbody>
</table>
with the R code to do the calculation
2</span>*</span>pnorm</span>(</span>-</span>23</span>,</span>sd</span>=</span>1.16</span>)</span></span></code></pre>


Figure 8.1 backtesting
2011-05-24T00:00:00+00:00
Philippe Mueller spotted a bug in the legend of Figure 8.1. It is an unwieldy figure, with almost too much going on, and hard to see in black and white. The color plot below is much clearer, and hopefully correct. Also, I called returns, volatility. Guess the pic was cursed. Finally, one could specify the probability, 1%. In any case, here is the correct.
 
</a></p>


Monte Carlo VaR with one basic asset
2011-05-24T00:00:00+00:00
item 2 on page 133 has an incorrect second index for the y. It should be
\(y_{t+1,i}\) not
\(y_{t+1,t}\)</p>


Typo in equation for ES for the normal
2011-05-07T00:00:00+00:00
I do thank Oliver Linton for spotting a typo in the equation of ES for the normal at the bottom of page 103 and top of page 104. The setup and derivation is correct, but somehow the \(\sigma\)  became  \(\sigma^2\). The correct equation (bottom page 103)
$$\text{ES}=-\frac{\sigma \phi(-\text{VaR}(p))}{p}$$
and the corresponding equation at the top of 104
$$\text{ES}=-\varphi \frac{\sigma \phi(-\text{VaR}(p))}{p}$$</p>

1%</td>	0.3886496</td> </tr>
2%</td>	0.08468295</td> </tr>
3%</td>	0.009703866</td> </tr>
5%</td>	1.630002e-05</td> </tr>
15%</td>	3.007448e-38</td> </tr>
23%</td>	1.721029e-87</td> </tr> </tbody> </table> with the R code to do the calculation 2</span>*</span>pnorm</span>(</span>-</span>23</span>,</span>sd</span>=</span>1.16</span>)</span></span></code></pre> Figure 8.1 backtesting 2011-05-24T00:00:00+00:00 Philippe Mueller spotted a bug in the legend of Figure 8.1. It is an unwieldy figure, with almost too much going on, and hard to see in black and white. The color plot below is much clearer, and hopefully correct. Also, I called returns, volatility. Guess the pic was cursed. Finally, one could specify the probability, 1%. In any case, here is the correct. </a></p> Monte Carlo VaR with one basic asset 2011-05-24T00:00:00+00:00 item 2 on page 133 has an incorrect second index for the y. It should be \(y_{t+1,i}\) not \(y_{t+1,t}\)</p> Typo in equation for ES for the normal 2011-05-07T00:00:00+00:00 I do thank Oliver Linton for spotting a typo in the equation of ES for the normal at the bottom of page 103 and top of page 104. The setup and derivation is correct, but somehow the \(\sigma\) became \(\sigma^2\). The correct equation (bottom page 103) $$\text{ES}=-\frac{\sigma \phi(-\text{VaR}(p))}{p}$$ and the corresponding equation at the top of 104 $$\text{ES}=-\varphi \frac{\sigma \phi(-\text{VaR}(p))}{p}$$</p>