Correcting an Idiosyncratic Risk Figure in a Portfolio Example
Summary
This post walks through a portfolio risk calculation using two stocks and a broad market ETF. It lays out market exposure from dollar positions and betas, idiosyncratic variance from asset-specific volatilities, total volatility, and annualized tracking error. The author’s question concerns a later statement that attributes a specific dollar amount of idiosyncratic risk to the stock positions, which does not match the preceding calculation.
The accepted answer reports that the book’s author acknowledged the stated figure was a typo and that the reader’s reasoning was correct. The post therefore serves mainly as an erratum, rather than a full derivation of the calculation. It flags a discrepancy in the text but does not independently rework every step or resolve the reader’s separate concern about the market-volatility amount.
Key ideas
- The example separates portfolio market exposure from asset-specific risk.
- Idiosyncratic variance is calculated from position values and individual idiosyncratic volatilities.
- The answer says the later stated idiosyncratic-risk figure was a typo.
- The post does not fully rederive the calculation or settle every concern raised.
Tags
Full text
# Idiosyncratic risk calculation - Advanced Portfolio Management (Giuseppe Paleologo)
# Idiosyncratic risk calculation - Advanced Portfolio Management (Giuseppe Paleologo)
I'm struggling with the walkthrough of a calculation within this text. For anyone with the book it's an example from section 3.4.2. I will go through the steps here and show where I am getting lost - any help appreciated!
We are given this table below
| Field | SYF | WMT | SPY |
| Beta | 1.2 | 0.7 | 1 |
| Daily Market Vol (%) | | | 1.4 |
| Daily Idio Vol (%) | 1.2 | 0.5 | 0 |
| Net Market Value | 10M | 5M | 10M |
We are then given the formula below $$ PNL_{port} = (NMV_{SYF}\times\beta_{SYF}+ NMV_{WMT}\times\beta_{WMT}+NMV_{SPY}\times\beta_{SPY})m+ (NMV_{SYF}\times\epsilon_{SYF}+NMV_{WMT}\times\epsilon_{WMT}+NMV_{SPY}\times\epsilon_{SPY}) $$
I have no problem here.
We then are given:
$$ \text{(porftolio market volality = (portfolio dollar beta)}\times\text{(market volatility)} $$ $$ \text{25.5M}\times1.4\% = \\\$36K $$
This is where my first problem lies, as to me that number should be ~360k. As usual I expect somehow this is my mistake and will carry on.
The next step is the idio variance
we are given
$$ \text{(portfolio idio variance)} = (NMV_{SYF}\times vol_{SYF})^2+(NMV_{WMT}\times vol_{WMT})^2+(NMV_{SPY}\times vol_{SPY})^2 $$
Therefore:
$$ \text{(portfolio idio volatility)} = \sqrt{(10\times1.2)^2+(5\times0.5)^2}=\\\$122K $$
no problems here.
Then we get the total vol as
$$ \text{(portfolio total vol)} = \sqrt{36^2+122^2}=\\\$127K $$
barring my issues with the 36k vs 360k i mentioned before, no problems.
we are then told about tracking error and how this will be
$$ \\\$122k\times\sqrt{252}\approx\\\$1.9M $$
So far, excluding hte 36k vs 360k again i am absolutely fine with whats going on here and how its been explained to me.
### The Problem
The section then stops, and we open up the new chapter with the following
"we had a Portfolio with a large amount of market risk, but we also have $6.7M of idiosyncratic risk coming from SYF and WMT."
Where is this $6.7M number coming from?
## Answer by oronimbus (score 6, accepted)
https://quant.stackexchange.com/a/81331
The author admitted on $\mathbb{X}$ (feeling fancy for using Latex on this) that this is a typo and that your reasoning is correct. See this post which was submitted by a user a couple of weeks ago. A second edition of the book will follow at some point in the future.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.