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# Delta normal VaR

#### afterworkguinness

##### Active Member
Hello,
I came across a practice question (not from BT) for which I can't make sense of the answer and am hoping somebody can shed some light on:

Question:
A portfolio has a current market value equal to $5,334,500 with a daily variance of .0002. Over the years the portfolio has increased its proportionate holdings of equity securities. What is the annual 5% VaR assuming 250 trading days in a year. Answer given Convert daily variance to standard deviation: sqrt(.0002) = 0.1414 Daily VaR = 1.65(0.01414)=2.33% Annual VaR = .0233xsqrt(250) = 36.89% My question: This is clearly a delta normal VaR question, why does the answer given not use delta in the VaR calculation: 1.65(0.01414)(5,334,500). It is not stated that it is a portfolio of some option with x delta so I assume it is a portfolio of all linear instruments (also the statement about equities seems to hint at this too) Thanks #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber The sentence "over the years the portfolio has increased its proportionate holdings of equity securities" is odd. You don't need a delta, really, because the question gives you the variance of the portfolio market value directly; e.g., even if the portfolio contains an option position, apparently, any component variances are aggregated into the portfolio variance. So, as the answer shows, it's the same method you would use to compute the VaR of all equities portfolio, or even a single position in a stock. If the question instead said something like, "portfolio includes an option position with a value of$X million and delta of 0.6, while underlying asset price has variance of S," you would use delta [and covariance between positions].

Delta-normal VaR, as an approach, while the favored terminology by GARP (b/c Jorion uses this term) can be confusing. It does not mean that option delta must be used. It means that the risk factors (e.g., in the question above, to keep it real simple, if we assume the portfolio consists entirely of a position in a single stock, the risk factor is the price of that stock):
1. [normal] are normally distributed (or multivariate normal); hence our ability to use the 1.65, if the risk factor here was not normal, we'd need a different deviate, and
2. [delta] the portfolio/position exposures, to the risk factor(s), are expressed linearly
"Delta" is just meant to convey the linear exposure; it really is more accurately the first partial derivative (which happens to be delta, if there happens to be options, but is duration for bonds, etc etc). Put another way, there is sort of a "delta" implicitly in the answer, it is the delta of a equity position when the risk factor is the equity price, the delta of which (dS/dS) is 1.0:

VaR = 1.65 deviate * (0.01414 volatility) * (1.0 delta: dP/dS) * (5,334,500 position), where change in the portfolio (dP) = change in the equity position (dS)

I like this from Carol Alexander MRA Vol IV, where she explains why she prefers to call this approach Normal Linear VaR rather than delta-Normal VaR
"In fact, the parametric linear VaR models have been given many different names by many different authors. Some refer to them as the analytic VaR models, but analytic expressions for VaR may also be derived for non-linear portfolios. Other authors call normal linear VaR the delta–normal VaR, but we do not actually need to assume that risk factors are normally distributed for this approach and the use of the term ‘delta’ gives the impression that it always refers to a linearization of the VaR for option portfolios. The parametric linear VaR model is only applicable to a portfolio whose return or P&L is a linear function of its risk factor returns or its asset returns. The most basic assumption in the model is that risk factor returns are normally distributed, and that their joint distribution is multivariate normal, so the covariance matrix of risk factor returns is all that is required to capture the dependency between the risk factor returns." -- C.A., MRA IV, p 42.

#### afterworkguinness

##### Active Member
Thanks for clearing that up. The notes I have from university gave a different approach which is the method I stated above.

#### afterworkguinness

##### Active Member
Hi David, I have a follow up question. In you answer you said delta for the linear portfolio is actually 1, but when creating a delta neutral portfolio we would for example short 20 stocks to bring delta down by 20 ?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi afterwork, I think i said that, in the case of the question above where the portfolio is all equities, the delta is implicitly 1.0; just as the delta of a single share is 1.0. This refers to the so-called percentage delta. To hedge, you really need to (or at least want to) neutralize the position delta.

So, in the example above, if the position delta = 1.0 percentage delta * N = N position delta (where N = \$5,334,500/share price = number of shares), then any (-N) position delta will delta neutralize; e.g.,
• short N shares
• short C call options with %delta of D, where -C*D(c) = -N
• long P put options with %delta of -D, where +P*-D(p) = -N
see note here http://www.bionicturtle.com/forum/threads/l1-t4-7-dynamic-delta-hedging.4839/#post-12869 on position vs % greeks

#### afterworkguinness

##### Active Member
So we use percentage delta when calculating var using delta normal VaR and position delta when hedging ?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Yes, correct. It follows from a more general idea: our basic analytical measure is a first partial derivative (e.g., percentage delta), but dy/dx is insufficient by itself to hedge, or to generally work with to alter portfolio composition. The analog in bonds is:
• we use modified duration, a function of the 1st partial derivative (basically -1/P * "delta") to compute bond VaR,
• but to hedge or neutralize, we need dollar (aka, value) duration

#### [email protected]

##### New Member
Hi All,

I too had similar question on Var by normal approach .

I was analysing a bank portfolio and used below formulae.

(LOS * Standard deviation )/Sqrt(252/Holding Period)

I computed the results for 1 day and 10 day holding period .Would be happy if you can help me with below queries .

1)The Var results returned I hope it's in percentage and not a number .

2) how can we back test the results by chi-sqaure method , More specifically how to calculate the frequency of returns exceeding Var numbers .

Should we use a portfolio value or the returns itself for comparison against Var numbers .

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@[email protected] I can't quite process your setup; e.g., what is LOS? I get the time scaling to per square root rule. We typically backtest with the binomial; although chi-squared is used to test variance. If you want to share calcs with Excel, it would be easier to help. Thanks,