P1 Focus Review: 6th of 8 (Valuation)

[David Harper CFA FRM]

P1 Focus Review 6th of 8: Valuation

• The 6th (of 8) Part 1 Focus Review video (valuation) is located here at http://www.bionicturtle.com/how-to/video/2012.p1.-focus-review-6

Concepts:

• Parametric value at risk (VaR)
• Historical simulation (HS) Var
• Fixed income valuation

Parametric Value at Risk (VaR)
Parametric (aka, analytical) value at risk (VaR) is the measurement epicenter of the FRM. Your exam will probably contain several parametric VaR questions. The first question (Question 1) from GARP's 2012 practice is typical:

"1. You have been asked to estimate the VaR of an investment in Big Pharma Inc. The company’s stock is trading at USD 23 and the stock has a daily volatility of 1.5%. Using the delta-normal method, the VaR at the 95% confidence level of a long position in an at-the-money put on this stock with a delta of -0.5 over a 1-day holding period is closest to which of the following choices?"

Make sure you practice several of these sort of parametric VaR questions, but with respect to our basic asset classes:

• VaR of a stock/futures position (linear, one asset)
• A two-asset stock portfolio VaR (linear, two assets)
• VaR of an option position (non-linear)
• VaR of a bond position (non-linear)

The common theme in our use of analytical (parametric) VaR is that we tend to employ the first (and maybe the second) partial derivative due to the application of a truncated Taylor Series: ΔY = dY/dX*ΔX + 0.5*d^2Y/dX^2*(ΔX)^2 ....

The first term, used by itself, gives us the two most important inputs into linear VaR:

• Option delta as first derivative with respect to change in the underlying stock price, dc/dS (as in the above question)
• Bond duration as the first derivative with respect to yield change, but multiplied by -1/P, such that modified duration = dP/dY*-1/P

And if we include the second term, we get the two most important inputs into a two-term non-linear VaR:

• Option gamma
• Bond convexity

You will almost certainly need to be able to translate a delta-normal VaR horizon/confidence; for example, you should easily be able to translate a 95% x-day VaR of $X into its 99% y-day VaR equivalent. Related, you will certainly need to apply the square root rule, SRR (recall it assumes i.i.d., which is by definition violated by autocorrelated returns).

Question P1.E1.24 is a good example of the need to be able to perform VaR translations:

"GARP 2011.P1.E1.24. Assume that portfolio daily returns are independently and identically normally distributed. Sam Neil, a new quantitative analyst, has been asked by the portfolio manager to calculate the portfolio Value-at-Risk (VaR) measure for 10, 15, 20 and 25 day periods. The portfolio manager notices something amiss with Sam’s calculations displayed below. Which one of following VARs on this portfolio is inconsistent with the others?

a. VAR(10-day) = USD 316M
b. VAR(15-day) = USD 465M
c. VAR(20-day) = USD 537M
d. VAR(25-day) = USD 600M"

Historical simulation (HS) VaR
HS VaR has appeared in pretty much every test. A typical HS question will appear like GARP's 2011.P1.E1.22:

"22. You are the risk manager of a fund. You are using the historical method to estimate VaR. You find that the worst 10 daily returns for the fund over the period of last 100 trading days are -1.0%, -.3%, -0.6%, -0.2%, -2.7%, -0.7%, -2.9%, 0.1%, -1.1%, -3.0%. What is the daily VaR for the portfolio at the 95% confidence level?"

Now, if the list of 100 losses is sorted, which is the 95% HS VaR: the fifth (5th) worst or the sixth (6th) worst? You should be able to identify either as the correct answer because both are justified.

• Historically, in the case, GARP has identified the 5th worst as the correct answer (consistent with Jorion)
• But, as we have carefully alerted GARP, the 6th worst is technically correct, too (Dowd's answer, and in my opinion, superior). So, as of 2012, the most likely sequence of losses will be one that repeats the same two proximate values such that it does not matter! For example, if n = 100, and the loss set = {-100,- 99, -98, -97, -96, -96, -92 ...}, the 95% VaR is 96 in either case. This is how I expect you will see it.

Fixed income valuation
The early Tuckman chapters are highly testable. A key difference, however, between Tuckman and the exam is: Tuckman assumes semi-annual discounting/compounding throughout; i.e., bonds that pay semi-annual coupons so the compound frequency is semi-annual. However, the exam could ask you to employ:

• Annual (in my understanding, this was the default assumption on the last exam)
• Semi-annual (like Tuckman), or even
• Some continuous applications

The other fixed income valuation topics you need to know include:

• Pricing a bond given a yield or spot rate curve (high testability!)
• Extracting an implied forward rate given two spot rates or two bond prices (the most common question in this sub-topic, almost guaranteed!)
• Compute a yield-to-maturity
• Modified duration and Macalauy duration (guaranteed to be tested)
• Dollar value of '01 (DV01)
• Using the DV01 to hedge a bond's price (market) risk

 

[paiva85]

Hi David,

I just want to clarify that for the question on slide 7 in the video that the correlation of 0.25 is a given input (highlighted in yellow). It wasn't stated in the question and it wasn't clear to me what formula would have allowed me to solve for the correlation value.

Thanks,

Devin

 

[David Harper CFA FRM]

Hi Devin,

Yes, in modifying the question, I omitted the correlation assumption, apologies; it cannot be answered with a correlation assumption (as you suggest). "Diversified VaR" implies use of correlation assumption. Here is GARP's source question: http://forum.bionicturtle.com/threads/question-16-diversification-for-a-var-valuation.2108/

Note after the table, it asks:

The correlation between the two returns is 0.25. From a risk management perspective, what is the gain from diversification for a VaR estimated at the 95% level for the next 10 days? Assume there are 250 trading days in a year.

So, the original question is quite good: it wants you to find the "Undiversified VaR" which does not require a correlation assumption (alas, GARP's answer forgets to include duration, so the answer is incorrect for not returning a "price VaR" but instead a "yield VaR," is why i did not just re-use the whole question ... too much going on i figured)

Thanks!

 

[paiva85]

Got it. Thanks!

 

[Goatie]

T bond futures gain if rates decrease, correct?