Jun 02

FRM 2008 Episode #6: Market C (VaR, Fx, CFaR)

by David Harper, CFA, FRM, CIPM

toc2

In this issue

Note about learning spreadsheets

For this episode, I added seven learning spreadsheets to the member page. As I have been building/refining/annotating these for virtually all of the tricky quantitative concepts, there are now many spreadsheets. So I highlighted in yellow the more critical/relevant subset. Yellow signifies an important or archetypal concept; yellow says "please review me, I contain testable ideas." Non-yellow says "I can be ignored" if your schedule does not allow.

About Episode #6 (VaR, FX, CFaR)

books

Please find links to earlier episodes (#1 to #5) at the end of this note. This episode is called Market C. Because we follow the sequence of GARP's 2008 FRM Study Guide, this is the final episode on the Market Risk discipline (next starts Credit Risk with Credit A). As usual, I would like to offer a few tips in regard to this episode. This episode reviews: Jorion's value at risk (VaR), Saunder's foreign exchange risk, and Stulz' cash flow at risk (CFaR).

 

Value at Risk (VaR)

vectorportvariance

In the 2008 FRM curriculum, VaR is located primarily in two places. First, basic VaR methods (this episode). Second, more advanced portfolio VaR tools (e.g., marginal VaR, component VaR) are reviewed in the Investment Discipline.

Please make sure you understand the parametric VaR that underlies much of the basic material; i.e., VaR = (volatility) * (critical value associated with confidence level) * (multiplier scales for the square root of the time delta). When we move from two-asset portfolio to three- or more-asset portfolio, the variance-covariance matrix is introduced. For a refresher, look at this early bird practice question. You can click-through to an EditGrid spreadsheet that shows the matrix math; i.e., the math that performs the portfolio volatility calculation displayed above. A few tips, let's discuss on the forum if you have doubts here:

  • The covariance matrix contains variances on its diagonal; i.e., covariance (A,A) = variance(A)
  • The covariance matrix embeds the correlation matrix (i.e., the covariance matrix itself is a product of volatilities and correlation matrix); and
  • The correlation matrix contains 1.0s on its diagonal; i.e., correlation (A,A) = 1.0.

var_map1

In regard to VaR mapping, I hope you try the practice question below which adapts Jorion's example in the text: mapping a two-bond portfolio to its cash flow vertices. This is good practice for it employs bond concepts from Tuckman. A difficulty with Jorion Chapter 10 and 11 is that it builds on prior (unassigned) chapters (Chapter 8 in particular is recommended).

For example, the Price Risk of a Bond (%) is introduced. Here is the concept in three steps. If you grasp this sequence, you are in good shape; if you do not, please study further or let's discuss on the forum. This is an archetypal FRM concept:

  1. (dP/P, % price change) = -D (duration) * (dy, yield change). In words, duration links yield change to price change but only as a linear approximation (first derivative, good only locally, no convexity captured)
  2. volatility [dP/P] = duration * volatility [dy]. In words, since price is a linear function of yield under the duration-based approximation, we can express price volatility as a linear function of yield volatility. Yield is the risk factor.
  3. VaR [dP/P] = duration * VaR [dy]. Since parametric VaR merely scales (multiplies by) volatility, we can now say the price risk of the bond [i.e., VaR(dP/P)] is a linear (duration-based) function of the yield VaR.

 

Foreign Exchange Risk

fx_risk1

The practice question below exposes most of what you need to know for the Saunder's FX chapter. The "problem" is simply that, if FX exposure is left unhedged, foreign currency appreciation/depreciation contributes directly to asset returns. Two "solutions" are offered:

  • On-balance-sheet hedge: match the foreign currency asset exposure with a similar liability. If the foreign currency depreciates and erodes returns, so will the funding cost of the liability be reduced (the cost of funds) and the balance sheet will naturally hedge.
  • Off-balance-sheet hedge: company hedges by entering into a foreign currency forward contract

 

Cash flow at Risk (CFaR)

cfar_1

The Stulz chapters contain some really elegant ideas. Alas, the prose is thick and makes the ideas hard to find. In my opinion, here are the essential ideas:

Price and quantity risk: Export Inc. is based in America and exports computers to Switzerland. As an exporter, Export Inc incurs two risks:

  • Price risk = currency movement (exchange rate);
  • Quantity risk = the number of computers sold in the foreign country.

The correlation between price (exchange rate) and quantity (number sold) determines the optimal hedge ratio. For example, consider the natural hedge (i.e., no futures contracts required) offered by perfectly negative correlation between price and quantity: currency depreciation creates fewer currency-translated dollars but this is offset by greater quantity of computers sold. I put some extra care in annotating this EditGrid spreadsheet.

Competitive scenarios: the anchor concept for these competitive scenarios is demand elasticity. Motor Inc is a British car maker exporting to the U.S. The first-order question is whether demand is perfectly elastic (perfect competition): if demand is perfectly elastic, Motor Inc cannot raise the dollar price of its cars to "offset" dollar depreciation. Under "less than perfect" elasticity of demand (e.g., inelastic), the demand curve is not flat, and Motor can adjust the price.

Impact of small project on VaR or cash flow at risk (CFaR): This is one of the more detailed calculations in the entire FRM exam. I uploaded an EditGrid spreadsheet for you to study. A few key steps:

  • The key to the VaR impact of a small trade is the asset's beta with the portfolio. To add a high-beta asset is to increase portfolio VaR. This is the quintessential one-factor idea that surfaces often; e.g., CAPM is a one-factor model, Basel II IRM is a one-factor (ASRF) model. There is only the single factor, idiosyncratic risk is assumed diversified away.
  • The net gain from trade is just the "absolute VaR" we have elsewhere seen (by absolute, I mean a VaR that includes the expected return and is appropriate for longer horizon. Or, if you like, absolute VaR is the only VaR and relative VaR is a special case where the mean return is zero): net gain = return change - VaR change. Or, it is the same thing: - (return change) + VaR change.
  • The final nuance here is that the VaR impact is not the dollar impact of a VaR change. Multiply VaR impact by (marginal cost of VaR per dollar of VaR) to get the dollar impact of the VaR. This translates, say, a $100 VaR increase into it's bottom line impact; e.g., $100 * marginal cost of $0.30 = $30 dollar impact of the VaR increase.

Impact of large project on cash flow at risk (CFaR): This is just a variant on the two-asset (Markowitz) portfolio volatility calculation. The firm is a cash flow, the big project is a cash flow. The volatility of combined cash flows is a function of their covariance. Again, I annotated the spreadsheet step-by-step.

New Learning Spreadsheets Added

learningxls

These "learning worksheets" can be accessed in three ways.

  1. Simply view in the browser,
  2. To open directly into Excel! Select File > Export As > Excel (.xls),or
  3. Most have a downloadable "native" Excel file (XLS) associated with the entry.

For this episode, I added:

  • Two-asset value at risk (VaR). This is a brief, annotated application of classic Markowitz (mean-variance) portfolio theory.
  • Delta-normal value at risk (VaR). This is a three-factor VaR calculation; I replicated Jorion's Table 15.6 from the FRM Handbook. It is the matrix equivalent of the two-asset portfolio VaR. I color-coded the steps to make this easier to follow. This may take a bit of time to study but I think it will payoff as several quantitative building blocks are used.
  • VaR Mapping a bond portfolio. I replicated Jorion's bond mapping example in Chapter 11 (VaR Mapping, Tables 11-2 to 11-4). You won't need to do this for the exam, but this gives a tactical understanding of principal versus duration versus cash flow mapping.
  • Hedging foreign exchange risks (Saunders FX reading). These are the spreadsheets the illustrate (i) an unhedged balance sheet, (ii) an on-balance sheet hedge, and (iii) an off-balance sheet hedge
  • Hedging cash flow with price and quantity risk (Stulz). The
  • VaR impact of a small project (Stulz). In my experience, this one is a bit tough for everybody. I spend a little extra time here in the screencast. But if you can work through the spreadsheet, you'll see how this brings together several elements.
  • VaR impact of a large project (Stulz). Also a bit tedious but instructive. As this uses CAPM and beta, which are reviewed in the investment discipline (V), you might save for later.

Screencast Tutorial

toc2

Paid member access the screencast in the member section. In addition to the viewable screencast:

  • You can downloadable the underlying Power Point slides (in PDF format). For this episode, there is a single 92 page deck.
  • An ipod format (.m4v)
  • A downloadable version of the screencast in a .zip file. (Save to new directory on local and launch the .html file.)

Non-members can sample the start of the screencast tutorial here.

 

Practice Questions

Here are some fresh, compound questions I wrote to engage you in this episode.

Question #1 (Jorion Chapter 10)

Three value at risk (VaR) methods are reviewed: Delta-normal, historical simulation and Monte Carlo simulation. Which are true of the following? (answer could be zero, one, two or all three methods)

Among the VaR methods, which...

(i) Can conduct a full portfolio revaluation?
(ii) Requires a (parametric) distributional assumption?
(iii) Is LEAST appropriate for a portfolio that contains many embedded derivatives (e.g., options)?
(iv) Is computationally fast?
(v) Handles fat tails?
(vi) Is easy (simplest to implement)?
(vii) Suffers from sampling variation?
(viii) Is the most powerful?
(ix) Handles correlations?

Question #2 (Jorion Chapter 11)

Assume a two-bond portfolio: bond #1 is a $100 million issue 5-year 6% coupon issue; bond #2 is a $100 million 5-year zero coupon issue. The yield curve is flat at 5% for all maturities. The yield value at risk (yield VaR) is 1% for all maturities.

(i) What is the 5-year (price) returns VaR?
(ii) Which risk factor(s) are mapped under principal mapping?
(iii) Which risk factor(s) are mapped under duration mapping?
(iv) Which risk factor(s) are mapped under cash flow mapping?
(v) Could we construct a barbell portfolio with similar duration? If so, how would its convexity compare?

Question #3 (Saunders FX)

A U.S. bank is funded with $100 million in U.S. dollar-denominated liabilities (CDs). It invests $50 million (50%) in U.S. dollar denominated loans and the remaining $50 million in British (pound sterling denominated) assets (i.e., assets of $50 + $50 million = liabilities of $100 million). At the start of the year, the spot currency exchange rate is $2 per 1 GBP ($2/1 GBP or 0.5 GPB/1 dollar, near its current level).

The bank loans at 6% in the U.S. and 9% in the U. K. (i.e., return on assets) and deposits/liabilities earn 5% in the U.S. and 7% in the U. K. (i.e., cost of funds to the bank). The following questions refer to a simple single period.

(i) If the currency exchange rate does not move, what is the bank's return on investment (ROI)?
(ii) If the pound sterling depreciates (dollar appreciates) to $1.90/GBP (0.53 GBP/$), what is the unhedged ROI?
(iii) Given the same pound depreciation/dollar appreciation, what is the bank's ROI if the bank employs an on-balance sheet hedge?
(iv) Illustrate an off-balance sheet hedge if the bank can take a position in a forward currency contract, where the forward price is a $0.05 discount (2.5%) from the current spot.

Question #4 (Stulz)

(This is tedious but several concepts are employed.) Assume a $100 million portfolio that contains three equally-weighted assets (all weights = 1/3 = 33.3%). Asset #1 has expected return and volatility = 10%; asset #2 has expected return and volatility = 20%; asset #3 has expected return and volatility = 30%. Pairwise correlations are 0.5 (i.e., correlation between each pair of assets = 50%).

The manager wants increase his/her expected return by: selling asset #1 and buying asset #3, in an amount equal to 10% of the portfolio.

(i) What is the expected return and volatility of the portfolio?
(ii) What are the three asset betas with respect to the portfolio (this is not easy!)?
(iii) Given asset betas of 0.42 (asset #1), 0.96 (asset #2) and 1.62 (asset #3), what is the VaR impact of the trade?
(iv) If the marginal cost of VaR (per dollar of VaR) is $0.50, what is the expected net gain from the trade?
(v) Conceptually, the trade moved out of a lower-expected-return into a higher-expected-return asset. Isn't that always an improvement?
(vi) Conceptually, if the negative VaR impact is greater than the positive change in expected return, is the "net gain from trade" necessarily negative?

Question #5 (Stulz)

Everything here is simple one-period. Assume a firm generates $100 million in cash flow with volatility of $20 million. The riskless rate is 4%; the equity premium is also 4%. The firm is evaluating a LARGE PROJECT: investment of $50 million with end-of-single-period payoff = $70 million. The project will have cash flow volatility of $10 million. The project has a beta (with respect to market) of 1.6 and a correlation with the firm's cash flow of 0.5.

(i) What is the project's cost of capital (COC) using CAPM?
(ii) Based on the project's COC, what is the net present value (NPV) of the project?
(iii) The NPV is positive. Therefore, shouldn't the firm necessarily invest in a positive NPV project?
(iv) After investing in the project (i.e., the pre-project firm becomes firm plus project), what is the volatility of the new firm's cash flow?
(v) What is the new firm's cash flow at risk (CFaR) with 99% confidence?
(vi) If the firm's cost of CFaR is $0.80 per dollar of CFaR, what is the project's NPV adjusted for CFaR?

 

My answers to these questions

Previous newsletters

Here are links to Episodes #1 through #5:

Thanks very much.

David Harper, CFA, FRM, CIPM
Founder
www.bionicturtle.com

David-Harper_100w

Comments

  1. Be the first to leave a comment!

Leave a Comment