Financial Risk Manager

Credit A - Question 8 (Portfolio EL and UL)

2008-06-15T22:42:53-08:00

Question:

Assume a two-asset portfolio. The first exposure is as above ($10 million outstanding; $10 million unused commitment; usage given default (UGD) = 50%; probability of default (PD) = 1%; loss given default (LGD) = 50%; standard deviation of LGD = 50%. The second exposure is the same EXCEPT its LGD = 25% and the standard deviation of its LGD = 25%. Default correlation between the two exposures is 10%.

(i) What is the portfolio’s expected loss (EL)?
(ii) What is the portfolio’s unexpected loss (UL)?
(iii) What are the risk contributions, respectively, of each exposure?
(iv) What do the risk contributions sum to?
(v) What is the interpretation of the risk contribution? Why are the sum of unexpected losses (UL + UL) less than portfolio UL? Under what condition would (UL) + (UL) = portfolio UL?

Answer:

(i)
Portfolio EL is the sum of component ELs. In this case,
Exposure #1 EL = ($15 MM AE)(1% EDF)(50% LGD) = $75,000
Exposure #2 EL = ($15 MM AE)(1% EDF)(25% LGD) = $37,500
Portfolio EL = $112,500

(ii)
Unexpected loss (UL) = SQRT[(EDF)(variance of LGD) + (LGD^2)(variance of EDF)]*(Adjusted exposure)
Exposure #1 UL = SQRT[(1%)(50%^2) + (50%^2)(9.95%^2)]*$15 MM = $1,058,005
Exposure #2 UL = SQRT[(1%)(25%^2) + (25%^2)(9.95%^2)]*$15 MM = $529,003
Portfolio UL = SQRT[UL#1^2 + UL#2^2 + (2)(default correlation)(UL#1)(UL#2)]. In this case,
Portfolio UL = SQRT[$1.58 MM ^ 2 + $0.529 MM ^2 + (2)(10%)(1.058 MM)(0.529 MM)] = $1.229 Million

(iii)
Risk contribution #1 = (UL#1)*(UL#1 + UL#2 * correlation)/Portfolio UL. In this case,
Risk contribution #1 = $1.058 MM * ($1.058 MM + $0.529 MM * 10%)/($1.229 MM) = $956,115
Risk contribution #2 = $0.529 MM * ($0.529 MM + $1.058 MM * 10%)/($1.229 MM) = $273,176

(iv)
The sum to portfolio unexpected loss (UL). In this case,
$956,115 + $273,176 = $1,229,291

(v) Another partial first derivative! Why are we not surprised, risk is about sensitivities of exposures to underling factors.
Risk contribution is the incremental risk that the exposure of a single asset contributes to the portfolio’s risk.
I wrote the question incorrectly, I meant:
The portfolio UL will be less than the sum of component (individual) unexpected losses. The reason is imperfect correlation.
If the default correlation is 100% (1.0), then UL#1 + UL#2 = Portfolio UL. But for any default correlation < 100%,
UL#1 + UL#2 > Portfolio UL.

Market C - Question 2 (Jorion Chapter 11)

2008-06-01T20:16:39-08:00

Question:

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?

Answer:

(i)
Returns VaR = (Modified Duration) * (Yield VaR).
In the case of the 5-year zero coupon bond, modified duration = 5/(1+5%) = 4.762, so returns VaR = 4.76%

Note: Jorion uses annual compounding. It is consistent with Tuckman to use semiannual compounding:
Modified duration = 5/(1+5%/2) = 4.88 such that returns VaR = 4.88%

(ii), (iii) and (iv)

See spreadsheet:

Principal mapping maps to the average maturity of the portfolio which is five (5) years.
Duration mapping maps to portfolio duration. Portfolio duration is the weighted average duration of the portfolio. In this case, average of 4.26 and 4.76 = 4.51 That’s under Jorion’s annual compounding. Alternatively, it is consistent with Tuckman to use semi-annual compounding. In which case, portfolio duration = AVERAGE[4.3,4.88] = 4.6.
Cash flow mapping maps to the five cash flow vertices.

(v)
Since portfolio duration is the weighted average of duration components, it is easy to construct a barbell portfolio with the same duration. The component durations could be [3,6] or [2,7] or [1,8] as their average duration would be 4.5 in each case (assuming equal weights).
But the convexity would be higher for the portfolios with the longer maturity as convexity generally scales with the square of maturity; e.g., the [1,8] portfolio would have the greatest convexity.

Market C - Question 3 (Saunders FX)

2008-06-01T20:18:08-08:00

Question:

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 GBP/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 5% discount from the current spot.

(i)
In this case, the bank’s ROA = 7.5% (50% @ 6% and 50% @ 9%) and its cost of funds = 5%. So it’s ROI = 7.5%-5= 2.5%. In short, absent currency impacts, the bank maintains its spread.

(ii)
In this case, pound sterling depreciation erodes returns. ROI = -0.22%:

(iii)
The on-balance hedge implied $50 million in pound sterling ($50 = 25 million GBP) denominated liabilities fund the assets. The below show the same pound sterling depreciation to $1.90/GBP but it does not matter; the dollar could depreciate instead. Because the $50 million = 25 million GBP in liabilities will fluctuate up/down along with the assets. The ROI will always be positive.

(iv)
In the case, the forward foreign currency contract maintains the spread regardless of whether the spot erodes:

Key points:
* Un-hedged is a currency mis-match between assets & liabilities. The un-hedged exposure is a “double-edged sword” that can contribute a loss (if assets in the foreign currency depreciate) or a profit (if assets in the foreign currency appreciate)

* An on-balance sheet (currency) hedge matches assets with liabilities in regard to currency (do not confuse with duration matching). In the case of foreign currency depreciation, erosion to asset returns is offset (mitigated) by lower cost of funds.

* An off-balance sheet (currency) hedge “locks-in” the currency exchange rate with forward contract(s)

OpRisk A - Question 3 (LDA at Work)

2008-07-27T16:50:02-08:00

Question:

(i) Where does Deutsche Bank use EXTERNAL data and why?
(ii) The “short story” version the DB LDA process is: for each cell, they compound two different distribution types, one of which is piece-wise. Explain what this means: what is a cell, what is compounding two distributions, and what is piece-wise?
(iv) How is the tail (i.e., losses greater than $50 MM) modeled?
(v) How do dependencies (dependencies = a more encompassing type of correlation but that allows for non-linear and indirect “correlations") enter in the DB approach?
(vi) How does insurance enter into the DB approach

Answer:

(i)
This is a key theme in OpRisk. Especially in regard to the loss tail, typically a single firm’s (bank’s) internal history does not yield enough data. As the authors of the DB paper say,
“We believe that loss data is the most objective risk indicator currently available. However, even with perfect data collection processes, there will be some areas of the business that will never generate sufficient internal data to permit a comprehensive understanding of the risk profile. This is the reason why internal data is supplemented by external data and generated scenarios”

(ii)
The cell represents losses of a GIVEN TYPE within a certain BUSINESS LINE; i.e., the matrix is event type/business line
Compounding refers to combining the frequency distribution with the severity distribution to produce a single loss distribution.
Piece-wise refers to constructing (or mixing) a distribution by “slicing” it into separate distributions. In the DB instance, the loss distribution has three segments: losses from 10K to 1 million, losses from 1 million to 50 million, and losses above 50 million.

(iii) Mistake, there is no (iii) in the question.

(iv) Extreme value theory (EVT) is employed to model the *parameteric tail.* Specifically, the GPD distribution is used via peaks-over-threshold (POT).

(v)
DB’s LDA model does model dependencies among frequency distributions: “a natural because such dependencies are reflected in the date of occurrence of these losses frequencies in respective cells will be positively correlated”

But DB does not assume positive correlations between severity distributions; i.e., severities assumed independent. DB calculated linear and rank correlations of monthly severity time series and found no severity correlations (except for losses caused by the same event). Modeling severity correlations is more difficult than modeling frequency correlations

(vi)
The insurance model in the loss distribution approach (LDA) consists of two main components:
1. A quantitative model of the individual insurance contracts,
2. A mapping from the OR event types to the insurance contracts.

Insurance does not impact the frequency distribution (i.e., insurance does not tend change whether and when loss events happen) but insurance DOES reduce the severity (i.e., as the insurer absorbs some of the losses)

Credit B - Question 4 (Culp on Securitization)

2008-06-30T12:32:07-08:00

Question:

Adapting Culp’s example, assume a bank wants to securitize a $100 million portfolio of credit-sensitive assets that earns an interest rate of 100 basis points over LIBOR. Assume LIBOR is 5%. The senior expenses of the SPE are 10 basis points. The SPE issues three classes of securities: The senior tranche with a face value of $60 million and a coupon of LIBOR + 50 basis points; a mezzanine tranche with a face value of $30 million a coupon of LIBOR + 100 basis points; the equity tranche constitutes the rest of the capital structure and receives the realized excess spread.

(i) Where is the overcollateralization (O/C)?
(ii) What is the internal credit enhancement (internal C/E) provided to the senior (tranche) debt?
(iii) How much is the net excess spread?
(iv) List examples of EXTERNAL credit support
(v) Give an example of structural liquidity risk
(vi) Give examples of liquidity support

Answer:

(i)
Overcollateralization is when the assets of the structure exceed liabilities. In this case, the junior-most equity tranche has still has some enhancement. In the above, $100 assets - $60 senior debt - $30 mezzanine = $10 O/C which is the equity tranche.

(ii)
The internal credit enhancement here = subordination + O/C
(note: there can be other internal credit enhancements, they are just not named)
internal enhancement = 30/100 (i.e., subordination) + 10/100 (i.e., O/C) = 30% + 10% = 40%

(iii)
100(L+100)-(100)(10 bps)-60(L+50bps)-30(L+100bps) = 10(L+300),
so net excess spread = 3%

Or, see above, in dollars the net excess spread ($) = $0.80,
and $0.80/$10 = 8%, and
8% - 5% LIBOR = +3% spread

(iv)
Insurance, wraps, guaranties.
Letter of credit.
Credit default swap (CDS)
Put option on assets

(v)
On the liabilities side, the SPE may issue coupon-bearing debt (investors paid with regular coupons).
But on the asset side, the portfolio of credit-sensitive assets (which has been securitized) is likely to generate cash flows according to a different pattern; e.g., trade receivables are not interest bearing, mortgage loans may default/prepay.

Note the structural cash flow mismatch (liquidity risk) is not itself credit risk (it may become credit risk).

(vi)
Internal liquidity support:
* Liability design and maturity structure
* Reserves

External liquidity support:
* letters of credit with recourse
* Asset swaps

Market A - Question 3

2008-05-05T11:06:33-08:00

Question:

(A variation on Hull’s Example 6.6)

You manage a $10 million bond portfolio with anticipated duration, in two months, of 8.0 years. You decide to hedge with T-bond futures. The current futures price is 95 with duration of 9.0 years at maturity (each T-bond contract is for delivery of $100,000 face value).

(i) What is the hedging transaction?
(ii) Have you fully immunized the portfolio?

Answer:

(i)
Number of contracts is given by

[(portfolio value)(duration of portfolio at hedge maturity)]/[(Price of futures contract)(Duration of futures contract)]

In this case,

[($10 MM)(8)]/[(95,000)(9)] = 93.57

So, we short 93 or 94 contracts

(ii)

No, a duration hedge does not fully immunize the portfolio (i.e., hedge the portfolio against interest rate changes).
Duration is a first-order linear approximation: to match duration is to hedge only small, parallel shifts in interest rates. First, rates can change in non-small ways; duration does not capture the curvature (convexity) of the rate curve. Second, rate (yield) curve can have non-parallel shifts; i.e., twists in curvature or changes in shape (humpedness).

Credit A - Question 5 (de Servigny LGD)

2008-06-15T22:40:25-08:00

Question:

(i) Unlike PD which can be estimated with a single figure, de Servigny says LGD should be captured by a distribution. Why? Which distribution is typical?
(ii) What are the key determinants of LGD/recovery according to de Servigny?
(iii) An unconditional model of expected loss (EL) might say EL = PD * LGD. Explain how collateral impacts this formula and how PD, LGD and collateral may be influenced by macroeconomic conditions.
(iv) According to de Servigny, which of the rating agencies incorporates LGD?
(v) How might we argue that Basel II is “procyclical” in regard to PD and/or LGD?

Answer:

(i)
According to de Servigny, the key difference between PD and LGD (note: LGD = 1 - recovery, so that statements about LGD are statements about recovery) is that recovery is harder to estimate with a single figure. Recovery depends on not only quantifiable factors, but “fuzzy” (not quantifiable) factors such as the bargaining power of debtors and creditors. A distribution is more appropriate, therefore, to capturing this uncertainty or range of outcomes.

In regard to the distribution, “most portfolio credit risk models...assume that the recovery rate follows a beta distribution. This type of parametric distribution is very appealing, as it offers a lot of flexibility.”

(ii)
Factors impacting the recovery rates (1-LGD) of bonds include:

* Seniority
* Collateral
* Industry: 1. Physical asset obsolescence (higher obsolescence leads to lower recovery), 2. Industry growth (higher growth leads to greater recovery), 3. Industry concentration (lower concentration [more competitive] leads to greater recovery).
* Macroeconomic conditions
* Jurisdiction
* Bargaining power

Note he also gives four factors that may lead to suboptimal loan recovery:
* Capital structure: Debt structure of the firm may cause so-called â€œcreditorâ€™s runâ€
* Bargaining power of certain debtors to extract concessions
* Lazy banking: Senior debt holder may reduce monitoring of firm or trigger premature bankruptcy
* Bank control may sub-optimize firm performance

(iii)

A collateral asset serves as a guarantee in the event of default. Collateral is therefore a key determinant of recovery. Greater collateral, therefore lowers LGD and lowers EL.

However, de Servigny makes the point that the value of collateral, LGD and PD are all economically sensitive. Citing research, he even suggests that LGD and PD may be correlated (linked) by the value of collateral; e.g., during a recession, the value of collateral falls, such that both PD and LGD increase as the value of the collateral falls. This is a warning that “lazy banking” practices should not mistakenly assume collateralized assets do not need to be monitored.

(iv)
According to de Servigny,
“S and P perceives its ratings primarily as an opinion on the likelihood of default” while
“Moody’s ratings tend to reflect ... opinion on the expected loss”

Therefore,
S and P is analogous to PD
Moody’s is analogous to PD * LGD

(v)
“Linking capital requirements to PDs may induce all banks to overlend in good times and underlend in bad time, thereby reinforcing credit and economic cycles.” This is an especially acute concern for point-in-time measures like Merton. Make sure you understand why a Merton model is more susceptible to the procylicality argument.

This arguments can be combined with arguments about the mutual correlation of PD & LGD to macroeconomic cycles, such that expected loss metrics (EL = PD * LGD) are even more procyclical than PD. That is, especially if measured point-in-time, the PDs and LGD will tend to rise during a recession, which in turn makes the bank set aside more economic capital, which in turn may cause the bank to “underlend” during a recession.

Quant A - Question 7

2008-03-23T21:55:01-08:00

Question:

Assume a portfolio of two equally-weighted assets that happen to also have equivalent VaRs (value of risk) of $100 each; i.e., VaR(A) = $100 and VaR(B) = $100.

(i) If VaR(A + B) > VaR(A) + VaR(B), what risk measure criteria is violated?
(ii) If we assume normality, and the correlation between A & B is 1.0, what is VaR (A+B)? (an unfair question at this junction, not yet covered)
(iii) If we assume normality, and the correlation between A & B is zero, what is VaR (A+B)? (also unfair)

Answer:
(i) In this case, the VaR is not sub-additive, which renders the VaR not coherent.
(ii) If correlation = 1.0, then portfolio VaR = $100 + $100 = $200
(iii) if correlation = zero (i.e., independence), portfolio VaR = SQRT[100^2 + 100^2] = $141 (note diversification benefits)

matrix math

2008-11-08T09:26:08-08:00

David
You show matrix math in several screencast and how to do it on excel. I can do that but I can’t do it by hand. Do I need to know how to do it by hand?

FRM Exam in different times zones on D-Day

2008-11-08T10:28:01-08:00

Hi David,

A small though crossed my mind today. On the D-Day (15th Nov), how GARP is able ensure confidentiality of the question paper when the exam is conducted in centres across the world with different time zones. Or the papers across different time zones are different ones?

from
simhan