Financial Risk Manager

Associate PRM Exam

2009-06-02T15:43:11-08:00

Hi All,

I am going to be taking the APRM exam next month and I was wondering if anybody had any information/sample questions? Please let me know.

Thanks in advance.

Cheers!

-S

Bonds

2009-12-16T07:23:42-08:00

Hi Dave

Finding it difficult to understand the difference between yield and YTM. Please assist.

Studying

2009-12-16T07:21:13-08:00

Hi Dave

Im fairly new to the financial world but i intend to write FRM 1 in May 2010. Any useful tips for an amateur like me?

2009 FRM EXAM (FULL) - ACTUAL EXAM

2009-11-22T04:07:59-08:00

Hi David,

2009 FRM Full examination was relatively satisfactory compared to 2008 exam. The afternoon session was more satisfactory than the morning one. There were a couple of questions ( for eg, valuation of interest rate swap - morning session and the basel ii ima model var question) in which none of the four choices matched with the answer I got. As a whole, it was a decent performance but nothing can be said till Jan 5. Your guidance and materials were too good and helped me very much in the actual exam. Would like to get the views of FRM exam takers.

Regards
Manoj.

2009 Full I question 16

2009-11-02T09:08:45-08:00

Hi David,

I wonder how I should understand “daily volatility of the yield is 1%”? is it 1% of the yield or just 1%? If it is the former, I think the answer should be (D1*V1 + D2*V2)*1.645*(yield * vol) = 0.08225

For the latter case, it is hard to imagine the yield is so volatile on a daily basis..

Very confused. Pls help.

Thanks..

Question 1 (portfolio var)

2009-06-29T11:00:19-08:00

This practice question is imprecise. Notice it does not explicitly address the time dimension. (Some will argue the answer given is incorrect). Instructive, I hope - David

Question E1.1 [source 2009 Full Exam 1.1]
Given the information provided in the table below, what is the portfolio VaR, at the 99% confidence level, of the following 100 million CHF (Swiss francs) equally weighted investment portfolio?

a. 27.96 million CHF
b. 22.77 million CHF
c. 20.97 million CHF
d. 13.98 million CHF

My additional questions:

1e. The answer given in the sample exam is based on a one-year time horizon. Given we are computing an one-year time horizon, can we argue for an alternative calculation?
1f. If we incorporate the time dimension, what is the calculation for an alternative portfolio VaR?
1g. What are the individual VaRs of, respectively, the stock and bond position (CHF not % returns)? For convenience, please assume we want relative VaR for the remaining questions; i.e., disregard the expected future value of the positions.
1h. Based on 1g, can you quickly find the undiversified VaR?
1i. What is the VaR if the correlation is zero?
1j. [challenging!] What is the beta of the stock position with respect to the portfolio? What is the beta of the bond position with respect to the portfolio?
1k. Given 1j, compute the marginal VaRs of, respectively, the stock and bond positions.
1l. Given 1k, compute the component VaRs of, respectively, the stock and bond positions.

Answers:

Sample exam answer: CORRECT: B

The variance of the equally weighted portfolio is 0.5^2 * 0.18^2 + 0.5^2 * 0.06^2 + 2 * 0.5 * 0.5 * 0.1 * 0.18 * 0.06 = 0.0081 + 0.0009 + 0.0005 = 0.00954. The volatility is then 9.77%. The portfolio VaR or the risk budget is 2.33 * 9.77% * 100 million CHF = 22.77 million CHF. Reference: Allen et al. Chapters 2,3.
Note: this is a correct relative VaR, it disregards the expected growth in the portfolio

1e. The answer given in the sample exam is based on a one-year time horizon. Given we are computing an one-year time horizon, can we argue for an alternative calculation?

It is common to omit the expected return for short horizon; e.g., daily VaR (a trading perspective) tends to assume the daily expected return is near enough to zero so that it can be omitted. But for longer horizons, we often use an absolute VaR; some practitioners will argue absolute VaR is the correct VaR regardless (But Jorion on p .110 points out that relative VaR is more conservative). In any case, we can view relative VaR as a special case of absolute VaR. Jorion’s absolute VaR is given by the following:

VaR% (confidence) = -[expected return]*[Time] + [volatility]*[deviate; e.g., 2.33 @ 99%]*[SQRT(Time)], or
VaR$ (confidence) =( -[expected return]*[Time] + [volatility]*[deviate; e.g., 2.33 @ 99%]*[SQRT(Time)] ) * Initial Wealth$

This is actually Kevin Dowd’s formulation (see Market Risk p 57) which I happen to prefer because it is robust against +/- signage errors that we’ve seen with alternatives. But, you can see the point is: for longer horizons, positive expected return mitigates the potential loss… Another way to think about this is: whereas RAPMs (e.g., Sharpe ratio) are risk-adjusted returns, VaR is return-adjusted risk!

1f. If we incorporate the time dimension, what is the calculation for an alternative portfolio VaR?
Portfolio expected value, 1 year [annual compounding but you can use continuous] = $50*(1+24%) + $50*(1+15%) = $119.5
Absolute VaR = -$19.5 + 22.7 = $3.22

1g. What are the individual VaRs of, respectively, the stock and bond position (CHF not % returns)? For convenience, please assume we want relative VaR for the remaining questions; i.e., disregard the expected future value of the positions.
Stock VaR = $50 MM position * 18% volatility * 2.33 = $20.94 MM
Bond VaR = $50 MM position * 6% volatility * 2.33 = $6.98 MM

1h. Based on 1e, can you quickly find the undiversified VaR?
Undiversified VaR = sum of individual VaRs = $27.92

1i. What is the VaR if the correlation is zero?
If correlation is zero, the covariance term drops and portfolio VaR =SQRT[Var[stock]^2+ VaR[bond]^2) = $22.07 million CHF

1j. [challenging!] What is the beta of the stock position with respect to the portfolio? What is the beta of the bond position with respect to the portfolio?
Beta (stock, portfolio) = Cov(stock, portfolio)/Variance(portfolio) = Cov(stock, 0.5*stock + 0.5*bond)/Variance(portfolio)
and Cov(stock, 0.5*stock + 0.5*bond) = Cov(stock, 0.5*stock) + Cov(stock, 0.5*bond) = 0.5*Cov(stock, stock) + 0.5*Cov(stock, bond) = 0.5*[Var(stock) + Cov(stock, bond)]. Such that,
beta (stock w/ respect to portfolio) = 0.5*[Variance(stock) + Cov(stock, bond)] / Variance(portfolio) = 1.755
beta (bond w/ respect to portfolio) = 0.5*[Variance(bond) + Cov(stock, bond)] / Variance(portfolio) = 0.245

1k. Given 1h, compute the marginal VaRs of, respectively, the stock and bond positions.
marginal VaR = beta * portfolio VaR/portfolio value, such that:
marginal VaR (stock) = 1.755 * $22.7/$100 = 0.399
marginal VaR (bond) = 0.245 * $22.7/$100 = 0.399

1l. Given 1i, compute the component VaRs of, respectively, the stock and bond positions.
Estimate of component VaR (stock) = marginal VaR * position = $19.9354
Estimate of component VaR (bond) = 2.7867
And, these should sum to the portfolio VaR; i.e., individual VaRs do not sum to portfolio VaRs, component VaRs do!

Merton model for credit risk (Question 8)

2009-07-02T16:05:25-08:00

Question:

E1.08 [source sample 2009 FRM Full Exam 1] You don’t have access to KMV’s data. Your boss wants you to tell him your estimate of the probability of default of a credit. To do so, you use the Merton Model because the credit you are considering has no systematic risk. In Merton’s Model, the distance to default (DD) and the expected default frequency (EDF) are:

a. positively and linearly related
b. negatively and linearly related
c. positively and non-linearly related
d. negatively and non-linearly related

my adds [some of these are tough]

E1.08e. Explain the Merton Model in a few brief sentences.
E1.08f. Cite two differences between Merton Model and Moody’s KMV.
E1.08g. Cite a few variables that would decrease the EDF?
E1.08h. Cite a disadvantage of this approach (i.e., equity-based model of default prediction).
E1.08i. [hard] The question implies that the Merton Model requires, or wants, an assumption that the credit has no systematic risk. Is this true?
E1.08j. [hard] As the relationship between DD & EDF is non-linear, can we be more specific: what is the distribution of DD? Reconcile this distribution with the lognormal property of stock prices.
E1.08k. The answer says the risk-neutral probability of default (PD) = 1 – N(d2). But de Servigny says PD = N(-d2). Which is correct?
E1.08l. If the distance to default (DD) is 2.0, what is Merton’s implied risk-neutral probability of default (PD)?
E1.08m. [hard] Under risk-neutral valuation, can we assume this PD applies in the real world?

Answers:

E1.08 [source] The risk neutral probability of default, EDF, in the Merton Model is 1‐ N(d2). The higher the distance to default, DD, the lower the risk neutral probability of default is. On the contrary, the lower DD, the higher EDF is. The relationship is non‐linear. When the DD is low, EDF, is high. If DD is imminent, EDF is high as well. Similarly, if DD is high, EDF is small and not imminent Reference: De Servigny and Renault, Measuring and Managing Credit Risk, Chapter 3.

E1.08e. Explain the Merton Model in a few brief sentences.

My short version: Merton model re-frames the (simple two-class) capital structure of an entire firm as a combination of two derivatives: shareholders are long a call option on the firm’s assets (with strike = face value of debt) and debtholders are short a put option on the firm’s assets. This implies that BSM OPM can be used to price the equity. Further, under the structural approach, a prediction of default is based on the probability that the future firm’s asset value will breach (fall below) the debt value; i.e., when asset value < debt value, then equity + debt < debt value, there is no equity cushion and default is predicted.

E1.08f. Cite two differences between Merton Model and Moody’s KMV.

1. Moody’s KMV does not use the total debt value for the default threshold (a.k.a., default point); they use (book) value of short-term debt plus some fraction of long-term debt.
2. Importantly, KMV does not parameterize the distance to default: - 3 standard deviations, for example, is not translated into EDF by a pure parametric distribution. Rather, it is emipircally based on default experience.

E1.08g. Cite a few variables that would decrease the EDF?

Future distribution (yes, that’s right, does not need to be normal or lognormal!)
Higher expected return in growth of firm’s assets;
Longer horizon;
Lower asset volatility;
Higher initial asset value and/or lower default point;

Although KMV has said that the critical variables are: asset value, future distribution, asset volatility, and level of default point; with expected growth and horizon length having “little discriminating power.”

You are understanding the Merton well to understand each of the above.

E1.08h. Cite a disadvantage of this approach (i.e., equity-based model of default prediction).

de Servigny cites their tendency for “false starts:” a fall in equity markets increases EDFs.
Further, de Servigny asserts the tendency for overreation makes the models highly pro-cyclical.

There can be other criticisms levied, based on Altman’s observations: “The EDF model has two fundamental differences with other approaches. First, it relies on the information in equity prices. Two, it does not try explicitly to be predictive. Whereas agency debt ratings are based upon trying to forecast future events, there are no real future forecasts in the EDF model. It simply looks at the current value of the ?rm relative to its default point and historical volatility. Thus, if it has predictive power, it is because the current value of the firm is a good prediction of future values. Since this value is derived from the firm’s equity market value, the EDF model is totally dependent on stock prices for its information content.”

E1.08i. [hard] The question implies that the Merton Model requires, or wants, an assumption that the credit has no systematic risk. Is this true?

No, the Merton does not require the absence of systematic risk; it does not requre a riskfree rate as an input. The phrase is misleading.
(the valuation of the equity takes riskfree rate as input, per the risk-neutral idea. But expected growth is used as an input in the default prediction).

E1.08j. [hard] As the relationship between DD & EDF is non-linear, can we be more specific: what is the distribution of DD? Reconcile this distribution with the lognormal property of stock prices.

The same properties are working here as in the Black-Scholes-Merton: the returns are normal and the firm (asset) levels are lognormal. The implied distribution of the distance to default—not it contains LN(firm value) and LN(default)!—is normal.

E1.08k. The answer says the risk-neutral probability of default (PD) = 1 – N(d2). But de Servigny says PD = N(-d2). Which is correct?
Both due to the symmetry of the normal: 1-N(d2) = N(-d2)

E1.08l. If the distance to default (DD) is 2.0, what is Merton’s implied risk-neutral probability of default (PD)?
=NORMSDIST(-2) = 2.275%; but this is the step KMV Moody’s does not perform. This, relying on “normal tails” understates the likely PD/EDF.
Therefore, Moody’s KMV EDF > Merton’s NORMSDIST(-2)

E1.08m. [hard] Under risk-neutral valuation, can we assume this PD applies in the real world?
No! Common mistake…The risk-neutral idea applies to the option value (i.e., the equity value), not to N(-d2) itself, the probability of default. Like in BSM, risk-neutral extension into real world applies to the option’s value, not to the “inside term” N(d2).

FRM 2009 Full I q46

2009-11-12T11:50:53-08:00

see the attached pdf for the problem. My question is, why is the Credit Loss of year 1 subtracted from the 200M in the Credit Loss calculation of year two. Any takers?

Question 26

2009-11-12T11:29:19-08:00

Basis Risk:
Spread Risk (when you long and simultaneously short)
1. same asset but long and short are different maturities
2. very similar instruments (positively correlated assets)

In the answers here, the long and short positions were in the exact same amounts. Was this a coincidence or should quality 3. read

3. long and short should be in the exact same amount of contracts?

Implied LGD from bond rates (Question 9)

2009-07-03T14:17:37-08:00

This is a good sample question from GARP: first, it is not too easy and not too hard; this IMO is about typical of the exam’s difficulty and (ii) it does not favor formula memorization: the cited reference is de Servigny, but I do not think you will find the formula in the referenced reading. A formula is not needed, we can apply logic and yet another instance of a no-arbitrage idea - David

Question

Full E1.09. Suppose the rate on Company A’s one-year zero-coupon bond is 10.0% and the one-year T-bill rate is 8.0%. Assume the T-bill is riskless and the probability of default of Company A’s bond is 10%. What is the LGD of Company A’s bond?

a. 18.18%
b. 81.82%
c. 20.01%
d. 79.99%

[my adds next, let’s beat this up!]

9b. [hard] Assume instead we are given that the LGD of Company’s A’s bond is 75% (using Basel II LGD for junior debt under foundation IRB approach). Assuming the credit spread remains 2% (10% - 8%), what is the implied probability of default (PD)?

9c. The answer assumes annual compounding. If we instead assume continuous compounding (i.e., risky bond returns 10% continuous and T-bill returns 8% continuous), what is the implied loss given default (LGD)?. Notice that compounding matters; semi-annual compounding would give a slightly different result.

9d. [hard] As Saunders writes, “Collateral requirements are a method of controlling default risk; they act as a direct substitute for risk premiums in setting required loan rates.” Let collateral (recovery) = 1 - Loss given default (LGD) and solve for the credit spread as a function of the probability of default (PD) and the recovery (c).

9e. Which continuous probability distribution—that is reviewed in the assigned Rachev (Fat-tailed and Skewed Asset Return Distributions)—are we most likely to find in use to characterize the loss given default (LGD) or recovery function?

9f. How are recovery rates (1-LGD) estimated?

9g. What are the most important determinants of recovery/LGD?

Answers:

9a.
An investor should be indifferent between:
(i) invest in riskless T-bill, or
(ii) invest in risky bond

Such that, if k = risky return and i = riskless return (using Saunders’ notation)
1+i = (probability of repayment * return if repayment) + (probability of default * return if default); i.e., weighted average return
1+i = [(1-PD) * (1+k)] + [PD*(1+k)*(1-LGD)]; i.e., we recover 1-LGD

solve for LGD:
1+i = [(1-PD) * (1+k)] + [PD*(1+k)*(1-LGD)]
PD*(1+k)*(1-LGD) = (1+i) - [(1-PD) * (1+k)]
(1-LGD) = (1+i) - [(1-PD) * (1+k)] / (PD*(1+k))
LGD = 1- [(1+i) - [(1-PD) * (1+k)] / (PD*(1+k))]

and here:
LGD = 1- [(1+8%) - [(1-10%) * (1+10%)] / 10%*(1+10%)] = 18.18%

Instead solve for PD:
1+i = [(1-PD) * (1+k)] + [PD*(1+k)*(1-LGD)]
I get PD = 2.42%

The no-arbitage indifference is now:
EXP(i) = (1-PD)*EXP[k] + PD*EXP[k]*(1-LGD)
solving for PD, I get 19.8%

1+i = [(1-PD) * (1+k)] + [PD*(1+k)*(1-LGD)]—>
1+i = (1+k)*[(1-PD) + [PD*(1-LGD)]
...divide both sides by [(1-PD) + [PD*(1-LGD)]:
(1+i)/[(1-PD) + [PD*(1-LGD)] = (1+k)
...since we want the spread (k-i), subtract (1-i) from both sides:
(1+i)/[(1-PD) + [PD*(1-LGD)] - (1+ i) = (1+k) - (1+i), or
(1+k) - (1+i) = k-i = (1+i)/[(1-PD) + [PD*(1-LGD)] - (1+i)
note: Saunders gives an equivalent expression where p = prob of repayment = 1- PD and gamma = recovery = 1 - LGD
so that that above: (1+k) - (1+i) = k-i = (1+i)/[(1-PD) + [PD*(1-LGD)] - (1+i)
= k-i = (1+i)/[gamma + p - p*gamma] - (1+i); Saunder’s version

Beta distribution (flexible)

9f. How are recovery rates (1-LGD) estimated?

Postdefault prices (if possible) or estimated recovery (e.g., PV of estimated cash flows)

9g. What are the most important determinants of recovery/LGD?

Seniority (arguably most important, but the others are not in any order)
Collateral
Macroeconomy & Business cycle (systemic risks)
Industry