L1.2010.GARP Practice Exam A

Level I (Questions 1-20) FRM 2010 Practice Questions Vol. I

By David Harper, CFA FRM CIPM

www.bionicturtle.com

FRM 2010 LEVEL I (QUESTIONS 1-20) 1 www.bionicturtle.com

Table of Contents

Question 1: Simulation methods [quantitative] 2

Question 2: Jensens Alpha measure [foundations] 3

Question 3: Creating Value [foundations] 4

Question 4: Market Structure [products] 6

Question 5: Bonds DV01 [valuation] 7

Question 6: Mean & standard deviation [valuation] 9

Question 7: Simple regression model [quantitative] 10

Question 8: Null hypothesis [quantitative] 12

Question 9: Hypothesis testing [quantitative] 14

Question 10: Option delta [valuation] 16

Question 11: Estimate of invoice price [products] 18

Question 12: Hedge [products] 20

Question 13: Market/Credit/Operational risk [foundation] 22

Question 14: Models for estimating volatility [quantitative] 24

Question 15: Duration of the bond [valuation] 26

Question 16: Diversification for a VaR [valuation] 28

Question 17: Probability [quantitative] 30

Question 18: Bonds [valuation] 33

Question 19: Duration and convexity [valuation] 35

Question 20: Historical simulation [valuation] 37

FRM 2010 LEVEL 1 (QUESTIONS 1-20) 2 www.bionicturtle.com

These sample questions are for paid members of bionicturtle.com only! Anyone else is

using an illegal, pirated copy and also violates GARPs ethical standards.

Some of the questions may have a follow-up explanation. This would be located on the

forum: http://www.bionicturtle.com/forum/viewforum/45/

Question 1: Simulation methods [quantitative]

Which of the following statements about simulation is invalid?

a) The historical simulation approach is a nonparametric method that makes no specific

assumption about the distribution of asset returns.

b) When simulating asset returns using Monte Carlo simulation, a sufficient number of

trials must be used to ensure simulated returns are risk neutral.

c) Bootstrapping is an effective simulation approach that naturally incorporates

correlations between asset returns and non-normality of asset returns, but does not

generally capture autocorrelation of asset returns.

d) Monte Carlo simulation can be a valuable method for pricing derivatives and examining

asset return scenarios.

[Please note: the additional, follow-up questions were written by David Harper. The goal is to explore the topic in greater depth.]

1.2. Which VaR simulation approach can incorporate (handle) heavy-tailed or skewed asset

returns?

1.3. Briefly explain Linda Allens hybrid approach.

1.4. What is the motive and advantage of Jorions QMC method?

Answer: B

Explanation: Risk neutrality has nothing to do with sample size.

Topic: Quantitative Analysis, Subtopic: Simulation methods. Reference: Jorion, chapter 12.

1.2 Which VaR simulation approach can incorporate (handle) heavy-tailed or skewed asset

returns? All VaR methods, including both HS and MCS, can incorporate non-normal returns!

1.3 The hybrid approach blends HS and EWMA for the weighting scheme; Linda Allens hybrid is

still essentially a SIMULATION approach as a parametric form does not describe returns.

1.4 quasi Monte-Carlo (QMC) are faster, producing an error that shrinks at a faster rate,

proportional to close to 1/K rather than 1/SQRT(k)


Question 2: Jensens Alpha measure [foundations]

Portfolio Q has a beta of 0.7 and an expected return of 12.8%. The

market risk premium is 5.25%. The risk-free rate is 4.85%. Calculate

Jensens Alpha measure for Portfolio Q.

a) 7.67%

b) 2.70%

c) 5.73%

d) 4.27%

2.2. What is the portfolios Treynor measure?

2.3. Are Jensens and Treynor related?

2.4. What is the portfolios Sharpe measure?

2.5. What is a criticism of Jensens alpha?

2.6. Is Jensens alpha the same as Grinolds alpha?

Answer: D (4.27% or 4.28%).

Explanation: Jensens alpha is defined by: E(RP ) RF = P + P(E(RM) RF);

P = E(RP ) RF - P(E(RM) RF) = 0.128 - 0.0485 - 0.7 * (0.0525 + 0.0485 - 0.0485)= 0.0427

a. Incorrect. Forgets to subtract the risk-free rate for the excess market return.

b. Incorrect. Forgets to multiply the excess market return by beta.

c. Incorrect. Forgets to subtract the risk-free rate for both excess market return and excess

portfolio return.

d. Correct.

2.2 Treynor measure = (12.8% - 4.85%) / 0.7 = 0.114

2.3. Yes, both assume CAPM such that an exact linear relationship exists between them!

2.4. Not enough information! We need portfolio volatility.

2.5 Says Amenc, The Jensen measure is subject to the same criticism as the Treynor measure:

the result depends on the choice of reference index. In addition, when managers practice a

market timing strategy, which involves varying the beta according to anticipated movements in

the market, the Jensen alpha often becomes negative, and does not then reflect the real

performance of the manager.

2.6 No, Grinolds alpha (think hedge fund alpha) is a generalized version of Jensens alpha.


Jensens alpha presumes CAPM such that the only risk factor is the market (or equity) risk

premium, and alpha is the residual return that is unexplained only by the equity risk premium:

Jensens alpha = portfolio return (ERP * beta)

Grinolds pure alpha is the residual return that is unexplained by any and all common beta

factor exposures:

alpha = portfolio return return (exposure1*factor 1+ exposure2*factor2 + exposure3*factor3 +

.... + exposure_n*factor_n). Hopefully, you can see how we can view Jensens alpha as a

SPECIAL case of Grinolds alpha where there is only one common risk factor (i.e., the

equity risk premium per the market model)

Question 3: Creating Value [foundations]

A corporation is faced with the decision to choose between the two

following projects:

Project Investment Perpetual Annual Cash Flow Cash Flow at Risk

A 100 20 50

B 80 55 200

Assuming that there is no systematic risk and the projects are mutually exclusive, under what

circumstances would project A be selected over project B?

a) Project A should never be chosen because it requires a larger initial investment and

generates lower perpetual annual cash flows.

b) Project A could be preferred over Project B if Project As cash flows are negatively

correlated with the firms existing cash flows while the cash flows of Project B are highly

positively correlated with the firms existing cash flows.

c) Project A should be chosen if the opportunity cost of funds is low, and Project B should

be chosen otherwise.

d) Project A should be chosen if the net present value of the project is positive.

3.2. If the confidence level is 95%, what are the implied project cash flow volatilities?

3.3 Is Stulz CFaR a relative or absolute VaR?

3.4 The question says assuming there is no systematic risk What does that imply about beta?

Are there any non-zero betas to be found?


Answer: B.

Explanation: Project A should be chosen only if the cash flow at risk of the project has low or

negative correlation with the other projects the company currently has or plans. The overall

cash flow position of the firm has to be evaluated as a result.

3.2. If the confidence level is 95%, what are the implied project cash flow volatilities?

Stulz CFaR = cash flow volatility * normal deviate. In this case, CFaR = CF volatility * 1.645, such

that:

Project A CF volatility = -50/-1.645 = $30.4, and

Project B CF volatility = -200/-1.645 = $121.6

3.3 Is Stulz CFaR a relative or absolute VaR?

It is a relative VaR because it is given by Prob[E(C) C > CFaR] = 5%

i.e., it is not a shortfall relative to zero (an absolute loss) but rather a shortfall relative to

EXPECTED cash flow

3.4 The question says assuming there is no systematic risk What does that imply about beta?

Are there any non-zero betas to be found?

The question implies the market betas of the project are zero (i.e., no systematic risk). However,

this does not imply that PROJECT BETAS (i.e., the beta of the project with respect to company

cash flow) is zero. In fact, please note that Project As cash flows are negatively correlated with

the firms existing cash flows necessarily implies a NEGATIVE PROJECT BETA for project A

because:

Project Beta = Covariance (Project, Firm) / Variance (Firm) = Volatility (Project)*Volatility

(Firm) * Correlation (Project, Firm) / Variance (Firm) = Volatility (Project)*Correlation (Project,

Firm) / Volatility (Firm).

And since Volatility () must be positive, a negative Correlation () implies a negative project beta!


Question 4: Market Structure [products]

If the lease rate of commodity A is less than the risk-free rate, what is the

market structure of commodity A?

a) Backwardation

b) Contango

c) Flat

d) Inversion

4.2 Is the CONVERSE a true statement: Does a contango necessarily imply the lease rate is less

than (


Question 5: Bonds DV01 [valuation]

Sarah is a risk manager responsible for the fixed income portfolio of a

large insurance company. The portfolio contains a 30-year zero coupon

bond issued by the US Treasury (STRIPS) with a 5% yield. What is the

bonds DV01?

a) 0.0161

b) 0.0665

c) 0.0692

d) 0.0694

5.2. The question says, The portfolio contains a 30-year zero coupon bond issued by the US

Treasury Does the US Treasury issue STRIPS?

5.3. This question assumes semi-annual compounding (this assumption should be stated!). What

is the price of this bond assuming semi-annual compounding?

5.4. What is the bond price assuming continuous compounding? (You must be able to do this

calculation with ease!)

5.5. Assuming semi-annual compounding, what is, respectively, the Macaulay duration and

Modified duration of this bond?

5.6. Assuming continuous compounding, what is, respectively, the Macaulay duration and


5.7. Assuming semi-annual compounding, solve for the bonds DV01 as a function of the bonds

PRICE and MODIFIED DURATION. (Please focus on this question, important relationship!)

5.8 [Bonus, beyond L1] This question assumes a particular version of DV01 (by default). Can we

specifically name this DV01?

Answer: B

Explanation:

The DV01 of a zero-coupon is

DV01 = 30 / 100 (1 + y/2)2T+1 100 (1 + 5%/2)61 = 0.0665

Topic: Valuation and Risk Models

Subtopic: DV01, duration and convexit

Reference: Tuckman, Chapter 5

(the source explain appears to have typos but gets to correct answer). As follows:

1/10,000 * 1/(1+y/2) * [T * 100/(1+y/2)^(2T)]

= 1/10,000 [T * 100/(1+y/2)^(2T+1)]


= T /[100*(1+y/2)^(2T+1)]

= 30/[100*(1+5%/2)^(60+1) = $0.06662

5.2. The question says, The portfolio contains a 30-year zero coupon bond issued by the US

Treasury Does the US Treasury issue STRIPS?

No. The government does not directly issue STRIPS; they are formed by investment

banks or brokerage firms, but the government does register STRIPS in its book-entry

system. They cannot be bought through TreasuryDirect, but only through a broker.

http://en.wikipedia.org/wiki/United_States_Treasury_security

5.3. This question assumes semi-annual compounding (this assumption should be stated!). What

is the price of this bond assuming semi-annual compounding?

N= 30*2 = 60

I/Y = 5/2 = 2.5

PMT = 0

100 = FV

CPT PV = $22.73

5.4. What is the bond price assuming continuous compounding? (You must be able to do this

calculation with ease!)

PV = 100 * EXP(-5%*30) = $22.31

notice the CC bond price is necessarily lower than the s.a. bond price

5.5. Assuming semi-annual compounding, what is, respectively, the Macaulay duration and


The Mac duration is 30 years (Mac duration = T for a zero coupon bond).

Modified duration = 30/(1+5%/2) = 29.2683

5.6. Assuming continuous compounding, what is, respectively, the Macaulay duration and


The Mac duration is 30 years (Mac duration = T for a zero coupon bond).

Under the special case only of continuous compounding, Modified duration = Macaulay

duration; so, in this case only, Mod duration = 10, also! Can you show why this is the case?

5.7. Assuming semi-annual compounding, solve for the bonds DV01 as a function of the bonds

PRICE and MODIFIED DURATION. (Please focus on this question, important relationship!)

DV01 = Price * Modified Duration / 10,000; in this case,

DV01 = $22.73 * 29.26829 / 10,000 = 0.06652

5.8 [Bonus, beyond L1] This question assumes a particular version of DV01 (by default). Can we

specifically name this DV01?

This question assumes, as does the FRM typically, a YIELD-BASED DV01 which means that

the one basis point change is a change in the YIELD-TO-MATURITY (YTM). DV01

generalizes to include other 1 bps shifts; e.g., spot/zero curve, forward curve.


Question 6: Mean & standard deviation [valuation]

Currently, shares of ABC Corp. trade at USD 100. The monthly risk neutral

probability of the price increasing by USD 10 is 30%, and the probability

of the price decreasing by USD 10 is 70%.What are the mean and standard

deviation of the price after 2 months if price changes on consecutive

months are independent?

Mean Standard Deviation

a. 70 11.32

b 70 12.96

c. 92 11.32

d. 92 12.96

6.2 What is skew of price distribution after two months?

6.3. Is the final price distribution light- or heavy-tailed?

6.4. What is the value of a two-month option if strike price is $100 (i.e., strike = stock) and

riskless rate is 4%?

6.5 Solve for the implied (annualized) volatility

Answer: D

Explanation:

Develop a 2 step tree. Mean = 9% (120) + 42% (100) + 49% (80) = 92

Variance = 9% (120 - 92)2 + 42% (100 - 92)2 + 49% (80 - 92)2 = 168 Thus, standard deviation =

12.96

6.2 Skew = third moment about mean / cube of standard deviation (note: sample skew not

required). Skew = 0.62

6.3. Kurtosis = fourth moment about mean / standard deviation^4

Kurtosis = 2.38 or Excess kurtosis = -0.61

As a binomial distribution, this is a LIGHT-TAILED DISTRIBUTION

(i.e., as n increases, kurtosis approaches 3 and the binomial converges on normal)

6.4. What is the value of a two-month option if strike price is $100 (i.e., strike = stock) and riskless rate is 4%? Backward induction (@4% riskless rate) under two-step implies option price = $1.79 6.5 Solve for the implied (annualized) volatility. We can approximate (albeit a shift from discrete to continuous) by employing Hull 11.13: u = EXP(volatility*SQRT(delta time)); volatility = LN(u)/SQRT(delta time); and in this case volatility = LN(1.1)/SQRT(1/12) = 33%


Question 7: Simple regression model [quantitative]

Which of the following statements about the ordinary least squares

regression model (or simple regression model) with one independent

variable are correct?

i. In the ordinary least squares (OLS) model, the random error term is assumed to have

zero mean and constant variance.

ii. In the OLS model, the variance of the independent variable is assumed to be positively

correlated with the variance of the error term.

iii. In the OLS model, it is assumed that the correlation between the dependent variable and

the random error term is zero.

iv. In the OLS model, the variance of the dependent variable is assumed to be constant.

a) i, ii, iii, and iv

b) ii and iv only

c) i and iv only

d) i, ii, and iii only

7.2. When we move from the regression with one independent variable to a multiple regression

model (i.e., two or more independent variables), the assumptions underlying the CLRM are

essentially the same except for the ADDITION of one key assumption. What is the additional

assumption and what do we call its violation?

7.3 Which answer given in the explanation for 7.1 (source) is technically incorrect?

7.4 If our regression meets the CLRM assumptions, what is the salient implication on regression

analysis?

7.5 How many one-word VIOLATIONS of the CLRM model assumptions can you name?


Answer: C

Explanation:

i. Is correct. In Simple Linear Regression model, the random error term is assumed to be

stationary. It means that the variance of random error term must be constant, or by using

another term: it is assumed that there is no heteroskedasticity in linear regression model.

ii. Is incorrect. In Simple Linear Regression model, the independent variable and the error term

have constant variances.

iii. Is incorrect. The dependent variable is allowed to be correlated with the error term.

iv. Is correct. In Simple Linear Regression model, the variance of the dependent variable is

assumed to be constant. Thus, the correct option is option C.

Topic: Quantitative Analysis, Subtopic: Linear regression, Reference: Gujarati, Ch 7, pp. 140-145.

7.2. When we move from the regression with one independent variable to a multiple regression

model (i.e., two or more independent variables), the assumptions underlying the CLRM are

essentially the same except for the ADDITION of one key assumption. What is the additional

assumption and what do we call its violation?

Gujarati A8.6 is the additional assumption: No exact collinearity exists between any two

explanatory variables. The violation is called multicollinearity.

7.3 Which answer given in the explanation for 7.1 (source) is technically incorrect?

The Explain for (iii) says In Simple Linear Regression model, the independent variable and the

error term have constant variances but it should say either:

In Simple Linear Regression model, the assumption is that the error term has constant variance

In Simple Linear Regression model, the assumption is that the error term is uncorrelated with

the explanatory variable

but the requirement that the independent variable [must] have constant variance is

awkward at a minimum: the independent variables can be stochastic (as long as they are

uncorrelated with the error)

7.4 If our regression meets the CLRM assumptions, what is the salient implication on regression

analysis?

The key implication is that we produce DESIRABLE ESTIMATORS (slope, intercept). Specifically,

the estimators are BLUE: best linear unbiased estimator.

7.5 How many one-word VIOLATIONS of the CLRM model assumptions can you name?

If not Linear in PARAMETERS (not variables!) then: NONLINEAR violation!

If not constant variance, then: Heteroskedastic violation!

If covariance (u,u) is non-zero, then: Autocorrelation violation!

IF collinearity between X1, X2, then: imperfect (not deadly) or perfect (deadly!)

Multicollinearity violation!


Question 8: Null hypothesis [quantitative]

Bob tests the null hypothesis that the population mean is less than or

equal to 45. From a population size of 3,000,000 people, 81 observations

are randomly sampled. The corresponding sample mean is 46.3 and

sample standard deviation is 4.5. What is the value of the appropriate

test statistic for the test of the population mean, and what is the correct

decision at the 1 percent significance level?

a) z = 0.29, and fail to reject the null hypothesis

b) z = 2.60, and reject the null hypothesis

c) t = 0.29, and accept the null hypothesis

d) t = 2.60, and neither reject nor fail to reject the null hypothesis

8.2 Where is the technical mistake in the answer given?

8.3 What is the 99% confidence interval for the true population mean?

8.4. If we want to create a confidence interval for the population variance, what distribution is

employed?


Answer: B

Explanation:

a. is incorrect. The denominator of the z-test statistic is standard error instead of standard

deviation. If the denominator takes the value of standard deviation 4.5, instead of standard error

4.5/sqrt(81), the z-test statistic computed will be z = 0.29, which is incorrect.

b. is correct. The population variance is known and the sample size is large (>30). The test

statistics is: z = (46.3-45)/(4.5/(sqrt(81)) = 2.60. Decision rule: reject Ho if zcomputed >

zcritical. Therefore, reject the null hypothesis because the computed test statistics of 2.60

exceeds the critical z-value of 2.33.

c. is incorrect because z-test (instead of t-test) should be used for sample size (81) >= 30

d. is incorrect because z-test (instead of t-test) should be used for sample size (81) >= 30

8.2 Where is the technical mistake in the answer given?

The population variance/standard deviation is not given; consequently the TEST STATISTIC

uses the sample standard deviation (4.5). Please note this itself JUSTIFIES the use of the

STUDENTS T DISTRIBUTION (technically, because a d.f. has been consumed). The Z-statistic is

justified if we use the population standard deviation.

Therefore, an ACCEPTABLE ANSWER is also: t = 2.6 and REJECT THE NULL (i.e., one-tailed

critical t @ 99% = 2.374).

Please note: technically the computed test statisticbecause it used the sample sigmais a

students t variable not a Z-variable.

However, this is a LARGE SAMPLE (n=81) and on that groundsi.e., that the students t with

large d.f. approximates (is asymptotic to) the normalwe can use the normal!

To recap the bottom line: by using the sample variance/sigma, the students T is

technically correct and justified (i.e., known population variance justifies the normal!),

but because the sample is large, we are okay to use the students T because as the sample

size increases the students t is approaching the normal.

8.3 What is the 99% confidence interval for the true population mean?

If students t, then 46.3 +/- (2.639)*(4.5/SQRT(81)) = 44.98 =< pop mean =< 47.62

If Z, then 46.3 +/- (2.576)*(4.5/SQRT(81)) = 45.01 =< pop mean =< 47.59

did you notice the change from a one-tailed null hypothesis to a TWO-TAILED confidence

interval?

8.4. If we want to create a confidence interval for the population variance, what distribution is

employed?

The chi-squared distribution


Question 9: Hypothesis testing [quantitative]

Which one of the following four statements about hypothesis testing holds

true if the level of significance decreases from 5% to 1%?

a) It becomes more difficult to reject a null hypothesis when it is actually true.

b) The probability of making a type I error increases.

c) The probability of making a type II error decreases.

d) The failure to reject the null hypothesis when it is actually false decreases to 1%.

9.2 If the level of significance decreases from 5% to 1%, the probability of a Type II error

increases from 95% to 99%. True or false?

9.3 Is there any way to simultaneously decrease the probability of both a Type I and Type II

error?

9.4 What are the normal deviates, respectively, for a 99% VaR (1% significance) and a 95% VaR

(5%) significance? Can you use these deviates in a sentence that expresses them in terms of

Type I/II error

9.5. (bonus) In the Basel II IMA back-test framework (i.e., green/yellow/red traffic light), what is

a Type I error?


Answer: A

Explanation:

Type I error: The rejection of the null hypothesis when it is actually true.

Type II error: The failure to reject the null hypothesis when it is actually false. The significance

level is the probability of making a type I error.

a. is correct. Decreasing the probability level makes it more difficult to reject the null when it is

true.

b. is incorrect. Decreases the probability of making a type I error.

c. is incorrect. All else being equal, the decrease in the probability of making a Type I error

comes at the cost of increasing the probability of making a Type II error.

d. is incorrect. Increases the probability of making a Type II error, in other words, the

probability of failing to reject the null hypothesis when it is actually false decreased

9.2 If the level of significance decreases from 5% to 1%, the probability of a Type II error increases

from 95% to 99%. True or false?

False. If the significance level is 1%, then the probability of a Type I error is 1%. And, due to the

necessary trade-off, a low significance level does imply a increase in the probability of a Type II

error (i.e., to accept, or fail to reject, a false null) HOWEVER this probability is not equal to 1

significance.

9.3 Is there any way to simultaneously decrease the probability of both a Type I and Type II error?

Yes, but only by increasing the sample size. For a given sample size, the trade-off is unavoidable.

9.4 What are the normal deviates, respectively, for a 99% VaR (1% significance) and a 95% VaR

(5%) significance? Can you use these deviates in a sentence that expresses them in terms of Type

I/II error?

One-tailed normal deviate @ 99% = 2.33

One-tailed normal deviate @ 99% = 1.645

i.e., VaR is always one-tailed! Memorize these deviates, they are commonly tested!

In the case of a 99%, we could say:

If the VaR model is accurate, the probability of a loss in excess of 2.33 standard deviations is 1%

which is the probability of a Type I error; i.e., the left-tail rejection region is 1%, or even better

If the VaR model is accurate, we do expect a loss of AT LEAST 2.33 standard deviations in 1% of

instances, which is the probability of a Type I error; i.e., the left-tail rejection region is 1%,

notice this latter emphasizes the fact the VaR is not giving us information about the tail

(losses in excess of the VaR threshold)

9. 5. (bonus) In the Basel II IMA back-test framework (i.e., green/yellow/red traffic light), what is a

Type I error?

The null is the banks VaR model is accurate. A Type I error is to reject an accurate model (red

zone) and a Type II error is to accept an inaccurate model (green zone).


Question 10: Option delta [valuation]

Mr. Black has been asked by a client to write a large put option on the

S&P 500 index. The option has an exercise price and a maturity that are

not available for options traded on exchanges. He, therefore, has to

hedge the position dynamically. Which of the following statements about

the risk of his position are not correct?

a) He can make his portfolio delta neutral by shorting index futures contracts.

b) There is a short position in an S&P 500 futures contract that will make his portfolio

insensitive to both small and large moves in the S&P 500.

c) A long position in a traded option on the S&P 500 will help hedge the volatility risk of the

option he has written.

d) To make his hedged portfolio gamma neutral, he needs to take positions in options as

well as futures.

[my adds]

10.2. In regard to answer (d) above, explain why a futures contract is insufficient to neutralize

gamma.

10.3 Assume Mr. Black wrote 10,000 put options with strike price at the current index price

(ATM). Given an estimate for percentage delta and position delta; and use these estimates to

illustrate the impact of an index price decline on the position.


Answer: B

Explanation:

The short index futures makes the portfolio delta neutral. It does not help with large moves,

though.

Topic: Valuation and Risk Models, Subtopic: Greeks, Reference: Hull, Chapter 17.

10.2. In regard to answer (d) above, explain why a futures contract is insufficient to neutralize

gamma.

The gamma of a futures contact is approximately equal to zero: the delta of a futures contract

equals EXP(rate*time), so the delta starts (at longer maturities) as 1.X and only slightly

converges to 1.0 as maturity decrease. Given that delta is almost a constant, the gamma is zero

and therefore, futures cannot neutralize gamma. Please note: neither a position in the

underlying (in this case, the index) nor an forward/futures contract on the same can neutralize

gamma (both are linear).

10.3 Assume Mr. Black wrote 10,000 put options with strike price at the current index price (ATM).

Given an estimate for percentage delta and position delta; and use these estimates to illustrate the

impact of an index price decline on the position.

A put option has a NEGATIVE percentage delta; if ATM, the put delta is generally in the

neighborhood of -0.4 (but it depends on the inputs!). Please note: delta of put= delta of call 1 =

N(d1) 1.

If we illustratively assume percentage delta = -0.4, then position delta of this SHORT position is

given by:

Position delta = -0.4 percentage delta * -10,000 written options = +4,000

... notice the short position is captured by a NEGATIVE in the QUANTITY. Here, the negatives

cancel to create a positive position delta; i.e., an increase in the underlying (index) produces a

gain for the short put

Now we might say,

Given a 1-unit drop (-1) in the index price, the position losses (-4000) or,

Given a 1-unit increase (+1) in the index price, the position gains (+4000) or,


Question 11: Estimate of invoice price [products]

On March 13, 2008, William Tell, a fund manager for the Rossini fund,

takes a short position in the March Treasury bond (T-bond) futures

contract. He plans to deliver the cheapest-to-deliver Treasury bond with

a coupon of 4.5% payable semiannually on May 15 and November 15 (182

days between), a conversion factor of 1.3256, and a face value of USD

100,000. The delivery date is Friday, March 15 (121 days after November

15 coupon payment date). The settlement price for the cheapest-to-

deliver Treasury bond on March 13 is 68 2/32. Which of the following is

the best estimate of the invoice price?

a) USD 90,118.87

b) USD 91,719.53

c) USD 92,367.75

d) USD 95,619.47

11.2 Which day count convention is assumed here?

11.3 Would the invoice price be higher or lower if the day count convention were instead,

respectively, 30/360 and Actual/360?

11.4 Assume the quoted price of the CTD bond above is $100.00 (i.e., the bond with a conversion

factor of 1.3256). Assume further that another eligible-for-delivery bond has a quoted price of

$110.00 with a conversion factor (CF) of 1.45. In this case, which bond is cheapest-to-deliver?

11.5 Why are there several bonds (a basket) that can be delivered in the CBOT Treasury bond

futures contact, as opposed to a single bond?

11.6 In regard to factors that determine the cheapest to deliver (CTD) bond, what bonds are

favored when bond yields are high (low)?


favored when the yield curve is upward-sloping (downward-sloping)?


Answer: B

The invoice is based on a settlement price of 68 2/32 or 68.0625. The accrued interest is

calculated on the basis of the number of days since the last coupon payment date, November 15,

and the delivery date, March 15. That is 121. During the current six-month period between

coupon payment dates, November 15 to May 15, there are 182 days. Thus the accrued interest

on USD 100,000 face value of the bond is 121/182 * USD 100,000 * 0.045/2 = USD 1,495.88

Explanation:

The invoice price is USD 100,000 * 0.680625 * 1.3256 + USD 1,495.88 = 91,719.53

Topic: Financial Markets and Products. Subtopic: Cheapest to deliver bond, conversion factors

Reference: Bruce Tuckman, Fixed Income Securities, 2nd Edition. This is incorrect reference:

should be Hull Chapter 6

11.2 Which day count convention is assumed here?

Actual/Actual (121/182) is the convention for U.S. Treasury bonds.

11.3 Would the invoice price be higher or lower if the day count convention were instead,

respectively, 30/360 and Actual/360?

If 30/360, then AI = 120/180 * USD 100,000 * 0.045/2 = $1,500. HIGHER.

If Actual/360, then AI = 121/180 * USD 100,000 * 0.045/2 = $1,512. HIGHER.

11.4 Assume the quoted price of the CTD bond above is $100.00 (i.e., the bond with a conversion

factor of 1.3256). Assume further that another eligible-for-delivery bond has a quoted price of

$110.00 with a conversion factor (CF) of 1.45. In this case, which bond is cheapest-to-deliver?

The net cost = (quoted bond price) (settlement price * CF).

In the case of the original bond above, net cost =$ 100 - $68.0625*1.3256 = $9.78

In the case of the second bond, net cost = $110 - $68.0625*1.45 = $11.31

Therefore, the original remains the CTD.

Please note the logic of CTD:

The short must purchase the bond with cost = Quoted bond price + AI

Then delivers the bond in order to receive = (Settle price * CF) + AI

So net cost = cash paid cash received

= Quoted bond price + AI [(Settle price * CF) + AI] = Quoted bond price - (Settle price * CF)

11.5 Why are there several bonds (a basket) that can be delivered in the CBOT Treasury bond

futures contact, as opposed to a single bond?

Tuckman says, The design of bond futures contracts purposely avoids a single underlying

security. One reason for this is that if the single underlying bond should lose liquidity, perhaps

because it has been accumulated over time by buy-and-hold investors and institutions, then the

futures contract would lose its liquidity as well. Another reason for avoiding a single underlying

bond is the possibility of a squeeze. To illustrate this problem, assume for the moment that only

one bond were deliverable into a futures contract. Then a trader might be able to prot by

simultaneously purchasing a large fraction of that bond issue and a large number of contracts. As

parties with short positions in the contract scramble to buy that bond to deliver or scramble to


buy back the contracts they have sold, the trader can sell the holding of both bonds and

contracts at prices well above their fair values. But by making shorts hesitant to take positions,

the threat of a squeeze can prevent a contract from attracting volume and liquidity.


favored when bond yields are high (low)?

Hull: A number of factors determine the cheapest-to-deliver bond. When bond yields are in

excess of 6%, the conversion factor system tends to favor the delivery of low-coupon long-

maturity bonds. When yields are less than 6%, the system tends to favor the delivery of high-

coupon short-maturity bonds.


favored when the yield curve is upward-sloping (downward-sloping)?

Hull: when the yield curve is upward-sloping, there is a tendency for bonds with a long time to

maturity to be favored, whereas when it is downward-sloping, there is a tendency for bonds

with a short time to maturity to be delivered.

Question 12: Hedge [products]

The yield curve is upward sloping, and a portfolio manager has a long position in 10-year

Treasury Notes funded through overnight repurchase agreements. The risk manager is

concerned with the risk that market rates may increase further and reduce the market value of

the position. What hedge could be put on to reduce the positions exposure to rising rates?

a) Enter into a 10-year pay fixed and receive floating interest rate swap.

b) Enter into a 10-year receive fixed and pay floating interest rate swap.

c) Establish a long position in 10-year Treasury Note futures.

d) Buy a call option on 10-year Treasury Note futures.

12.2 What additional risk(s) in the total position (i.e., long note plus hedge trade) are not

necessarily hedged. Please think about this before looking at the answer

12.3 Assume the portfolio manager has $1 million invested in the long position with an expected

duration of 7.0 years. If he/she hedges with T-bond (T-note) futures with contract price of 90.00

and duration of 8.0 years, what is the duration-based hedge?

12.4 In the original question, assume the portfolio manager hedges with a 5-year swap where

the notional equals the principal invested. What is the problem with such a hedge?


Answer: A

Explanation:

a. is correct. An increase in rates will increase the value of the hedge position and offset the loss

in value from the Bond position.

b. is incorrect. An increase in rates will decrease the value of the hedge position and add to the

loss in value from the Bond position.

c. is incorrect. An increase in rates will decrease the value of the futures position and add to the

loss in value from the Bond position.

d. is incorrect. An increase in rates (all else equal), will decrease the value of the call option and

add to the loss in value from the Bond position.

Topic: Financial Markets and Products. Subtopic: Futures, forwards, swaps and options

Reference: John Hull, Options, Futures, and Other Derivatives, 6th Edition (New York: Prentice

Hall, 2006) Chapter 7 - Swaps

12.2 What additional risk(s) in the total position (i.e., long note plus hedge trade) are not

necessarily hedged. Please think about this before looking at the answer

First, the long T-Note is funded by a (short-term) overnight repo; we can call this funding

liquidity risk.

Second, the swap involves a counterparty so the hedge transaction incurs counterparty risk.

(arguably, under an alternative taxonomy, we could argue the repo funding is subject to market

liquidity risk. And, as counterparty risk is a sub-class of credit risk, we could subsume both

under credit risk.)

Third, there remains BASIS RISK between the underlying T-note exposure and the swap hedge;

dont forget basis risk!

12.3 Assume the portfolio manager has $1 million invested in the long position with an expected

duration of 7.0 years. If he/she hedges with T-bond (T-note) futures with contract price of 90.00

and duration of 8.0 years, what is the duration-based hedge?

The number of contracts = $1 million / (90 * $100,000 face) * (7/8) = 9.72.

The PM would SHORT approximately 10 contracts.

Note answer (c) would be correct if it said establish a SHORT position in 10-year Treasury

note futures

12.4 In the original question, assume the portfolio manager hedges with a 5-year swap. What is the

problem with such a hedge?

The long Treasury note position likely has a duration > 7 but the swaps duration will be less

than 5; i.e., the swap can be treated as two bondsa fixed and a floaterand the duration of

floater is roughly zero (or time to next coupon) such that the swaps duration is approximately

equal to the fixed legs duration.

So, in this case, when interest rates increase, the underlying position will lose more value than

will be offset by the increase in the swaps value.


Question 13: Market/Credit/Operational risk [foundation]

Jennifer Durrant is evaluating the existing risk management system of

Silverman Asset Management. She is asked to match the following events

to the corresponding type of risk. Identify each numbered event as a

market risk, credit risk, operational risk, or legal risk event.

Event

1) Insufficient training leads to misuse of order management system.

2) Credit spreads widen following recent bankruptcies.

3) Option writer does not have the resources required to honor a contract.

4) Credit swaps with counterparty cannot be netted because they originated in multiple

jurisdictions.

a) 1: legal risk, 2: credit risk, 3: operational risk, 4: credit risk

b) 1: operational risk, 2: credit risk, 3: operational risk, 4: legal risk

c) 1: operational risk, 2: market risk, 3: credit risk, 4: legal risk

d) 1: operational risk, 2: market risk, 3: operational risk, 4: legal risk

13.2 Where is the MISTAKE in the answer given?

13.3 Please try to identify THREE RISKS that are not included among market, credit or

operational risks.

13.4 Is legal risk an operational risk?

13.5 Which of the risks listed above are meant to be covered by Basel II capital requirements?

13.6 [this does not have a correct answer, but please think about it]. Where do we classify

LIQUIDITY RISK vis--vis Credit, Market & Operational risks?


Answer: C

Explanation: a, b and d are incorrect. c is correct.

1. Insufficient training leads to misuse of order management system is an example of

operational risk.

2. Widening of credit spreads represents an increase in market risk.


Question 14: Models for estimating volatility [quantitative]

Which one of the following four statements on models for estimating

volatility is incorrect?

a) In the RiskMetrics EWMA model, some positive weight is assigned to the long-run

average variance rate.

b) In the RiskMetrics EWMA model, the weights assigned to observations decrease

exponentially as the observations become older.

c) In the GARCH (1, 1) model, a positive weight is estimated for the long-run average

variance rate.

d) In the GARCH (1, 1) model, the weights estimated for observations decrease


14.2 What is the most likely weight of the most recent observation (observation = squared

return) in the RiskMetrics EMWA model?

14.3 True or false: EWMA and GARCH (1,1) are both parametric approaches to VaR. If true, what

is their advantage/disadvantage as parametric methods?

14.4 True or false: In the multivariate density estimation (MDE) approach, the weights estimated

for observations decrease exponentially as the observations become older.

14.5 True or false: In Lindas Allens hybrid approach, the weights estimated for observations

decrease exponentially as the observations become older.


Answer: A

Explanation:

a. is incorrect. The RiskMetrics model does not involve the long-run average variance rate in

updating volatility, in other words, the weight assigned to the long-run average variance rate is

zero.

b. is correct. In the RiskMetrics model, the weights assigned to observations decrease


c. is correct. In the GARCH (1, 1) model, some positive weight is assigned to the long-run

average variance rate.

d. is correct. In the GARCH (1, 1) model, the weights assigned to observations decrease


Topic: Quantitative Analysis, Subtopic: EWMA, GARCH models, Reference: Hull, Chapter 21.

14.2 What is the most likely weight of the most recent observation (observation = squared return)

in the RiskMetrics EMWA model

In the infinite series, if we are estimating todays volatility (n) the weight applied to period (n-t)

= (1-lambda)*lambda^(t-1). The most recent weight (at t=1), assuming a lambda of 0.94, is given

by: (1-94%)*94%^(1-1) = 6%*1 = 6%.

14.3 True or false: EWMA and GARCH(1,1) are both parametric approaches to VaR. If true, what is

their advantage/disadvantage as parametric methods?

True: EWMA and GARCH are both parametric: historical (empirical) data is used to compute a

variance/volatility. VaR is then computed per the distributional assumption (the historical data

is no longer needed).

Advantages of parametric methods include parsimony (no dataset) and analytical tractability

(e.g., we can scale standard deviation per the square root rule)

The salient disadvantage, for risk purposes, is arguably the inability of any distribution to

adequately characterize the tail (and extreme tail). More broadly, the distributional assumption

is unlikely to fully characterize the actual behavior of returns (e.g., skewed, fat-tailed, unstable).

Note this problem extends BEYOND NORMAL distribution; of course the normal does not

characterize returns.

14.4 True or false: In the multivariate density estimation (MDE) approach, the weights estimated

for observations decrease exponentially as the observations become older.

False (except if by coincidence): in MDE a state vector (current versus historical) determines the

weights

14.5 True or false: In Lindas Allens hybrid approach, the weights estimated for observations

decrease exponentially as the observations become older.

TRUE! The hybrid is essential a simulation method, but unlike simple historical simulation (HS),

the historical returns are weighted in declining fashion per EWMA. (the hybrid refers to a

hybrid between HS and EWMA, but ultimately this is a NON-PARAMETRIC SIMULATION

approach).


Question 15: Duration of the bond [valuation]

The table below gives the closing prices and yields of a particular liquid

bond over the past few days.

Day Price Yield

Monday 106.3 4.25%

Tuesday 105.8 4.20%

Wednesday 106.1 4.23%

What is the approximate duration of the bond?

a) 18.8

b) 9.4

c) 4.7

d) 1.9

15.2 What is the mistake in the question (why is the data virtually impossible)?

15.3. What is the implied dollar duration of the bond?

15.4 What is the implied DV01 of the bond

15.5 What is the implied convexity of the bond [this can actually be observed without

calculations!]


Answer: B

Explanation:

The duration can be approximated from the price changes. (106.3 - 105.8)/106.3/.0005 = 9.4

(106.3 - 106.1)/106.3/.0002 = 9.4

Topic: Valuation and Risk Models. Subtopic: DV01, duration and convexity. Reference: Tuckman,

chapter 5

15.2 What is the mistake in the question (why is the data virtually impossible)?

Notice that the price of the bond increases as the yield increases!

This implies negative duration which is only possible for an interest-only (IO) security.

15.3. What is the implied dollar duration of the bond?

The dollar duration (DD) is the slope of the tangent line. In this case, DD

= (106.1 105.8) / (4.23% - 4.20%) = 1,000

(note: consistent with the original flaw, the DD is positive but typical bonds have negative dollar

durations as the slope of the tangent line is negative).

15.4 What is the implied DV01 of the bond

DV01 = P*D/10,000 = 10.6.1*-9.43/10000 = -$0.10

Please note the calculation is unnecessary: we can observe the bond price changes $0.20 when

the yield shifts by two basis points (2 bps) from 4.23% to 4.25%. So, we can observe that a ONE

BASIS POINT SHIFT implies ~ $0.10 price change.

15.5 What is the implied convexity of the bond [this can actually be observed without calculations!]

The dollar duration/DV01 is exactly the same from 4.20% to 4.23% as it is from 4.23% to

4.25%; i.e., the DVO1 is CONSTANT at $0.01! Convexity is the rate of change of the (dollar)

duration (the 2nd derivative of price with respect to yield = the 1st derivative of the 1st

derivative). Convexity is therefore ZERO.

Mathematically, we can confirm:

Dollar convexity = (1000 dollar duration @ 4.25% - 1000 dollar duration @ 4.23%)/(4.25% -

4.23%)

= 0/0.02% = 0;

Convexity = dollar convexity / $106.1 = 0.


Question 16: Diversification for a VaR [valuation]

Bond Yield Maturity (Yrs) STDEV of Yield Annual Exposure

A 5% 2 5% USD 25.00

B 3% 13 12% USD 75.00

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.

a) 76,500

b) 283,000

c) 382,300

d) 1,413,000

16.2 The question incorrectly (or imprecisely at least) solves for the yield VaR (i.e., the worst

expected change in yield). Per Jorion, we would instead typically solve for a return- or price-

based VaR. What is the better answer to the question if we compute VaR in terms of price/value

at risk?

16.3. What is the individual VaR of Bond B?

16.4. If the correlation changes to 1.0, what is the difference between diversified and

undiversified VaR?

16.5. Define the marginal VaR (Bonus: Calculate marginal VaRs)

16.6. Define component VaR (Bonus: Calculate component VaRs)


Answer: B

Explanation:

1. Calculate the undiversified VaR

VaRundiv = 1.645 * 5%*(10/250) * 25 + 1.645 * 12% 10/250 *75 = 0.4113 + 2.9610 = 3.3723

2. Calculate the diversified VaR

1.645 0.25 2 * 5% 2 + 0.75 2 * 12% + 2 * 0.25 * 0.75 * 5% * 12% * 0.25 * (10/250) * 100 =

1.645 * 0.0939 * 10/250 * 100 = 3.0893

3. Difference is 0.283

Topic: Valuation and Risk Models. Subtopic: VaR for fixed income securities. Reference: Allen,

Boudoukh, Saunders, Understanding market, Credit and Operational Risk: The Value at Risk

Approach, Chapters 2, 3

16.2 The question incorrectly (or imprecisely at least) solves for the yield VaR (i.e., the worst

expected change in yield). Per Jorion, we would instead typically solve for a return- or price-based

VaR. What is the better answer to the question if we compute VaR in terms of price/value at risk?

Volatility (dP/P) = Duration * volatility (yield); i.e., Jorion 8.14

If we assume semi-annual compounding, then duration of bonds is:

Duration (bond A) = 2/(1+5%/2) = 1.95,

Duration (bond B) = 2/(1+3%/2) =12.81,

Price volatility (bond A) = 5% yield volatility * 1.95 = 9.76%, and

Price volatility (bond B) = 12% yield volatility * 12.81 = 153.7%.

Per the formula above (portfolio variance), the DIVERSIFIED 10-day VaR = $38.1293, and

the UNDIVERSIFIED 10-day VaR = $38.723.

The difference is 0.59386 ($594,000)

16.3. What is the individual VaR of Bond B?

= 153.7% * SQRT(10/250) * 1.645 * $75 = $37.92

16.4. If the correlation changes to 1.0, what is the difference between diversified and undiversified

VaR?

Zero. At perfect correlation, benefit of diversification reduces to zero.

16.5. Define the marginal VaR (Bonus: Calculate marginal VaRs)

Marginal VaR = deviate * Covariance (Asset, Portfolio)/Volatility (portfolio);

Marginal VaR (Bond A) = 1.645 * 0.0305/$115.9 = 0.0433

Marginal VaR (Bond B) = 1.645 * 0.1.7810/$115.9 = 2.5275

16.6. Define component VaR (Bonus: Calculate component VaRs)

Component VaR = marginal VaR * position. In this case,

Component VaR (Bond A, time scaled) = 0.0433 marginal VaR * $25 * SQRT(10/250) = $0.22

Component VaR (Bond A, time scaled) = 2.5275 marginal VaR * $75 * SQRT(10/250) = $37.91

And note that the SUM of COMPONENT VaRs equals the diversified VaR:

$0.22 + $37.91 = $38.1293


Question 17: Probability [quantitative]

Assume that a random variable follows a normal distribution with a mean

of 100 and a standard deviation of 17.5.What is the probability that this

random variable is between 82.5 and 135?

a) 68.0%

b) 81.9%

c) 82.8%

d) 95.0%

17.2. Which of Gujaratis sampling distributions (FRM assigned Chapter 4) converge on a normal

distribution?

17.3. In significance tests of the regression coefficients like slope (i) what justifies the

assumption of normal distribution and (ii) what fact anyhow often precludes its use?

17.4. (please dont peek, give this some thought) Assume we are given NO INFORMATION about

the distribution. Now, what is the probability that the random variable is between 65 and 135?

17.5 What is meant by the normals property of LOCATION-SCALE INVARIANCE (source: FRM

assigned Rachev)?

17.6 What is meant by the normals property of SUMMATION STABILITY (source: FRM assigned

Rachev)?

17.7 What is meant by the normals property of DOMAIN OF ATTRACTION (source: FRM

assigned Rachev)?


Answer: B

Explanation: Prob (-1* < X < 2*) = (1 - 0.0228) - 0.1587 = 0.8185

a. is incorrect. Almost 68% of the observations will be within the interval from one standard

deviations below the mean to one standard deviations above the mean, which is within the

interval [100 - 17.5; 100 + 17.5].

b. is correct. 82.5 = 100 - 17.5 and 135 = 100 + 2 * 17.5. So, the percentage is 34% on the left

hand side of the mean, plus 95%/2 on the right hand side of the mean.

c. is incorrect. Almost 95% of the items will lie within the interval from two standard deviations

below the means to two standard deviations above the mean, that is within the interval [100 - 2

*17.5;100 + 2 * 17.5].

d. is incorrect. Assumes wrongly that 97.5% of the observations will be within [100 - 2 *

17.5;100 + 2 * 17.5].

Topic: Quantitative analysis, Subtopic: Probability Distributions, Reference: Damodar N Gujarati,

Essentials of Econometrics, 3rd Edition (New York: McGraw-Hill, 2006), chapter 4, pp. 80-84

17.2. Which of Gujaratis sampling distributions (FRM assigned Chapter 4) converge on a normal

distribution?

All of them!

Students t is heavy-tailed but approximates normal as d.f. > 29 (n=30).

Chi-square is non-negative and positively skewed but, for large d.f., also converges to normal

(Per CLT, as chi-square is sum of independent random variables)

F-distribution is (also, like chi-square) non-negative and skewed, but similarly approaches

normal for large d.f.

(note the binomial and Poisson also approach normal as, respectively, N and lambda, increase

toward infinity)

17.3. In significance tests of the regression coefficients like slope (i) what justifies the assumption of

normal distribution and (ii) what fact anyhow often precludes its use?

(i) The central limit theorem justifies the normal assumption;

(ii) But in practice we rarely know the POPULATION VARIANCE, so we use the sample variance

which consumes a d.f. and requires the students t distribution

17.4. (please dont peek, give this some thought) Assume we are given NO INFORMATION about the

distribution. Now, what is the probability that the random variable is between 65 and 135?

Chebyshevs Inequality says that P[X is WITHIN +/- 2 sigma] is AT LEAST 1 - 1/(2^2) = 3/4;

similarly,

The P [X is WITHOUT +/- 2 sigma] is LESS THAN OR EQUAL TO 1/(2^2) = 1/4

So, we can say, the probability the random variable is between 82.5 and 135 GREATER THAN OR

EQUAL TO 75%. (at most, 25% lie outside this interval)


17.5 What is meant by the normals property of LOCATION-SCALE INVARIANCE (source: FRM

assigned Rachev)?

Cconsider the random variable Y, which is obtained as Y = aX + b. In general, the distribution of Y

might substantially differ from the distribution of X, but in the case where X is normally

distributed, the random variable Y is again normally distributed with parameters and . Thus, we

do not leave the class of normal distributions if we multiply the random variable by a factor or

shift the random variable.

17.6 What is meant by the normals property of SUMMATION STABILITY (source: FRM assigned

Rachev)?

If you take the sum of several independent random variables, which are all normally distributed

with mean (mu) and standard deviation (sigma), then the sum will be normally distributed

again. For reflection, why is this relevant for financial series?

17.7 What is meant by the normals property of DOMAIN OF ATTRACTION (source: FRM assigned

Rachev)?

Rachev: The last important property that is often misinterpreted to justify the nearly exclusive

use of normal distributions in financial modeling is the fact that the normal distribution

possesses a domain of attraction. A mathematical result called the central limit theorem states

thatunder certain technical conditionsthe distribution of a large sum of random variables

behaves necessarily like a normal distribution. In the eyes of many, the normal distribution is

the unique class of probability distributions having this property. This is wrong and actually it is

the class of stable distributions (containing the normal distributions), which is unique in the

sense that a large sum of random variables can only converge to a stable distribution.


Question 18: Bonds [valuation]

The following table gives the prices of two out of three US Treasury notes

for settlement on August 30, 2008. All three notes will mature exactly

one year later on August 30, 2009. Assume annual coupon payments and

that all three bonds have the same coupon payment date.

Coupon Price

2 7/8 98.40

4 1/2 ?

6 101.30

Approximately what would be the price of the 4 1/2 US Treasury note?

a) 99.20

b) 99.40

c) 99.80

d) 100.20

18.2. If the actual (observed) price of the 4 1/2 is $100.00, is the bond trading cheap or rich, and

what might explain?

18.3 Does the law of one price allow for arbitrage opportunities?

18.4 Which of the bonds carries the highest yield (YTM; assume semi-annual compounding as

this refers to Tuckman!)?

18.5 (bonus) Can different yields be explained under the no-arbitrage assumption, or is than an

internal inconsistency (error) in the question?


Answer: C

Explanation: 2.875% * x + 6.25% *(1 - x) = 4.5% X = 52%

The portfolio that has cash flows identical to the 4 1/2 bond consists of 52% of the 2 7/8 and

48% of the 6 1/4 bonds. As this portfolio has cash flows identical to the 4 1/2 bond, precluding

arbitrage, the price of the portfolio should equal to 52% * 98.4 + 48% * 101.30 or 99.80

Topic: Valuation and Risk Models

Subtopic: Bond prices, spot rates, forward rates

Reference: Tuckman, Chapter 1

18.2. If the actual (observed) price of the 4 1/2 is $100.00, is the bond trading cheap or rich, and

what might explain?

If the market price is greater than the predicted (model) price, the bond is trading rich; i.e., $100

> $99.80.

Technical factors (e.g., supply/demand) can explain be they represent omitted model factors. In

Tuckmans discussion of Treasuries that trade rich/cheap he tends to explain with LIQUIDITY;

i.e., rich bonds are more demanded (or less supplied) and cheap bonds are less demanded (or

more supplied).

18.3 Does the law of one price allow for arbitrage opportunities?

No.

The law of one price states that absent confounding factors (e.g., liquidity, special financing

rates, taxes, credit risk), two securities (or portfolios of securities) with exactly the same cash

flows should sell for the same price.

Arbitrage is technically an instantaneous risk-free profit. Therefore, the law of one price implies

no arbitrage.

18.4 Which of the bonds carries the highest yield (YTM; assume semi-annual compounding as this

refers to Tuckman!)?

The bond-equivalent yield of the first bond = 2 * RATE(2,2.875/2, -98.4,100) = 2 * 2.265 =

4.530%

In calculator terms:

N = 2,

PV = -98.4,

PMT = 2.875/2,

FV = 100

CPT I/Y = 2.2647% (semiannual)

Similarly, the yield of the second bond = 4.711% and the third bond = 4.902%.

So, the third bond offers the highest yield-to-maturity (YTM).

18.5 (bonus) Can different yields be explained under the no-arbitrage assumption, or is than an

internal inconsistency (error) in the question?

Yes, because the yield essentially impounds the rates into a single flat yield curve.

But the six month rate can be different from (lower than) the annual rate. Put another way, a

non-flat yield curve can explain different yields (YTM).


Question 19: Duration and convexity [valuation]

A newly issued non-callable, fixed-rate bond with 30-year maturity

carries a coupon rate of 5.5% and trades at par. Its duration is 15.33

years and its convexity is 321.03.Which of the following statements about

this bond is true?

a) If the bond were to start trading at a discount, its duration would decrease.

b) If the bond were to start trading at a premium, its duration would decrease.

c) If the bond were to start trading at a discount, its duration would not change.

d) If the bond were to remain at par, its duration would increase as the bond aged.

19.2. What is the bonds modified duration?

19.3. What is the bonds DV01? Use this DV01 in a sentence to give the DV01 a definition.

19.4. Use duration and convexity to approximate the change in bond price given a 1% drop in

yield.

19.5 [tough] If this bond were to start trading as a discount, would an increase in maturity imply

an increase in duration and DV01?


Answer: A

Explanation:

a. is correct. At higher interest rates, the bond/price relationship is closer to linear than it is

when rates are low. So, the new duration would be lower than 15. Alternatively, one can think of

duration as a weighted average of the times when cash flows are made, where the weights are

the percentage of the total value of the bond. When rates rise, the present values associated with

the later payments are relatively smaller and the duration falls.

b. is incorrect because it is the exact opposite of a, the correct answer.

c. is incorrect. It fails to recognize the logic stated in a.

d. is incorrect because duration is mainly a function of duration and, all else constant, duration

would decrease as the bonds maturity shortened.

Topic: Valuation and Risk Models. Subtopic: Duration and convexity. Reference: Bruce Tuckman,

Fixed Income Securities, 2nd edition, Chapter 5

19.2. What is the bonds modified duration?

Assuming semi-annual compounding, modified duration = Macaulay duration / (1 + yield/2).

Because the bond trades at par, the yield = coupon = 5.5%. Therefore, modified duration =

15.33/(1+5.5%/2) = 14.92

19.3. What is the bonds DV01? Use this DV01 in a sentence to give the DV01 a definition.

DV01 = Price * Mod Duration/10000. In this case,

DV01 = 100 * 14.92/10000 = $0.15 (i.e., $0.1492)

If the yield (YTM) drops by one basis point (bps), in a parallel shift of the yield curve, the

bond price will increase by approximately $0.15

...note we are generally referring to a yield-based DV01 as in the rate shocked is a YTM.

19.4. Use duration and convexity to approximate the change in bond price given a 1% drop in yield.

Estimate change = duration impact + convexity adjustment. In this case,

=(-) (14.92)(-1%) + (321.03)*(-1%)^2*(1/2) = +14.92% + 1.62% = +16.54%

please note the convexity adjustment is always a positive addition; i.e., duration alone will

understate the actual bond price in both up and down yield movements (the convexity gap, for a

bond without embedded options, is always positive)

19.5 [tough] If this bond were to start trading as a discount, would an increase in maturity imply

an increase in duration and DV01?

Duration (Mac or modified) is always an increasing function of maturity, so for the discounted

bond, an increase in maturity would indeed increase the duration.

However, DV01 is infected by a price effect in addition to a duration effect. For the original par

bond, maturity extension will increase duration; but as a discounted bond, extensions to

maturity DECREASE PRICE (think pull to par but in reverse!). Therefore, the impact on DV01 is

ambiguous.


Question 20: Historical simulation [valuation]

Rational Investment Inc. is estimating a daily VaR for its fixed income

portfolio currently valued at USD 800 million. Using returns for the last

400 days (ordered in decreasing order, from highest daily return to

lowest daily return), the daily returns are the following: 1.99%, 1.89%,

1.88%, 1.87%,, -1.76%, -1.82%, -1.84%, -1.87%, -1.91%.

At the 99% confidence level, what is your estimate of the daily dollar VaR

using the historical simulation method?

a) USD 14.08mm

b) USD 14.56mm

c) USD 14.72mm

d) USD 15.04mm

20.2 Is the answer given the only correct answer? If not, give alternative(s).

20.3 We could also calculate the historical volatility (X%) of the 400 days series and compute a

99% VaR as given by 2.33*X%. What is the disadvantage of such a parametric approach?

20.4 As a risk manager, you perceive recent days (which have been more volatile) to be more

relevant to the VaR estimate than distant days (which were relatively calm). Propose a

modification to the approach.


Answer: B

Explanation: VaR = 1.82% * 800 = 14.56 million

20.2 Is the answer given the only correct answer? If not, give alternative(s).

No, the answer given is not the only correct answer. Per Dowd, the 1% VaR HS is the

observation given by (1-confidence)*n + 1. In this case, the 99% VaR refers to the 5th highest

loss observation (i.e., if sorted where losses are positive). Dowds logic is simply that

observations in the tail constitute 1%; in this case, the four worst losses constitute the 1% tail.

please note this helps us state VaR in the following way: at least 1% of the time, we expect

losses to be worse than 1.76%

under Dowds approach, therefore, answer (a) of 14.08 MM is also correct!

Further, the answer given by =PERCENTILE (array, 1%) is also acceptable although difficult to

quickly calculate. PERCENTILE (this array, 1%) = 1.7606%

20.3 We could also calculate the historical volatility (X%) of the 400 days series and compute a

99% VaR as given by 2.33*X%. What is the disadvantage of such a parametric approach?

2.33 is the deviate implied by a normal distribution at under the one-tailed 99% VaR; i.e., 2.33 is

a normal deviate. The first problem, then, is that we have IMPOSED a NORMAL distribution on

data that is likely non-normal (e.g., may be kewed or heavy-tailed)

20.4 As a risk manager, you perceive recent days (which have been more volatile) to be more

relevant to the VaR estimate than distant days (which were relatively calm). Propose a

modification to the approach.

You can use the hybrid approach (Linda Allens term) which is the exact same as Dowds AGE-

WEIGHTED HISTORICAL SIMULATION. In this approach, the weight assigned to historical return

days will decline, per EWMA, in constant proportion so that more recent days receive more

weight. The 1% is then unlikely to be the 4th or 5th worst observation as it would be under

simple HS.

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L1.2010.GARP Practice Exam A