Q. σ2 of return of stock P= 100.0σ2 of return of stock Q=225.0

Quantitative Analysis

MomentsProb. & Prob. distribution

Hypothesis

Testing

Correlation & Regression

Sampling Volatility Estimation Monte Carlo Simulation

Mean Skewness Kurtosis

Mean: ∑xi/n

Mode:-Value that occurs most frequently

Median:-Midpoint of data arranged in ascending order

Avg. of squareddeviations from mean

Population varianceσ2 =[∑Ni=1 (Xi -µ)2]/N

Sample variances2 =[∑ni=1 (Xi -Xmean

-)2]/(n-1)

Var(ax+by)=a2Var(x)+ b2Var(y)+2abCov(x,y)

Standard deviation = √Variance

•Positively : mean> median> mode• Negatively : mean< median< mode• Skewness of Normal = 0

•Leptokurtic: More peaked than normal (fat tails); kurtosis>3•Platykurtic: Flatter than a normal; kurtosis<0• Kurtosis of Normal = 3•Excess Kurtosis = Kurtosis - 3

Q. If d istributions of returns from financial instruments are leptokurtotic. How does it compare with a normal distribution of the same mean and variance?

Ans. Leptokurtic refers to a distribution with fatter tails than the

Variance

• No. of ways to select r out of n objects: nCr = n!/[r!*(n-r)!]

Probability

• No. of ways to arrange r objects in n places: nPr =n!/(n-r)!

Counting principlesSum rule and

Bayes’ TheoremProperties

)()()( BAPBAPBPc ∩+∩=

)(*)/()(*)/()( ccAPABPAPABPBP +=

)P(B*)P(A/BP(B)*P(A/B)

P(B)*P(A/B)P(B/A)

cc+=

AB

•Outcome only between [a, b] •P(outside a & b) = 0

Cumulative density function (cdf) for Uniform distribution :F(x)=0 For x<=aF(x)=(x-a)/(b-a) For a<x<bF(x)=1 For x>=b

Continuous uniform

distribution

Normal Distribution (ND)

Ques - The R.V. X with density function f(X) = 1 / (b - a) for a < x < b, and 0 otherwise, is said to have a uniform distribution over (a, b). Calculate its mean.

a b

If Z is a standard normal R.V. An event X isdefined to happen if either -1< Z < 1 or Z >1.5. What is the prob. of event X happening ifN (1) =0.8413, N (0.5) = 0.6915 and N (-1.5) =0.0668, where N is the CDF of a standardnormal variable

Ans The sum of areas shown in two figuresArea 1 = 1-2*(1- N(1)) = 1-2*(0.1587)Area 2 = 0.0668 , Total Area = 0.7514

-1 +1 1.5

-4 -3 -2 -1 0 1 2 3 4

68% of Data

95% of Data

99.7% of Data

Standardized RV is normalized mean = 0, σ = 1.

Z-score: # of σ a given observation is from population mean.Z=(x-µ)/σ

Q. At a particular time, the market value of assets of the firm is $100 Mn and the market value of debt is $80 Mn. The standard deviation of assets is $ 10 Mn. What is the distance to default?Ans. z = (A-K)/σA = (100-80)/10 = 2

Probability

Distribution

ContinuousDiscrete

Binomial Poisson

Only 2 possible outcomes: failure or success.P(x)=nCx*px(1-p)n-x

Fix the expectation λ=np.P(x)=λxe-λ/x! if x>=0P(x)=0 otherwise

Q. The number of false fire alarms in a suburb of Houston averages 2.1 per day. What is the (apprximate) probability that there would be 4 false alarms on 1 day?Ans. P(X=x) = (λxe-x)/x!X= 2.1, x = 4P(2.1) = 0.1

Binomial Random VariableE(X)=n*pVar(X)=n*p*(1-p)=n*p*q

AB

SE (σx) of the sample mean is σ of the dist. of sample means

Sampling

Central limit theorem

VisualizeFRM-Part I

σ of return of stock Q=225.0Cov (P,Q) =53.2Current Holding $1 mn in P.New Holding: shifting $ 1 million in Q and keepingUSD 3 million in stock P. What %age of risk (σ), is reduced?Ans. σP=√[w2σA

2 + (1-w)2 σB2 +2w(1-w)Cov(A,B)]

w= 0.75c2 = 100*(0.75)2 + 225*(0.25)2 +2*0.25*0.75*53.2 σP= 9.5 o ld σ = √100 = 10Reduction = 5%

Ans. Leptokurtic refers to a distribution with fatter tails than the normal, which implies greater kurtosis.

• P(A) = # of fav. Events/ # Total Events• 0 < P(A) <1, P(Ac)=1-P(A)•P(AUB)=P(A)+P(B)-P(A∩B)=P(A)+P(B) If A,B Mutually exclusive•P(A│B)= P(A∩B)/P(B)• P(A∩B)=P(A│B)P(B)=P(A)P(B)If A,B Independent

Q. The subsidiary will default if the parent defaults, but the parent will not necessarily default if the subsidiary defaults. Calculate Prob. of a subsidiary & parent both defaulting. Parent has a PD = .5% subsidiary has PD of .9%Ans. P(P∩S) = P(S/P)*P(P) = 1*0.5 = 0.5%

Q. ABC was inc. on Jan 1, 2004. Its expected annual default rate of 10%. Assume a constant quarterly default rate. What is the probability that ABC will not have defaulted by April 1, 2004?Ans. P(No Default Year) = P(No default in all Quarters)= (1-PDQ1)*(1-PDQ2)*(1-PDQ3)*(1-PDQ4)PDQ1=PDQ2=PDQ3=PDQ4=PDQP(No Def Year) = (1-PDQ)4P(No Def Quarter) = (0.9)4 = 97.4%

)P(B*)P(A/BP(B)*P(A/B)P(B/A)

cc+

Ans. Since the distribution is uniform, the mean is the center of the distribution, which is the average of a and b = (a+b)/2

P(2.1) = 0.1

Q. A portfolio consists of 17 uncorrelated bonds. The 1-year marginal default prob. of each bond is 5.93%. If spread of default prob. is even over the year, Calculate prob. of exactly 2 bonds defaulting in first month?Ans. 1-month default rate =1- (1-0.593)1/12 = 0.00508Ways to select 2 bonds out of 17 = 17C2 = 17*16/2P(Exactly 2 defaults) = (17*16/2)*(0.00508)2*(1-0.00508)15 = 0.325%

Q. 25 observation are taken from a sample of known variance. Sample µ =70 and population σ = 60. You wish to conduct a two - tailed test of null hypothesis that the mean is equal to 50. What is most appropriate test statistic?Ans. Standard Error of mean (σx) = σ/√(n)= 60/√25 = 12Degrees of freedom = 24Use t- statistic = (x - µ)/ σx = (70 - 50)/12 = 1.67

•Known pop. Var. σx= σ/ √(n)•Unknown pop. varsx= s/ √(n)

Central limit theorem

As Sample Size increases Sampling Distribution Becomes Almost Normal

regardless of shape of population

Null hypothesis:H0

Alternative

Hypothesis: Ha

One tailed testTwo tailed

test

Actually tested Hypothesis

Hypothesis that the researcher wants to

reject

Concluded if there is significant evidence to reject

H0

Test if the value is greater than or less than K

H0;µ<=K vs. Ha: µ>K

Test if the value is different from K

H0; µ=0 vs. Ha: µ≠0

Z & T test P- value 2 Mean Test

H0: µ1 = µ2 vsHa: µ1≠µ2

If n <30 and unknown σ, use t -Test

Given H00.250.25

Hypothesis

Testing

Correlation Coefficient (CC)

Only the linear correlation ,-1 < CC < 1, Appropriate Test structure:

Simple Linear

Regression

Regression

coefficient

Coefficient of

Determination( R2)

%age of total var. in Y explained by XLR model: Yi=b0+b1Xi+Ei

Y = Dependent variable, estimated

Correlation & Regression

Residual Diagnostic

Inference Based on

Sample Data

Real State of Affairs

H0 is True H0 is False

H0 is True Correct decision Confidence level = 1- α

Type II error

P (Type II error) = β

H0 is False Type I error Significance level = α*

Correct decision

Power = 1-β

*Term α represents the maximum probability of

committing a Type I error

Type 1 error: rejection of H0 when it is actually true

Type 2 error :Fail to reject H0 when it is actually false

Given H0

true, Prob. of obtaining

value of test statistic at least as

extreme as the one that was actually observed.

Q. If standard deviation of a normal population is known to be 10 and the mean is hypothesized to be 8. Suppose a sample size of 100 is considered. What is the range of sample means in which hypothesis can be accepted at significance level of 0.05?Ans sx = σ/√n = 10/√100 =1z = (x-µ)/ σx

= (x-8)/1

At 95% -1.96<z<1.96 ; So 6.04<x<9.96

Q. A stock has initial price of $100. It price one year from now is given by S = 100 x exp(r), where the rate of return r is normally distributed with mean of 0.1 and a standard deviation of 0.2. What is the range of S in an year with 95% confidence?

Ans 100e(0.1-1.96*0.2) < S < 100e(0.1+1.96*0.2)

74.68 < S < 163.56

Do not reject H0 Reject H0

α

χχχχ2αααα

χ2

H0: σ2 ≤ σ02

HA: σ2 > σ02

Upper tail test:

F

α/2

Fα/2Reject H0Do not

reject H0

H0: σ12 – σ2

2 = 0HA: σ1

2 – σ22 ≠ 0

Tests for a SinglePopulation Variances

Tests for a twoPopulation Variances

Chi-Square test F test

H0: σ2 = cHA: σ2 ≠ c

2

22

σ1)s(n −

=χ

H0: σ12 – σ2

2 = 0HA: σ1

2 – σ22 ≠ 0

22

21

ss

F =

Hypothesis Tests

for Variances

μc-μn=$1,000

Reject H0

α= 0.025

0

0.05

0.1

0.15

0.2

-10 -5

α= 0.025

Reject H0 Do not reject H0

$19,000

Critical value

0

0.05

0.1

0.15

0.2

- -5

-1 < CC < 1, if CC = 0, X & Y are uncorrelated rx,y = cov(x,y)/σxσy=√R2

Appropriate Test structure: H0:b1=0; Ha:b1≠0Test: tb1=(b^1-b1)/sb^1

Decision Rule:

reject H0 if t>+t critical or if t< -tcritical

R2 =( SSR / SST )=1-( SSE / SST)=explained variation/total variation

Yi = Dependent variable, estimated value of Yi, given value of Xi

Xi = independent variable b0 =intercept term; represents Y if X =0b1 = slope coefficient; measures change in Y for 1 unit change in X

The error variable must be normally distributed,The error variable must have a constant varianceThe errors must be independent of each other.

www.edupristine.com© Pristine Pages 1 of 6

ω =Weighted long run variance= γVL

VL=Long run avg. variance= ω/ (1-α-β)

α+β+γ=1

α+β<1 for stability so that γ is not -ve

2

1

2

1

2

−− ++=nnn

u βσαωσ

Q. GARCH model is estimated as follows:

On a particular day ‘t’; actual return was -1% & the std. deviation estimate was 1.8%.

Calculate the volatility estimate for next day (t+1) and long-term average volatility.

222

1 85.012.0000005.0ttt

σµσ ++=+

EWMA GARCH Implied Volatility

Q.Using a daily RiskMetrics EWMA model with a decay factor λ = 0.95 to develop a forecast of the conditional

variance, which weight will be applied to the return that is 4 days old?

Ans. The EWMA RiskMetrics model is defined as ht = λ*ht-1 + (1- λ)*rt-1. For t=4, and processing r0 through the

equation three times produces a factor of (1-0.95)*0.953 = 0.043 for r0 when t =

The implied volatility of an option contract is the volatility implied by the market price of the option based on an option pricing model.

Where,

λ=Persistence factor/Decay Factor

1- λ= Reactive factor

2

1

2

1

2)1( −− −+=nnnuλλσσ

Volatility Estimation

Distribution of Possible Future Values

Common Starting Point

Drift

Shocks Shocks

Time

Geometric Monte Carlo

Monte Carlo Simulation

Financial markets

Forward and Future

OptionsFixed Income

Securities

Repo rate is the rate at which the banks can borrow money from the central bank of the country in order to avoid scarcity of funds.LIBOR is a daily reference rate based on the interest rates at which banks offer to lend unsecured funds to other banks in the London wholesale money market.The n-year zero coupon rate is the rate of in terest earned on an investment that starts today and lasts for n years.The Yield Curve describes the yield differential among treasury issues of differing maturities.

It involves option pricing, factor affecting option pricing, American option, European option, put-call-parity, volatility smiles , Greeks, and Exotic option

Future & Forward Prices

Day count conventions

Foreign Currency Risk

Actual/actual: T-bond30/360: US corporate & municipal bond.Actual/360: T-bills & other market instrument

EB

C

Time value of money

A

Swap

D

Hedging using futures

Basis Risk: Arises out of two reasonsa) The properties of the underlying under the

contract and the asset to be hedged are different

b) The maturity date of the future contract is different than the date at which asset is to be sold or bought

Basis = Spot price to asset to be hedged -Futures price of the contract

Strengthening of Basis = Basis increase is good for short hedge and bad for long hedge

Weakening of basis = Basis declines is good for long hedge and bad for short hedge

Optimal hedge Ratio:

Where σS is the σ of spot price changes; σF is the σ of futures price changes; ρ is the correlation btw Spot & future pricesHedging with Futures: (β * P)/ A where P is the value of the portfolio , A is the value of the assets underlying one futures contract

A net long (short) currency position means a bank faces the risk that the FX rate will fall(rise) versus the domestic currency.

a) On-Balance Sheet hedging: matched maturity and currency foreign asset-liability book.

b) Off-Balance Sheet hedging: enter into a position in a forward contract

Q. A bank has a USD50mn portfolio available for investing. Cost of funds for the USD50mn is 4.5%. The bank lends 50% of the assets to domestic customers at an loan rate of 6.25%. The rest of the portfolio is lent to UK clients at 7%. The current exchange rate is USD1.642/GBP. At the same time, the bank sells a forward contract equal to the expected receipts one year from now. The forward rate is USD1.58/GBP. What is the weighted average return to the bank .

Ans. The return from domestic customers is

F

S

F

FSCovh

σσρ

σ*

),(2

* ==

Commodities

FB

of (1-0.95)*0.953 = 0.043 for r0 when t = 4.

Ans. Volatility estimate for next day

VL = .017%, Also, variance estimate for t+1 = .000005 + 0.12*(-1%)^2 +

0.85*(1.88%)2 = 0.0317%

Volatility (std. deviation) estimate for t+1 = sqrt(0.0317%) = 1.782%

Ans. Long Term Volatility

In the GARCH model, 12% is the weight given to latest squared return (reactive factor). 85% is the

weight given to latest variance estimate (persistence factor). Therefore, 1-0.12-0.85 = 3% is

weight given to long-term average Volatility.

Therefore, 3%*VL = 0.000005 i.e. VL = 0.017%

Geometric Brownian Motion

dSt=µtStdt + σtStdz

St=asset pricedSt=infinitesimally small price changesµt=constant instantaneous drift termσt=constant instantaneous volatilitydz=normally distributed random variable

•Technique that converts uncertainties in input variables of a model into probability distributions•Combining the distributions and randomly selecting values from them, it recalculates the simulated model many times and brings out the probability of the output.

Monte Carlo Simulation

Q. The price of a 91-day T-bill is 8%. Find the dollar amount of interest paid over the 91 day period and the corresponding rate of interest.

Ans: Dollar in terest is $100*0.08*91/360= $2 .0222Rate of interest = 2.022/(100-2.0222) = 2.064 %

market instrument

Q. Under which scenario is basis risk likely to exist?

a) A hedge (which was initially matched to the maturity of the underlying) is lifted before expiration.

b) The correlation of the underlying and the hedge vehicle is less than one and their volatilities are unequal.

c) The underlying instrument and the hedge vehicle are d issimilar.

d) All of the above are correct.Ans. D

Q. Current S & P 500 future is 1,167 and the manager wants to reduce the Beta from 1.20 to .85. Value of the portfo lio is $5 mnand the index multiplier is 250, the stock index futures position taken is : Ans. Short 6 contracts

Q. In August a fund manager has $10 million invested in a portfolio of government bonds with a duration of 6.80 years and wants to hedge against interest rate moves between August and December. The manager decides to use Dec T-bond futures. The futures price is 93-02 or 93.0625 and the duration of the cheapest to deliver bond is 9.2 years. The number of contracts that should be shorted is:Ans:

Ans. The return from domestic customers is 6.25%, The return from UK customers, $25,000,000/1.642 = GBP 15,225,335GBP*1.07 = GBP 16,291,108.The bank sells a forward contract: GBP 16,291,108*1.58 = USD 25,739,951

Earnings (USD 25,739,951 - 25,000,000)/ 25,000,000 = 2.96%Weighted average return = 6.25%*0.5 + 2.96%*0.5 = 4.61%

7920.9

80.6

50.062,93

000,000,10=×

FC

FA

Bond Yields:

Coupon yield: Coupon payment (C) divided by the face value = C / F Current yield: Coupon payment (C) divided by the bond price = C / P0.

Yield to maturity: (YTM) is the discount rate which returns the market price of the bond, is also called IRR.

( )∑ ++

+=

T

Tt YTM)(1

F

YTM1

CMP

A

PVof a CF is: CT/(1+y)T

FV of the bond is: PV*(1+y)T

If the Rate is Semi-annual the PV is : CT/(1+ys/2)2T.

PV of an Annuity: C/y where y = YTM

Continuously compounded interest ratesRc=m ln(1+ (Rm/m))Rm=m(e Rc/m -1)where Rc : continuously compounded rate; Rm: same rate with compounding m times per year.Equivalent annual yield = [1+r/n] n

- 1 ; Or er - 1

Bond PriceC = coupon payment T = Time to maturity r = interest rate/required yield F = value at maturity,/par value

Bond portfolio structure

B

Barbell : manager uses bonds with short and long maturityBullet : manager buys bonds

( )∑= +

++

=T

1tTt0

r)(1

F

r1

CP

Financial Markets and Products

Ways to Terminate a Commodity Swap1) Physical Delivery2) Square off position by entering into a

contract of equal size but an opposite position

3) Enter into an Exchange for Physical (EFP) agreement

4) Alternative Delivery Procedure

Law of One Price: Two assets (with the same liquidity, tax rates etc) that have the same cash flows, should have the same price.

C-STRIPS: Zero Coupon STRIPs derived from the coupon cash flows in a bond

Net Exposurei

= (FX Assetsi – FX Liabilitiesi) + (FX Boughti – FX Soldi)

= Net Foreign Assetsi + Net FX Bought

Retiring of Corporate Bonds before Maturity

1) Call and refunding provision1) Fixed Cost Call Provision2) Make-Whole Call Provision

2) Sinking FundsBonds are retired periodicallyAccelerated Sinking Funds grant the issuer to retire

FA FB FC FD

( )∑= ++1t

Tt YTM)(1YTM1

Q.A Fixed income instrument offers annual payment of $90 for 10 years. The current Value of the instrument is $ 950. Calculate YTM on this security.

Ans. Use Financial calculator , N=10, PMT = 90, PV = -950; CPT = 1/Y = 9.81%

Q. The EAY of a loan with a quoted rate of 8%, compounded quarterly is equivalent to the EAY of a loan with a continuously compounded quoted rate of:

Ans. For quarterly compounded rate , EAY = (1+ 0.08/4)^4 - 1 = 0.824For continuously compounded loan, we want to find the value of Rc that solves = ln(1.0824 ) = 0.0792

When Bond sells at a discount:YTM > coupon yield.

When Bond sells at a premium:coupon yield > YTM.

When bond sells at par:YTM = coupon yield.

YTM: Bond prices go down when the YTM goes up and vice-versa.Term to maturity – long maturity bonds have greater price volatility than short maturity bondsSize of coupon – low coupon bonds have greater price volatility than high coupon bonds

Clean and dirty price

Duration & convexity

Q. A US corporate bond (30/360 days convention with 10% coupon pays semiannually on Jan 1 and July 1. Assume that today is April, 1, 2005 and the bond matures on July, 1, 2015. Compute the Dirty price and Clean Price of the bond, if the required annual yield is 8%.

Ans. Use Calculator , N=21, PMT = 50, 1/Y= 4, FV=1000,CPT = PV = 1,140.29; Then 90 days later , on April, 1, 2005, the DP = 1,140.29 *(1.04)^.5 = $1,162.87CP = DP - AI (1000*.1*.25) = $1,162.87- $25 = 1,137.87

Clean price : Bond price without accrued interest

Dirty price : Includes accrued interest; Flat price (Clean price) = Full price ( Dirty price)- Accrued Interest

BA

Bullet : manager buys bonds concentrated in the intermediate maturity rangeIf a bullet and a barbell have the same duration, the barbell portfolio have greater convexity and is related to the square of maturity

Q. Which of the following is TRUE? i. A barbell portfolio will have a smaller

convexity than a bullet portfolio with the same duration

ii. The duration of a zero-coupon bond will be greater than the duration of a coupon bond of the same maturity.

iii. Duration and convexity are based, respectively, on the first and second derivatives of price with respect to yield.

iv. Convexity increases with the square of a bond's duration.

a) I and II. b) II and Ill. c)III only. d) I, III, and IV

Ans. B

Principal Only strips : receives principal payments; sold at d iscount; increase in value as prepayment increases; inverse relationship with interest rates

Interest Only strips: receives interest payments; investor want prepayments to be slow; positive relationship with interest rates.

Types of Bonds:

Call feature/bond – allows the issuer to redeem /pay-off the bond prior to maturity, usually at a premium

Retractable bonds – allows the holder to sell the bonds back to the issuer before maturity

Extendible bonds – allows the holder to extend the maturity of the bond

Sinking funds – funds set aside by the issuer to ensure that the firm is able to redeem the bond at maturity

Convertible bonds – can be converted into common stock at a pre-determined conversion price

Zero coupon Bond – does not pay any coupon during the tenure of the bond

High Yield Bond – low rating high risk bond with relatively high yield

Inflation Linked Bonds – allows the holder to mitigate risk against inflation

FD

Value of a Bond with an Embedded Option =

Option Free Bond Cost

+ Value of embedded Option

4) Alternative Delivery Procedure P-STRIPS: Zero Coupon STRIPs derived from the principal cash flow in a bond

Bonds lying above the yield curve are cheap.Bond lying below the yield curve are rich.You should buy cheap securities and sell rich securities.

The effect of the premium or discount of the bond prices decreases as the maturity date approaches. This is called Pull to Par

Boughti

Where, i is the ith currency.

Forex Volatility

Dollar Loss/Gain in a currency

= [Net exposure of foreign currency in dollars] * volatility of the $/foreign currency exchange rate

Interest Rate Paritiy

1 + rdom = F * [1+rfor]/S

Rates are expressed as the domestic/foreign exchange rate

Nominal Interest Rate Decomposition

Ri = rri +iei

Nominal Interest Rate = Real Interest Rate + Inflation Rate

Purchasing Power Parity

Accelerated Sinking Funds grant the issuer to retire more bonds3) Maintenance and Replacement Funds4) Redemption through sale of assets5) Tender Offers – not mentioned in the bond’s indenture

A poison put is an option given in a bond’s indenture to redeem the bond at par in case of a corporate restructuring.

Issuer Default Rate =

Number of Issuers that Default

Total Number of Issuers at the Beginning of the year

Dollar Default Rate =

Cumulative $ value of all defaulted Bonds

(Cumulative $ value of all issuance) X (weighted avg. no of years outstanding)

Cumulative annual Default Rate =

Cumulative $ value of all defaulted Bonds

Cumulative $ value of all issuance

Recovery Rate =

Market price at the time of default

Par value

Lease Rate

Where r is the effective annual interest rate

Contango:A market where the future prices are trading above spot prices.The forward curve is upward sloping.The lease rate is less than the risk free rate.

BackwardationA market where future prices trade below spot prices.The forward curve is downward slopingThe lease rate is more than the risk-free rate

Open InterestOpen interest is the number of long or short future contracts outstanding that have not been squared off.

When a company has a series of dates that face price risk, it can use:1) Strip HedgeUse many futures contacts each with a maturity that matches those dates2) Stack Hedge (Stack and Roll)Use the near-by most liquid futures contract and roll over at that contact’s maturity.

Pages 2 of 6

T Bill: The cash Price is: (100 - Y)* 360/n where Yis the Quoted Price(QP)in the market.

T-Bill Futures quoted price Z= 100 - (360/n)* (100 - Y)

The T-Bill futurescash invoice price=10,000[100 - (n/360)*(100 - Z)]

T Bond Cash price =QP +AI

T Bond Futures price = (quoted futures price)×(conversion factor) +AI

Eurodollar Future: is a future based on Eurodollardeposits. Contract Price = 10000*[100 - 0.25*(100 - Q)]

Cheapest-to-deliver ( CTD)

The party with the short position will have an option to deliver the CTD bond, a CTD bond is for which the following is the least:Quoted spot price - (Quoted futures price x Conversion factor)

CAD

-One party pays fixed and other pays depending on the floating reference rate (LIBOR is the reference rate)-At inception, the value of a swap is 0.-After inception, the value for the swap is the difference b/w the PV of the remaining fixed & floating rate payments.

V swap to pay fixed =Bfloat-Bfiixed

V swap to receive fixed =Bfixed- Bfloat

Exchange payments in 2 different currencies; payment can be fixed or floating ; If a swap has a positive value to one counterparty that party is exposed to credit risk.

Q. A bank entered into a 5-year $150 million annual-pay LIBOR-based interest rate swap three years ago as the fixed rate payer at 5.5%. The relevant discount rates (continuously compounded) for 1 year and 2-year obligations are currently 5.75% and 6.25%, respectively. A payment was just made. The value of the swap is closest to:

Ans. Fixed rate Coupon = $150* 0.055 = $8.25 million; Bfixed = 8.25e-0.0575 + 158.25e-0.0625*2 = $147.44Bfloating = $150 million; Value of Swap = $150 million - $147.44;

Q. Which of the following swap positions can be used to transform a floating-rate asset into a fixed-rate asset?

a) Receive the floating-rate leg and receive the fixed-rate leg of a plain vanilla interest-rate swap.

b) Pay the fixed-rate leg and receive the floating-rate leg of a plain vanilla interest-rate swap.

c) Pay the floating-rate leg and pay the fixed-rate leg of a plain vanilla interest-rate swap.

d) Pay the floating-rate leg and receive the fixed-rate leg of a plain vanilla interest-rate swap.

Ans. D

Co. Fixed Rate Floating RateA 4% L + 20B 5% L + 60In this example, Company A has absoluteadvantage in fixed and floating rate.Assume B & A wants to raise money in a fixedand floating rate respectively, However,comparatively B has to pay 1% higher than A onfixed rate but only 0.4% higher than A onfloating rate. Therefore B has comparativeadvantage in raising loan on floating rateinterest and A in fixed rate.

Comparative advantage Interest rate swap Currency Swap

Q. Which of the following regarding the payoff of a 1-period risky swap to a risk-free counterparty paying a fixed amount and receiving a variable is (are) TRUE?

i. The payoff profile is the same as a short position on a put option and a long position on a call.

ii. The payoff profile is the same as a long position on a put option and a short position on a call.

iii. If the correlation between the risky counterparty and the variable payment declines, the potential payoff is unaffected.

iv. If the correlation between the risky counterparty and the variable payment declines, the potential payoff is affected.

Q. Suppose the σ of short rate changes is 1.2%. Find the forward rate when the 8-year eurodollar futures price quote is 94. The time to maturity is 8 yrs, whereas the maturity of the rate underlying the futures is 8.25 years. Find the convexity adjustment and hence the forward rate.Ans:Forward rate = [6% pa/360 (qtrly)]/

BA

Duration

•Maturity increases, duration increases; •Coupon increases, duration decreases ; •Yield decreases, duration increases.Zero coupon bond :The duration is equal to the bond’s term to maturity. Therefore, the longest durations are found in stripped bonds or zero coupon bonds. These bonds have the greatest interest rate elasticity.

Characteristics of Duration

Convexity

Macaulay duration : Weighted average term to maturity of a bond’s cash flows.Modified duration (D*) : In case of n times compounded yield, the Macaulay Duration is not valid anymore & Modified duration is used

where r is the yield to maturity of the bond, and n is the number of cash flows per year.

DV01: Dollar value of basis point is the absolute change in the bond price from one basis change in yield DV01 = price at YTM0 - price at YTM1|

Q. Royal Bank has a $25 million par position in a 5-year, zero-coupon bond that has a market value of $19,059,948.

Convexity : Second derivative of the price/yield relationship. Price change for larger interest rates estimated by duration and convexity are more precise since convexity can capture the curvature

The convexity relationships imply that a larger price increase occurs with a yield decrease than a price decrease associated with an identical yield increase

Q. Evaluate, at the same yield , the investment that is expected to have the greatest convexity is

a. 10 year zero- coupon bondb. Portfo lio with a duration of 10 yrs that

contains a 5 year and a 15 year zero- coupon bond

c. 6% coupon bond of 10 year durationd. Callable 6% coupon bond of 10 year duration

Ans. B

Duration : First derivative of the price/yield relationship; the longer (shorter) the duration, more (less) sensitive the bond’s price to change in interest rates; can be used for linear estimates of bond price changes.

Q. Using a semiannual compounding, compute DV01 for 10 yrs, 5%

Duration hedging

Q. Ceteris paribus, the duration of a bond is positively correlated with the bond'sA. time to maturity. B. coupon rate.C. yield to maturity. D. all of the above.Ans. A

Q. Given the time to maturity, the duration of a zero coupon bond is higher when the discount rate isA. higher. B. lower. C. equal to the risk free rate.D. independent of the discount rate.

Ans. Duration of the Zero Coupon Bond is its term to maturity.

Q. Holding other factors constant, which one of the fo llowing bonds has the smallest price volatility?

nr1

DurationMacaulayD

*

+=

∆y*BV*2

)BV(BVDurationEffective

0

∆y∆y +− −=

2

0

0

y)(*P*2

2PPPionApproximatConvexity

∆−+

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Convexity*2

y)(∆y*D

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∆P 2

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Q .A fund manager has $10 mn invested in a portfolio of government bonds with a duration of 6.80 years and wants to hedge against interest rate moves between Aug and Dec. How many Dec T-bond futures should manager use.

Ans: The futures price is 93-02 or 93.0625 and the duration of the CTD bond is 9.2 years

= (10,000,000*6.8) / (93,062.5*9.2) = 79.

Q. If the quoted price for the June 2006 Eurodollar futures contract is 96.89, the value of one contract is closest to

Ans. $992,225

Value of Swap = $150 million - $147.44; declines, the potential payoff is affected.

a) I only. b) II only. c) II and III. d) I and IV. Ans. D

Q. Bank One enters into a 5-year swap contract with Mervin Co. to pay LIBOR in return for a fixed 8% rate on a nominal principal of $100 million. Two years from now, the market rate on 3-year swaps at LIBOR is 7%; at this time Mervin Co. declares bankruptcy and defaults on its swap obligation. Assume that the net payment is made only at the end of each year for the swap contract period. What is the market value of the loss incurred by Bank One as result of the default?

a. $1.927 million b.$2.245 million c.$2.624 million d.$3.011 million

Ans. C; At the new swap rate, the replacement cost on the swap is $1 million a year discounted at 7% for each of the 3 years, which is $2.624 million.

Ans:Forward rate = [6% pa/360 (qtrly)]/ [365/90*log(1+.06/4)] - ½*0.0122 *8 *8.25 = 5.563 %

that has a market value of $19,059,948. The modified duration of the bond is closest to:

Ans. YTM of the bond is , PV = -19,059,948, N= 10, PMT = 0, 5.5.%. Macaulay duration is equal to maturity for a zero-coupon bond, so modified duration = 5/(1+0.055/2) = 4.866 years. Please note: while inserting PV type it as a negative value.

Ans. B

Q. A bond has effective duration of 7.5 and a convexity of 104, if the yield rise by 82 bps, the price of the bond will:

Ans. % price change = [-duration *∆y*100] + [(1/2)*convexity*∆y^2 *100] =Decrease by 5.8%

DV01 for 10 yrs, 5% bond that is yielding 4.5%.

Ans. Price at 4.49%: N=10*2, PMT = 5/2, 1/Y= 4.49/2, FV=100, CPT = PV Price at 4.5%: N=10*2, PMT = 5/2, 1/Y= 4.5/2, FV=100, CPT = PV .

Please note: The PV is always negative value.

Duration Hedging:1. Hedge ratio: [DV01 (per $100 of in itial position)* beta]/ DV01 of

hedging instrument). 2. Hedge ratio: P*DP/(FC*DF);

where P = portfolio value; DP = Duration of Portfolio; FC = Future position with a contract; DF = Duration of future contract.

bonds has the smallest price volatility?a. 5-year, 20% coupon bond ; b. 5-year, 12% coupon bond c. 5 year, 14% coupon bond d. 5-year, 0% coupon bond

Ans. A. Higher the sensitivity of the bond to its interest rate, higher the volatility.

C

Futures Contracts:Agreement to buy or sell an asset for a certain price at a certain time. Its is traded on an exchange.

Forward Contracts : Forward contracts are similar to futures except that they are traded Over the Counter (OTC)

Spot rate: A t-period spot rate is the Yield to

For non yielding asset : f = (F0 - K) ert

For continuous yielding underlying : f = S0e-dt - Kert

For discrete dividend paying stock :f = S0 - I - Kert

K is delivery price in a forward contract F0 is forward price that would apply to the contract today

Investment AssetF0 = S0 ert, for non yielding assetF0 = S0 e(r-d)t , continuous dividend paying stockF0 = (S0 - I) ert , discrete dividend paying stock,

Foreign ExchangeF0 = S0 e(r-rf)t , Currency Forward CommodityF0 = S0 e(r-δ)t , for commodity with lease rentalsF0 = (S0 + M) ert, commodity with storage costs

Definitions Forward ValuationForward Pricing Interest Rate Futures

CA

E

Factors affecting an option price

Option price bounds

Rules for exercising American Option

Put Call Parity

Binomial Option pricing

Black Scholes-Merton Model

Option trading

strategies

Volatility smiles

P+S0=C+Xe-rT

Q. Consider a 1 year European call option with a Strike price of $27.50 that is currently valued at

Upper bound European/American call: c <=S0 ; C<=S0

Upper bound European/American put: p<= Xe-rT ; P<=XLower bound European call on a non dividend paying stock c >= max(S0-Xe-rT,0)

Currency option: implied volatility is lower for ATM option than it is for away from the money option.

Equity option: have volatility skew, the volatility decreases as

EDEC

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Forward Rate Agreement (FRA) is an agreement to pay or receive a certain rate between two future dates :

FRA = N(F0 - Ft) (T2 - T1) e -r(T2- t)

FRA = Value of the FRA to receive fixed rate at time tN = Notional Amount ; Ft = Forward rate between time T2 and T1 quoted at time tF0 = Rate between time T2 and T1 quoted at time 0r = Risk free rate

Arbitrage1. If F0>S0erT,borrow loan, buy spot, sell forward today, deliver asset,

repay loan at the end2. If F0<S0erT,Short sell the asset, invest the proceeds at risk-free rate,

buy forward today, collect loan buy asset under forward contract, deliver to cover short sale.

Q. FRA that settles in 30 days, $1 million notional, based on 90-day LIBOR, Forward rate of 5%, Actual 90-day LIBOR at settlement is 6%Ans. ∆I = (6% - 5%) * (90/360)* $1m = $2,500PV= 2,500 / (1 + (90/360)*6%) = $2,46

Spot rate: A t-period spot rate is the Yield to Maturity of on a Zero Coupon Bond that matures in t-years.

Forward Rates: Forward rates are interest rate between two dates in future as implied by the spot ratesF21 = (S2T2 - S1T1) / (T2 - T1) , whereF21 = Forward rate b/w time T2 and T1

S1 and S2 = Spot rate for maturity T1 and T2

respectively

Backwardation : Spot price is higher than the future price (high convenience yield compared to the cost i.e. rate of interest) 2nd Graph. Contagovice versa 1st Graph. The cost of carry is the storage cost plus the interest costs less the income earned

0 0

F0 = S0 e(r+λ-c)t , commodity with convenience yields

Consumption CommoditiesF0 <= (S0 + M) ert

Where r = annual interest rate, t = Time period d= % of annual dividend I = the PV of dividend received. rf = foreign currency domestic risk free rateM is the PV of storage costs δ= lease rate (cost of borrowing the commodity)c= % annual convenience yield (CY is the benefit of owing the consumable asset)λ= % annual storage cost

Short selling involves selling securities that is not owned.

Q. Current 1-year forward exchange rate is 1.200 USD /EUR. An American bank pays 2.4% annual interest rate on a 1-year deposit and a 4.0% annual interest rate on a 3-year USD deposit. A European bank pays a 1.5% annual interest rate for a 1-year deposit and a 2.0% annual interest rate for a 3-year EUR deposit. The forward exchange rate in USD per EUR for exchange 3 years from today is closest to:

Ans. The 2 year forward rate in US = √ [(1.04) ̂ 3 / 1.024] - 1 = 4.81%The 2 year forward rate in Europe = √ [(1.02)^3 / 1.015] - 1 = 2.25%The forward exchange rate in USD per EUR for exchange three years: 1.2 *(1.0481^2) / (1.0225^2) = 1.261

Forward Rate Agreements

Q. If 1 & 1.5 years’ spot rates are 1.8% and 2.2%, the 6-month forward rate on an investment that matures in 1.5 years is :

Ans. (2.2%*1.5 - 1.8%*1)/0.5 = 3%

Q. If the current USD/AUD rate is 0.6650 (1 AUD=0.6650 USD) and the risk-free rates for the USD and AUD are 1.0% and 4.5% respectively, what is the lower bound of a 5-month European put option on the AUD with a strike price of 0.6880?Ans. Lower bound = 0.6880 x [exp-(0.01x 5/12)] -0.6650 x [exp-(0.045x 5/12]]= 0.6880 x (0.9958) -0.6650 x (0.9814)=0.04

that is currently valued at $4.1 on a $25 stock. The 1 year risk free rate is 6%. Which of the following is the closest to the value of the corresponding put option

Ans: p=c + D-S0 + Xe –rt

=5

Q. The stock price is $25. A put option with a $20 strike price that expires in 6 months is available. N(-d1) = 0.0263 and N(-d2) = 0.0349. If the underlying stock exhibits an annual σ of 25%, & the continuously compounded risk-free rate is 4.5%, the BSM value of the put is:

Ans. p = Ke-rt N(-d2) - SoN(-d1) = $0.03

stock c >= max(S0-Xe ,0)Lower bound European put on a non dividend paying stock p >= max(Xe-rT-S0,0)

skew, the volatility decreases as the strike price increases, i.e. volatility used to price a low-strike-price option is higher than that used to price a high-strike-price option

EA

Q. According to Put Call parity for European options, purchasing a put option on ABC stock will be equivalent to

a) Buying a call, buying ABC stock and buying a Zero Coupon bond.

b) Buying a call, selling ABC stock and buying a Zero Coupon bond.

c) Selling a call, selling ABC stock and buying a Zero Coupon bond.

d) Buying a call, selling ABC stock and selling a Zero Coupon bond

Ans B

N(d1) is the delta of the option and therefore S0N(d1) represents the current price of delta. N(d2) is the probability that a call option will be exercised, 1- N(d2) is the probability that a put option will be exercised,

Variable c p C P

S0 + - + -

K - + - +

T ? ? + +

σ + + + +

r + - + -

D - + - +

Pages 3 of 6

ED

Covered call Protective Put Bull spread Bear spreadButterfly spread

Calendar spread

Long straddle StrangleStrips & Straps

Long stock plus short call

Long stock plus long put,

Purchases call option with low Strike price, & subsidized the purchase with sale of a

call option with a higher Strike price

3 different options. Buy 1 call with low exercise price, another with high exercise

price, & short 2 calls with an exercise price in b/w. Butterfly buyer is betting the

stock will stay near the price of the written calls

Similar to straddle except purchased option is out-of-the

money, so it cheaper to implement. Stock price have to move more to

be profitable

Bet on volatility, buy a call & a put of same exercise price & expiration

date. Profit is earned if stock price has a large change in either direction

EB

Early Exercise:• It is never optimal to exercise an American call on a non-dividend paying stock before its expiration

date• American Put can be optimally exercised early if they are sufficiently in-the money• An American call on a dividend paying stock may be exercised early if the dividend exceeds the

amount of forgone interest

EC

Binomial Pricing: At each step, it is assumed that the underlying instrument will move up or down by a specific factor (u or d) per step of the tree. So, if S is the current price, then in the next period the price will either be : Sup = S*u and Sdown = S *d; where U=e√σ and D=1/U. At each final node of the tree i.e. at expiration of the option the option value is simply its intrinsic, or exercise value. The following formula is applied at each node: European Option Payoff = [ p x Option up + (1-p) x Option down] x exp - r t

Q. Early exercise of an option is more likely for: i. European calls options on stocks paying large dividends. ii. American call options on stocks paying small dividends.iii. American call options close to maturity. iv. American put options on stocks paying large dividends

a) I and IV. b) II and IV. c) III only. d) III and IV. Ans. C

Q: Assume that a binomial interest-rate tree indicates a 6-month period spot rate of 2.5%, and the price of the bond if rates decline is 98.45, and if rates increase is 96. The risk-neutral probabilities respectively associated with a decline and increase in rates if the market price of the bond is 97 correspond to: Ans. [p*98.45 + (1-p)*96] / [1+ (0.025/2)] = 97

Delta Theta Gamma Vega Rho

Delta: estimates the change in value of an option for a unit change in stock price.A delta of 0.5 means that price of a call option will change by $.5 for $1 change in value of the stock.Call options delta :0 for deep out of money; 0.5 for at the money; 1 for deep in the money, Put option delta :- 1 deep in the money, - 0.5 for at the money and - 0 for deep out of money.

Delta of a option = δc/δs

Delta of a forward position is equal to 1 Delta of Future= ert

Theta: Time decay, most negative when option is at the money & close to expiration Theta is negative because as time passes the value of both calls and puts decreases..

Gamma: rate of change in delta as underlying stock price change ( also Convexity); largest when option is ATM, which indicates that option price changes very fast as Stock price changes. ITM options and OTM options have low gammas.As the maturity nears, the option gamma increases.Fixed coupon bonds, have positive convexity. Positive gamma is beneficial, it implies that value of the asset drops more slowly and increases more quickly. Long positions in options, ( calls or puts) , create positive gamma

Rho: sensitivity of the option price to changes in the Risk free rate. Largest for ITM option “ITM” calls and Puts are most sensitive to changes in the rates than “OTM”

Vega: sensitivity of the option price to changes in the volatility of the underlying stock, highest for long-term ATM options. close to 0 when option is deep ITM or OTM; Vega decreases with maturity.

EE

Q. True or Falsea.Theta affects the value of a call and a put in similar way. TrueQ. An existing option short position is delta neutral, but has a -5,000 gamma

Q. An investor is looking to create an options’ portfolio on XYZ stock that will have virtually zero Vega exposure while maximizing the ability to profit from increases in interest rates. If the current price of XYZ is $50, which of the following would accomplish his goals? I. Sell a call with Strike price (SP) 50 II. Buy a call with a SP of 25.III. Sell a put with a SP of $75. IV. Buy a put with a SP of $25.

a) I only. b) II and III.c) II and IV. d) III and IV.

Ans B

Delta Neutral Hedging:•To completely hedge a short call position , purchase no. of shares of stock=delta*no. of options sold.•Only appropriate for small changes in the value of underlying asset•Gamma can correct hedging error by protecting against large movement in asset price•Gamma neutral positions are created by matching portfolio gamma with an offsetting option position.

Q. At the money options close to the Long stock plus long put, also called portfolio

insurance

Strike price

2 options with different expiration. Sell a short dated option & buy a long dated option. Investor profits if stock price stays in a narrow range.

Purchase call with high strike price, short call with low strike price. Bear spread with puts involves buying put with high exercise price and selling put with exercise price . Both the option has same expiry date in bull and bear Spreads

written calls

Strip : when downside move is expected to be more likely, buy two puts and one call of same strike and expiry. Strap : when upside move is expected to be more likely , buy two calls and one put of same strike and expiry.

Q. An investor constructs a strap and buys 2 calls with Strike price of $40 at $3 each and One put with Strike price of $40 at $2. If the price of the underlying asset is $46, what is the Profit/Loss on the position? Ans. 6

Q. Long on a Call and short on a put on the same stock with higher strike price and same maturity is calleda. Calendar Spreadb. Butterfly Spreadc. Bear Spreadd. Bull SpreadAns. D

Q. In a butterfly spread� Both the upside & downside are unlimited� Both the upside and downside are limited� The upside is unlimited, but downside is limited� The upside is limited, but downside is limitedAns. B

a.Theta affects the value of a call and a put in similar way. Trueb.Theta is more pronounced when the option is “in the money”. Falsec.Theta usually decreases in absolute terms as expiration approaches. Falsed.It is possible for a European put option that is “in the money” to have a positive theta value. Truee.Rho for fixed income is small. falsef.Call option delta range from 0 to 1. Trueg.A Vega of .1 suggests that for 1% increase in volatility, the option price will increase by $.10. Trueh.Theta is the most negative for OTM options. falsei.Options are most sensitive to changes to volatility, when they are “At the money”. True

Q. An existing option short position is delta neutral, but has a -5,000 gamma exposure. An option is available that has a gamma of 2 and a delta of 0.7. What actions should be taken to create a gamma neutral position that will remain delta neutral? a.Go long 2,500 options and sell 1,750 shares of the underlying stock. b.Go short 2,500 options and buy 1,750 shares of the underlying stock. c.Go long 10,000 options and sell 1,750 shares of the underlying stock. d.Go long 10,000 options and buy 1,750 shares of the underlying stock

Ans: A, - Gamma means we are short on options, to create a gamma-neutral portfolio (5000/2) = 2,500 long option. However this will change the position from delta-neutral portfolio to 2,500*.7 = 1,750 long delta. So sell 1,750 shares to be gamma and delta neutrality.

Q. At the money options close to the maturity tend to have a higha.Rho b. Gamma

c. NPV d. VegaAns B

Q.Which of the following have –Deltaa.Strangle b. Straddle

c.Bear Spreadd. Bull Spread

Ans C

Foundation of Risk Management

Sources of Risk Tools for Risk ManagementRisk Management & value

creation Performance Measurement

Derivatives is the most popular tool used by Risk Managers for RM. Other tools include: •Stop-loss limit: Limit on the amount of losses in a position.•Notional Limit: Maximum amount to be invested in a asset.•Exposure limit: Exposure to risk factors like

By handling bankruptcy costs :∆ (Expected Value of firm) = ∆ (Present Value of firm) + ∆ (PV of bankruptcy costs) – Risk management cost.

Firms can use risk management to move their income across time horizon and reduce tax burden.

Business risk: Specific for the business house. Ex: Increase in the prices of cement for a construction company

Financial Risk: result of a firm’s financial market activities; volatility in various market related

Treynor Ratio: Is the excess return divided by per unit of market risk( Beta) in an investment asset [E(RP)-RF]/βp

Sharpe Ratio: Is the excess return divided by per unit of total risk in an investment asset: [E(RP)-RF]/σp, where

Rp = portfolio return, Rf = risk free return

Sortino Ratio (SR): Excess return divided by Semi

Risk & return Portfolio

Security market line (SML) & CAPM

- Expected return :E(RP)=∑n

i=1W iE(R i)- Variance for 2 asset portfolio σ2=w1

2σ12+w2

2σ22+2w1w2

ρ1,2σ1σ2

- Investors will only be compensated systematic risk since Unsystematic risk can be diversified.

Markowitz efficient frontier

- Combinations along the EF (Efficient Frontier) represent portfolios (explicitly excluding the risk-free alternative) for which risk for a given level of return is lowest

-Systematic risk (non-diversifiable risk or beta) : individual security’s risk that arises because of the positive covariance of the security’s return with overall market return’s. Beta (βa ) = Cov (ra, rp)/Var(rp)-Unsystematic risk (diversifiable risk): part of the volatility of a single security’s return that is uncorrelated with the volatility of the market portfolio.

Beta

Foundation of Risk Management

Capital Market Line (CML)

Alpha: measure of assessing an active manager's performance as it is the return in excess of a benchmark index.�αi < rf: the manager has destroyed value �αi = rf: the manager has neither created nor destroyed value �αi > rf: the manager has created value The difference αi − rf is called Jensen's alphaJensen’s α excess return of a stock, over its required rate of return as determined by CAPM: α = Rp – Rc ; where Rp= portfolio return, Rc = return predicted by CAPM

•Exposure limit: Exposure to risk factors like duration for debt instruments & Beta for Equity Investments.•VaR: maximum loss at given confidence level.

tax burden.

Reducing WACC: Also we can reduce the tax outgo by increasing interest outgo, but expected financial distress / bankruptcy costs because of leverage hamper the firm value beyond a level.

By Reducing The Probability Of Debt Overhang: Debt Overhang refers to situation where the amt of debt the firm is carrying prevents the shareholders from investing in +ive NPV projects

By Reducing The Problem Of Information Asymmetry: Information Asymmetry results in two problems:•Investors have to rely on mgmt estimates for profitability of new projects•Extent to which the performance is due to management decisions or external factors

in various market related instruments Ex: The recent market crash.

Types of Financial Risk

Market Risk: risk of value decrease due to change in prices of assets in the market.

Liquidity Risk: risk of not being able to quickly liquidate a position at a fair price.•Asset Liquidity: Large positions affecting asset prices. •Funding liquidity: Inability to honor margin calls, capital withdrawals. Ex: Lehman.

Credit risk: risk of loss due to counterparty default.•Sovereign Risk: Willingness and ability to repay.•Settlement Risk: Failure of counterparty to deliver its obligation•Exposure & recovery rate: Calculated on the happening of a credit event.

Operation risk: risk due to inadequate monitoring, systems failure, management failure, human error.•Model risk, people risk, legal and compliance risk

Sortino Ratio (SR): Excess return divided by Semi standard deviation(SSD) which considers only data points that represent a loss. More relevant when the distribution is more skewed to the left. (Rp – MAR) / SSD, MAR is minimum accepted return, Higher the SR, lower is the risk of large losses

Tracking error(TE) : TE = σEp (Std. dev. of portfolio’s excess return over Benchmark index) ; Where Ep = Rp –Rb ;Rp = portfolio return, Rb = benchmark return –Lower the tracking error lesser the risk differential between portfolio and the benchmark index TE Volatility(TEV) = ω = √ σA

2 - 2* ρAB* σA* σB+ σB

2

Relative Risk W= ω *PInformation ratio: is defined as excess return divided by TE. E(RP)-E(Rb)/TE

Q. Last 4 years, the returns on a portfolio were 6%, 9%, 4%, & 12%. The returns of the benchmark were 7%, 10%, 4%, & 10%. The minimum acceptable return is 7%. What is the portfolio's SR?0.4743

Q. Value of portfolio =100, Portfolio return σp = 25%Portfolio benchmark σB = 20%Correlation , ρPB =0.961Calculate TEV Ans: ω = √0.25^2 + 0.20^2-2*0.961* 0.25* 0.20 = 8%Relative risk = 8%*100 =8

Q.PV (Before edging) qqqq.qqq.Probability$200 0.10$300 0.20$400 0.30$500 0.40Debt $300Bankruptcy Cost $75PV (after hedging) Prob$200 0.00$300 0.25$400 0.30$500 0.45

Ans Debt value=probability*expected payment to debt i.e. 10% * 125+ 90% *300 = 282.5Equity value = probability * expected payment to equity i.e. 30% * 100+ 40%*200=110, Thus EV=392.5 2. If Hedging cost is 10 & after hedging PV are also shown as aboveDebt value =probability * expected payment to debt i.e. 100% *300=300Equity value= probability *expected payment to equity i.e. 0.30%*100+ 45%* 200 =120, Thus EV=420–10=4103. Incremental benefit=410-392.5=17.5

ρ1,2σ1σ2

- Correlation: ρ (X,Y) = cov(X,Y)/(σX *σY)- Lower the correlation greater the benefits from diversification

Q. E(RA) = 10%, σA = 20%, E(RB) = 10%, σB = 20%. Assume the weights to be 50 % for A & B. Calculate portfolio returns when :Case 1 : ρAB = 1 ; Case 2 : ρAB = 0,Case 3 : ρAB = -1Case 1. (0.5^2)*(0.2^2)+ (0.5^2)* (0.2^2)+2*0.5*0.5*0.2*0.2*1 =0.04Case 2 : (0.5^2)*(0.2^2)+ (0.5^2)* (0.2^2)+2*0.5*0.5*0.2*0.2*0 =0.02Case 3 : (0.5^2)*(0.2^2)+ (0.5^2)* (0.2^2)+2*0.5*0.5*0.2*0.2*-1 =0.00

can be diversified.- SML: indicates a return an investor should earn in the market for any level of Beta risk.-The equation of the SML is CAPM (return & systematic risk equilibrium relationship-CAPM: E(Ri)=RF+β i[E(Rmkt)-RF]- [E(Rmkt)-RF] is the risk premium

return is lowest- Risk-free asset has 0 variance in returns ,it is also uncorrelated with any other asset

CML: When a risky portfolio is combined with some allocation to a risk free asset, the resulting risk-return combinations will lie on a straight CML. All points along the CML have superior risk-return profiles to any portfolio on the Efficient FrontierEfficient Frontier

Ris

k f

ree

re

turn

Efficient Frontier

CML

SML

Rf

Efficient-market hypothesis: it is impossible to consistently outperform the market by using any information that the market already knowsThe three forms of market efficiency–weak-form efficiency : future prices cannot be predicted by analyzing price from the past–semi-strong-form efficiency : prices adjust to publicly available new information very rapidly and in an unbiased fashion–strong-form efficiency : prices reflect all information, public and private, and no one can earn excess returns

Efficient frontier: The optimal portfolios plotted along the curve have the highest expected return possible for a given amount of risk.

Fama And French Three Factor Model:•A factor model that expands on the capital asset pricing model (CAPM) by adding size & value factors in addition to the market risk factor in CAPM.•This model considers the fact that value and small cap stocks outperform markets on a regular basis. r = Rf + beta3 x ( Km - Rf ) + bs x SMB + bv x HML + alpha

The Arbitrage Pricing Theory (APT) : APT model points towards the relationship between factors and expected returns.CAPM is a special case of APT with only one factor exposure: market risk premium.Rn = ΣXn.k * bk + un ; Rn = Excess return for stock nXn = Exposure of Stock n to factor k ;bk = Factor return for factor k ;un = Stock n’s specific return`

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Case Studies

Types of Risk Management Failure

LTCM BaringMetallgesellschaft (MRM) Sumitomo

1. Risk metrics failure. Ex: MRM & LTCM2. Incorrect measurement of known risks. Ex: MRM &

LTCM.3. Ineffective risk monitoring. Ex: Barrings & Sumitomo4. Ineffective risk communication 5. Ignorance of significant known risks. Ex: MRM &

LTCM.6. Unknown risk.

•Yasuo Hamanaka - copper trader at Sumitomo manipulated copper prices on London Metal Exchange.•Fall in copper prices in June 1996 after revelation of Hamanaka’s unfair dealings led to ~2.6bn USD loss for Sumitomo•Positions were so large that company could not liquidate them completely •Hamanaka used his independence to trade in the market on behalf of the company and manipulated the copper prices by buying physical copper in large quantities and storing in the warehouse thereby creating lack of copper in the market •He sold put options to collect the premiums as he thought he can push the prices up & thus writing put options was not risky for him•Though, he never imagined that he could be susceptible to steep decline of copper prices•It had various risk exposures q.such as Operational Risk, Employee/ People Risk, Liquidity Funding Risk, Market Risk

•Nick Lesson, trader at Baring PLC, took concentrated positions

•It used Stack and roll hedging strategy•In 1991, it offered fixed price contract for supplying gasoline for 5 to 10 years. In order to hedge MG took long positions in near month futures and rolled the stack into next near month contract every time by decreasing the trade size gradually so as to match the stack with pending short position (in long term supply contracts).•MG bought futures on NYMEX to offset its forward commitments exposure with hedge ratio of one (every barrel was hedged). •As these derivatives were short-term thus MRM had to roll them forward every month-end or term-end till 5-10 years or the contract’s end. •Company was exposed on rising spot prices. It eventually lost more than USD 1.5bn in 1993.•It had various risk exposures q.such as Basis Risk, Market Risk, Funding Liquidity Risk.•LTCM was a hedge fund using highly leveraged

arbitrage trading activities in fixed income in addition to pairs trading. Before failing in 1998, it had given spectacular returns in 1995-97 periods (upto 40% post-fees). Post Russian default on its ruble denominated debt, LTCM lost more than 4bn USD in 4 months. •LTCM used proprietary mathematical models to engage in arbitrage trading in U.S., Danish, Russian, European and Japanese Govt. bonds.

Q. Which of the following reasons does not help explain the problems of LTCM in August and September 1998?

1. A spike in correlations2. An increase in stock index volatilities

Value at Risk

Value at risk

Factors affecting portfolio risk

Linear vs Nonlinearderivatives

Project & CF VAR

EWMAModel

GARCH Estimation Model

Methods of VAR calculation

Risk budgeting

Cash Flow at Risk: It is a measure of the expected cash flow at loss beyond a confidence level. If beta (β) of an asset is βX with the portfolio then the cash flow at risk (CFAR) = βX * CFAR of portfolio.

Project VAR: when considering a new project, you can explicitly calculate the dollar cost of the increase in CFAR and include it as an additional cost of the project.

σn2=ω+αr2n-1+β σ2n-1

Implicitly assumes variance reverts to a long run average level

Sum of (α+β) <= 1 for the model to be stationary

Q. Let h t be the variance at t and r2(t-1) the squared return at t - 1. Which of the following GARCH models will take the shortest time to revert to its mean?a. h t = 0.02 + 0.06r 2(t-1) + 0.93h t-1

b. h t = 0.03 + 0.04r 2(t-1) + 0.94h t-1c. h t = 0.04 + 0.05r 2(t-1) + 0.95h t-1

d. h t = 0.05 + 0.01r 2(t-1) + 0.96h t-1

Ans. (D) The speed of mean reversion is defined by α1 + β, which is lowest for d, it is .97

σn2= λ*σ2n-1+(1-λ)r2n-1

Where λ = weight on previous volatility and (1- λ) weight on squared return

Q. The current estimate of daily volatility is 1.5 %. The closing price of an asset yesterday & today are $30 and $30.5 respectively. Using the EWMA model with λ = 0.94, the updated estimate of volatility is?Ans. 1.5096

Assets concentrationAssets volatility Assets correlation Systematic risk

Q. A trader has an allocation equal to8% of the firm’s capital; the beta oftrader’s return with the return of thefirm is .90. The contribution of thetrader to the Firm’s VAR of $120million is:

Relationship b/w an underlying factor and the derivative’s value are linear in nature

A C

Q.A firm with existing projects have expected cash flow of $100 mn and cash flow volatility of $60 mn. New project with a cost of $30 mn and cash flow volatility of $20 mn. The correlation between two cash flows is 0.3. Calculate the volatility of the firm’s projects with new projects at 95% confidence level and the additional project cost due to the increased cash flow volatility, if the cost of cash flow volatility is $0.12.Ans. δ projects = sqrt (602 + 302 + 2*(.3)* 60*20) = $68.7 mnCFAR (at 5%) existing = 1.65*60 = $99 mnCFAR (at 5%) with new project = 1.65*68.7 = $113.4 mnThe additional project cost due to increased cash flow volatility is: ($113.4 mn - $68.7 mn)*.12=$1.73 mn

Q: Which of the following is false? a. EWMA approach of Risk Metrics is a particular case of a GARCH process.b. GARCH allows for time-varying volatility.c. GARCH can produce fat tails in the return distribution.d. GARCH imposes a positive

D

Types of VaR

B

VaR for Linear and Non Linear Derivatives1. Linear Assets: When the value of the delta is constant for all changes in the underlying. Example: Forwards,

futures.

Delta (1st derivative or duration in bonds) can be used to estimate the VAR for linear derivatives. The delta-normal approach (generally) does not work for portfolios of nonlinear securities. VAR Linear Derivative = Delta * VAR Underlying risk factor

•Nick Lesson, trader at Baring PLC, took concentrated positions Nikkei 225 derivatives for bank in Singapore International Monetary Exchange (SIMEX). He took arbitrage positions on Nikkei derivatives on different exchanges viz. Osaka, Tokyo & SIMEX. •Lesson was solely responsible for back & front office operations of Singapore. He used an error account hide his losses by fraudulently transferring funds to & from his error accounts•He kept on selling straddles on Nikkei futures with an assumption that Nikkei is under-priced. He took double long exposure on the same index from different exchanges. •He kept on building his positions even after Nikkei kept on falling, however after Jan’95 earthquake, he could not sustain his positions & failed to honor the margin calls•It eventually led to the collapse of Barings bank, when it was sold to ING for mere $1.60 only•It had various risk exposures qsuch as Operational Risk, Market risk, Employee/People risk

Russian, European and Japanese Govt. bonds. In 1998, LTCM’s positions were highly leveraged (1:28) with ~ USD 5: 130 billion of equity and assets.•LTCM’s model assumed maximum volatility of 20% annually. Based on its models, it was expected to losses more than ~500 million USD in once in 20 months.•It had its bet on convergence of Russian & American G-sec yield, which however diverged after Russian default.. Its failure led to a huge bailout by large commercial & merchant banks under the guidance of Federal Reserve•It had various risk exposures q.such as Model Risk, Funding liquidity risk, Sovereign Risk, Market Risk.

2. An increase in stock index volatilities3. A drop in liquidity4. An increase in interest rates on on-the-run

TreasuriesAns: D, An increase in interest rates on on-the-

run Trasuries

Q. A 6 month call option with a strike price of $10 is currently trading for $1.41, the market price of the underlying stock is $11. A 1% decrease in the stock to $10.89 results in a 6.35% decrease in the call option with a value of $1.32. If the annual volatility of the stock is s = 0.1975 and the risk free rate of return is 5%, calculate the 1 -day 5% VAR for this call option.Ans. The daily volatility is = 1.25% (0.1975 /√250); VARstock(5%) = 1.65 *1.25%= 2.06%;Delta of the call = 0.0635/.01 = 6.35 ; VARcall = ∆ VARstock = 6.35*2.06% = 13.1%,

million is:a. $7.8 mn b) $8.6mn c)$9.6 mn d) $10.8mnAns:.08*.9*120 million = 8.64 million,

Q. A bond of $10 mn, with modified duration of 3.6 yrs and annualized yield of 2%. calculate the 10 day holding period VaR of the position with 99% confidence interval, assuming there are 252 days in a year.Ans. VAR = $10,000,000* 0.02*3.6* [√10/ (√252)]* 2.33 = $334,186

d. GARCH imposes a positive conditional mean return.Ans. D

2. Non Linear Assets: When the value of the delta keeps on changing with the change in the underlying asset.

Examples: Options, Credit Derivatives, Swaps.

Taylor Approximation: large changes can be explained by the 2nd derivative i.e. gamma expected change in the delta of an option( or convexity in bonds).Taylor approximation is ineffective for callable bonds & mortgage backed securities.

How to measure VAR

•VaR (daily VaR) (in %) = ZX% * σ

- ZX% : the normal distribution value for the given probability (x%) (normal distribution has mean as 0 and standard deviation as 1)

Value at Risk (VaR) has become the standard measure that financial analysts use to quantify this risk.VAR represents maximum potential loss in value of a portfolio of financial instruments with a given probability over a certain horizon.

Example: The daily 5% VAR is $10,000, it indicates that there is only 5% chance that on any given day, the portfolio will experience a loss of $10,000 or more.

Value At Risk

A

VAR

Undiversified VAR: sum of the individual VARs for each risk factor. It assumes that all prices will move in the worst direction simultaneously, which is unrealistic.

Marginal VAR is the change in VaR of the portfolio with one unit change in the

Incremental VAR : The change in VAR from the addition of a new position in a portfolio.

Component VAR is the Amount a portfolio VAR would change by deleting either of the assets from a portfolio = DVAR *β

Types of VAR

Diversified VAR: accounts for diversification effects.DVARP=z* std dev* portfolio value

B

σport =√ wa2 σa

2 + wb2 σb

2+2wawb* σa* σb* correlation (a,b)

VaRport (daily VaR) (in %):=√ (wa

2 (%VaRa)2 + wb2 (%VaRb)2+2wawb*(VaRa)*(%VaRb)* σab)

$ VAR portfolio = √($ VARa2 + $ VARb

2 +2$ VARa *$ VARb* σ a,b )

VAR of uncorrelated positions: VAR portfolio = √ (VAR12 + VAR2

2 )

Q. If the assets has a daily σ of returns equal to 1.4% and asset has a current value of $5.3 mn, calculate the VAR ( 5%) on both percentage & dollar basis.

Ans. Z5%*σ = 1.65* 1.4% = 2.31%, and 0.0231* $5,300,000 = $122,430

Q. A portfolio is composed of 2 securities. Calculate VAR at 95% confidence level. Investment in security A & B are USD 1.5mn and 3 mnrespectively.Volatilityof security A & B are 7% & 3% respectively. Correlation A & B is 10%

Ans. σ portfolio = √(1/3)2 (7%) 2 + (2/3) 2 (3%) 2 + 2*(1/3)*(2/3)*10%* 7%*3% = 0.0316VAR = 1.65 * 0.0316 * 4,500,000 = 234,630

Q.If the value of stock is 100 and the value of the put option at 110 is 20. 10 units change in the underlying brings in change of 4 units change in the option premium. If the annual volatility is 0.25. Calculate daily VaR at 97.5% assuming 250 days?

Delta = 0.4 STDEV(annual) =0.25Days = 50 daily STDEV= 0.015811Z at 97.5% =1.96Options Value = 20 units VAR for option = 0.247923 units

deviation as 1)- σ : standard deviation (volatility) of the asset (or portfolio)

•VAR (X %) dollar basis = VAR (X %) * asset value•VAR for n days using 1day VAR : VAR(X%)n-days= (VAR(X%)1-days)*√n

loss of $10,000 or more.

VAR Benefits:

•Aggregatesall the risks in a portfolio into a single numberProvides an approach to arrive at economical capital.•Relates capitalwith the expected losses•Scaled to time

Mean μ=0

Approximately Normal Curve Representing VAR

The area under the normal curve for confidence value is:

unrealistic.VARP=√(VAR1

2+VAR22+2VAR1VAR2)

=VAR1+VAR2

one unit change in the components= DVAR *βA /portfolio value

a portfolio. assets from a portfolio = DVAR *β A

* weight of asset A. value=√(VAR1

2+VAR22)

Q.Weight of asset A & B are 0.6 & 0.4 in a portfolio. The value of the total portfolio is USD1 million and its σ is 0.060606; if the betas of asset A and asset B are 1.3 and 0.8 respectively, the respectively. What is the MVAR of Asset B and CVAR of Asset B at a 95%.Ans: DVAR =1.95*0.060606*1,000,000 = 99,999.90MVAR = 99,999.90*.8/1,000,000 = $0.08CVAR = 99,999.90*.8 *0.4 = $32,000.

Q. A portfolio has an equal amount invested in X and Y. The expected excess return of X is 9% and that of Y is 12%. The MVAR are 0.06 and 0.075 respectively. What should manager do to move towards the optimal portfolio?Ans. The Expected excess Return ratio for X and Y are 1.5 and 1.6 respectively. Therefore portfolio weight in Y should increase to move the portfolio towards the optimal portfolio.

Risk Budgeting

Risk Budgeting involves choosing and managing exposure to risk, 1st step is to determine the total amount of risk, as measured by VAR, Next is the optimal allocation of assets for that risk exposure.

Q. A Fund has $200 mn in assets and $180mn in liabilities. Expected return on the surplus, scales by assets is 4%, i.e. surplus is expected to grow by $8 mn over 1st year. The volatility of surplus is 10%. Use Z =1.65, what is the deficit with the loss associated with the VAR.Ans: Surplus = (200 - 180 ) = $20 mn, expected to grow by $8 mn to a value of $28 mn; VaR = 1.65* 20* .1 = $33 mnThe deficit is: ( 33 - 28) = 5 mn

Risk Budgeting with Active Mangers: is done using Tracking error ( Active Returns - benchmark return) & Information ratio (TE / volatility of managers TE) Weight of portfolio managed by manager i= IRi*(portfolio’s tracking error volatility)/

IRi*(manager’s tracking error volatility)

Q. Determine the optimal weight ratioTE vol Ratio IR

Manager A 5% .70Manager B 5% .50Benchmark 0% 0Portfolio 3% .82Ans. A=51%, B=37% and remaining 12% in benchmark

Funding Risk: is the risk that the value of the assets will not be sufficient to cover the liabilities of the fund

D

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MC Simulation: The price returns are subjected to simulation using certain Simulation Models to generate a set of random numbers which are mapped to particular statistical distributions and hence the tail events are calculated to arrive at the VaR; Pitfalls: Model risk

C

Historical Simulation: simply re-organizes actual historical returns, putting them in order from worst to best. Assumes history repeats itself.Pitfalls: Time variation in risk, unusual events

Delta-Normal or Variance-Covariance Method:assumes that the portfolio exposures are linear and that the risk factors are jointly normally distributed (ND); VAR(X%)=zx%*σ•Pitfall: non linearity , fat tails underestimate the occurrence of large observations because of its reliance on a ND

RiskMetrics approach is similar to the delta-normal approach. The only difference is that the risk factor returns are measured as logarithms of the price ratios, instead of rates of returns.

Local valuation: for Linear derivativesFull valuation: to take into account nonlinear relationships

Q: If you use delta-VAR for a portfolio of options, which of the following is always correct?a. It necessarily understates the VaR because it uses a linear approximation.b. It can sometimes overstate the VaR.c. It performs most poorly for a portfolio of deep ITM options.d. It performs most poorly for a portfolio of deep-OTM options.Ans: B, The delta-VAR could understate or overstate the true VAR, depending if the position is net long or short options, it is generally better for ITM options, because these have low gamma and for OTM options, delta is close to zero, so the delta-VAR would predict zero risk.

Q: Under usually accepted rules of market behaviour, the relationship bw parametric delta-normal VAR and historical VAR will tend to bea. Parametric VaR will be higher.b. Parametric VaR will be lower.c. It depends on the correlations.d. None of the above are correct.Ans: B, parametric VAR at high confidence levels will generally underestimate VAR

Q. Delta-normal, historical simulation, & MCS are methods available to compute VAR. If underlying returns are normally distributed (ND), then a. Delta-normal method VAR will be identical to the HS VAR.b. Delta-normal method VAR will be identical to the MCVAR.c. MCVAR will approach the delta-normal VAR as the number of replications increases.d. MCVAR will be identical to the HS VAR.Ans: C, In finite samples, the HS VAR will be in different from the delta-normal method, as the sample size increases, they converge when the returns are ND.

Worst Case Scenario (EVT): focuses on

Q. Calculate VAR for an S&P 500 futures contract using the HS approach. The current price is 935 and the multiplier is 250. The historical price data for the previous 300 days, What is the VAR of the position at 99%. Returns: -6.1%,-6%,-5.9%,-5.7%, -5.5%, -5.1%..........4.9%, 5%, 5.3%, 5.6%, 5.9%Ans: The 99% return among 300 observations would be the 3rd worst observation i.e. 5.9%; Therefore (935)*250* (0.059) = $13,791.

Q. Which of the following is NOT a drawback to stress testing?Worst Case Scenario (EVT): focuses on

distribution of worst possible outcomes given in an unfavourable event, expected loss is then determined from distribution. •EVT stresses on the extreme value.•EVT is calculation of the tail events & captures expected value of the fat-tail.•The expected shortfall is mean of the observations exceeding VaR value.•Back Testing: process of testing a trading strategy on prior time periods.

Stress testing: VAR tells the probability of exceeding a given loss but fails to incorporate the possible amount of a loss that results from an extreme event. Stress testing complements VAR by providing information about the magnitude of losses that may occur in extreme market conditions

Q: Which of the following methods would be most appropriate for stresstesting your portfolio?a. Delta-gamma valuationb. Full revaluationc. Marked to marketd. Delta-normal VARAns: B

to stress testing?a. Calculated losses may be extremely high relative to the 99% VAR significance level.b. Historical correlations mix normal and hectic periods.c. It identifies important factors not observed in historical data.d. The number of scenarios increases greatly with additional risk factors.Ans: C

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