Mind Map

VisualizeFRM-Part IQuantitative Analysis Probability Distribution AB Moments Prob. & Prob. distribution Sampling Hypothesis Testing Correlation & Regression Volatility Estimation Monte Carlo Simulation Discrete Continuous Continuous uniform distribution 68% of Data Normal Distribution (ND) Standardized RV is normalized mean = 0, σ = 1. Z-score: # of σ a given observation is from population mean. Z=(x- )/σ Mean Variance Skewness Kurtosis Probability Binomial Poisson AB •Outcome only between [a, b] •P(outside a & b) = 0 -4 -3 -2 -1 95% of Data 99.7% of Data 0 1 2 3 4 Mean: ∑x i/n Avg. of squared deviations from mean Sample variance s 2 =[∑ni=1 (Xi -Xmean ) 2]/(n-1) Population variance σ2 =[∑Ni=1 (Xi - ) 2]/N Mode:-Value that occurs most frequently •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 Properties Counting principles Sum rule and Bayes’ Theorem Only 2 possible outcomes: failure or success. P(x)=nCx *p x(1-p) n-x Fix the expectation λ=np. P(x)=λ x e-λ/x! if x>=0 P(x)=0 otherwise Median:-Midpoint of data arranged in ascending order Standard deviation = √Variance Var(ax+by)=a2Var(x)+ b2Var(y)+2abCov(x,y) • No. of ways to select r out of n objects: nC = n!/[r!*(n-r)!] r • No. of ways to arrange r objects in n places: nP =n!/(n-r)! r Cumulative density function (cdf) for Uniform distribution : F(x)=0 For x<=a F(x)=(x-a)/(b-a) For a<x<b F(x)=1 For x>=b If Z is a standard normal R.V. An event X is defined to happen if either -1< Z < 1 or Z > 1.5. What is the prob. of event X happening if N (1) =0.8413, N (0.5) = 0.6915 and N (-1.5) = 0.0668, where N is the CDF of a standard normal variable Binomial Random Variable E(X)=n*p Var(X)=n*p*(1-p)=n*p*q P( B ) = P( A ∩ B ) + P ( Ac ∩ B) P ( B ) = P ( B / A) * P ( A) + P ( B / Ac ) * P( Ac ) P(B/A) = P(A/B) * P(B) P(A/B) * P(B) + P(A/B c ) * P(B c ) 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. σ2 of return of stock P= 100.0 σ2 of return of stock Q=225.0 Cov (P,Q) =53.2 Current Holding $1 mn in P. New Holding: shifting $ 1 million in Q and keeping USD 3 million in stock P. What %age of risk (σ), is reduced? Ans. σ P =√[w2σA 2 + (1-w)2 σB 2 +2w(1-w)Cov(A,B)] w= 0.75 c 2 = 100*(0.75) 2 + 225*(0.25) 2 +2*0.25*0.75*53.2 σP = 9.5 old σ = √100 = 10 Reduction = 5% Q. If distributions 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 normal, which implies greater kurtosis. 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=PDQ P(No Def Year) = (1-PDQ)4 P(No Def Quarter) = (0.9)4 = 97.4% 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) = (λ x e-x )/x! X= 2.1, x = 4 P(2.1) = 0.1 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 -1 +1 1.5 Ans The sum of areas shown in two figures Area 1 = 1-2*(1- N(1)) = 1-2*(0.1587) Area 2 = 0.0668 , Total Area = 0.7514 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)/σ A = (100-80)/10 = 2 Sampling Central limit theorem 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 SE (σx ) of the sample mean is σ of the dist. of sample means •Known pop. Var. σ x = σ/ √(n) •Unknown pop. var s x = s/ √(n) • 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. 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 = 0.00508 Ways to select 2 bonds out of 17 = 17C2 = 17*16/2 P(Exactly 2 defaults) = (17*16/2)*(0.00508)2*(1-0.00508)15 = 0.325% As Sample Size increases Sampling Distribution Becomes Almost Normal regardless of shape of population Q. 25 observation are taken from a sample of known variance. Sample =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 (σ x ) = σ/√(n) = 60/√25 = 12 Degrees of freedom = 24 Use t- statistic = (x - µ)/ σx = (70 - 50)/12 = 1.67 Hypothesis Testing Correlation & Regression Alternative Hypothesis: Ha Con cluded if there is significant evidence to reject H0 Null hypothesis:H0 One tailed test Two tailed test Z & T test P- value 2 Mean Test Hypo thesis that the research er wan ts to reject Actually tested Hypothesis Test if the value is greater th an o r less than K H0; <=K vs. Ha: >K 0.25 Test if the value is different from K H0; =0 vs. Ha: ≠0 0.25 If n <30 an d un known σ, use t -Test Given H0 true, Pro b. of o btaining value of test statistic at least as extreme as the o ne that was actually observed. Hypothesis Tests for Variances H0: Ha: = 2 vs 1≠ 2 1 Simple Linear Regression Regression coefficient Coefficient of Determination( R2) Correlation Coefficient (CC) Residual Diagnostic Type 1 error: rejection of H0 when it is actually true Type 2 error :Fail to reject H0 when it is actually false -5 0.2 0.15 0.2 0.15 0.1 0.05 Critical value α= 0.025 0.1 0.05 α= 0.025 0 0 -10 -5 Inference Based on Sample Data H0 is True Real State of Affairs H0 is True Correct decision Confidence level = 1- α H0 is False Type II error $19,000 μc-μn=$1,000 Do not reject H0 Reject H0 LR model: Yi =b0+b1Xi +Ei Yi = Dep end ent variable, estimated value of Yi , given value of Xi Xi = independent variable b0 =intercept term; represents Y if X =0 b1 = slope coefficient; measures change in Y for 1 unit change in X Appropriate Test structure: H0:b1=0; Ha:b1≠0 Test: tb1=(b^1-b1)/sb^1 Decision Rule: reject H0 if t>+t critical or if t< tcritical %age of to tal var. in Y explained by X R2 =( SSR / SST )=1-( SSE / SST) =explained variation/total variation Only the linear correlation , -1 < CC < 1, if CC = 0, X & Y are uncorrelated rx,y = cov(x,y)/σ xσy=√R2 The error variable must be normally distributed, The error variable must have a constant variance The errors must be ind ependent of each other. Reject H0 H0 is False P (Type II error) = β Correct decision Type I error Significance level Power = 1-β = α* Q. A stock h as initial price o f $100. It p rice one year fro m now is given by S = 100 x exp(r), where the rate of return r is normally distributed with mean o f 0.1 and a stand ard deviation o f 0.2. Wh at 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 Q. If standard deviation o f a no rmal population is known to be 10 and th e mean is hypothesized to be 8. Suppose a samp le size of 100 is considered. What is the range of sample mean s in wh ich hyp othesis can be accepted at significance level o f 0.05? Ans s x = σ/√n = 10/√100 =1 z = (x- )/ σx = (x-8)/1 At 95% -1.96<z<1.96 ; So 6.04<x<9.96 *Term α represents the maximum probability of committing a Type I error Tests fo r a Sin gle Population Variances Chi-Square test H0: σ2 = c HA : σ2 ≠ c Tests for a two Population Variances F test H0: σ12 – σ22 = 0 HA : σ12 – σ22 ≠ 0 χ2 = (n − 1)s 2 σ2 Upper tail test: F= 2 s1 s2 2 H0: σ2 ≤ σ02 HA: σ2 > σ02 α H0: σ12 – σ22 = 0 HA: σ12 – σ22 ≠ 0 α/2 χ2 Do not reject H0 χ2α α Reject H0 Do not reject H0 Fα/2 Reject H0 F © Pristine www.edupristine.com Pages 1 of 6 When bond sells at par: YTM = coupon yield. The forward curve is upward sloping.AI (1000*.951 .12µt2 + 0. Future & Forward Prices Hedging using futures D GARCH Implied Volatility Common Starting Point FB Shocks Shocks Where.291. VL = 0.1*. Flat p rice (Clean price) = Full price ( Dirty price). Or er .e. EAY = (1+ 0.335GBP*1.291. Assume that today is April. should have the same price.951 Earnin gs (USD 25. sold at d iscount.000005 + 0. The bank lends 50% of the assets to domestic customers at an loan rate of 6. tax rates etc) that have the same cash flows.000 . on the first and second derivatives of price with respect to yield. The price of a 91-day T-bill is 8%. if the required annual yield is 8%. 1. The number of contracts that should be shorted is: Ans: 10 . no of years outstanding) Cumulative annual Default Rate = Cumulative $ value of all defaulted Bonds Cumulative $ value of all issuance Purchasing Power Parity When a company has a series of dates that face price risk. investor wan t p repayments to be slow. A barbell portfolio will have a smaller convexity than a bullet portfolio with the same duration ii. 2005. Backwardation A market where future prices trade below spot prices. Volatility estimate for next day VL = .0222) = 2.162. Cost of fun ds for the USD50mn is 4. 1-0. Th e return from domestic customers is 6. A is the value of the assets underlying one futures contract Q.2 years. PV of an Ann uity: C/y where y = YTM Continuously compounded interest rates Rc =m ln(1+ (Rm/m)) Rm=m(e Rc/m -1) wh ere Rc : continuously compounded rate.0317%) = 1. Monte Carlo Simulation Monte Carlo Simulation Geometric Brownian Motion Actual/actual: T-bond 30/360: US corporate & mun icipal bond.29 *(1. Q. Long Term Volatility In the GARCH model. 3%*VL = 0.Accrued In terest BA Forex Volatility Dollar Loss/Gain in a currency = [Net exposure of foreign currency in dollars] * volatility of the $/foreign currency exchange rate Open Interest Open interest is the number of long or short future contracts outstanding that have not been squared off.50 9 . we want to find the value of Rc that solves = ln(1.782% •Technique that converts uncertainties in input variables of a model into probability distributions •Combining the distributions and randomly selecting values from them. Use Financial calculator . The forward curve is downward sloping The lease rate is more than the risk-free rate Recovery Rate = Market price at the time of default Par value 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. The current Value of the instrument is $ 950.000. variance= ω/ (1-α-β) α+β+γ=1 α+β<1 for stability so that γ is not -ve The implied volatility of an option contract is the volatility implied by the market price of the option based on an option pricing model.85*(1.739. Then 90 days later . c) The un derlying instrument and the hedge vehicle are d issimilar. D Q. b) II and Ill.25.5 + 2. volatility smiles . Duration and co nvexity are based. 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 Interest Rate Paritiy 1 + rdom = F * [1+rfor]/S Rates are expressed as the domestic/foreign exchange rate Q. increase in value as p repayment increases. Bond lying below the yield curve are rich. 1. Which of the following is TRUE? i. 12% is the weight given to latest squared return (reactive factor). Rm: same rate with compounding m times per year. GARCH model is estimated as follows: σ t2+1 = 0. Compute the Dirty price and Clean Price of the bond. You should buy cheap securities and sell rich securities. III.08/4)^4 .000005 i. The return from UK customers. What is the weighted average return to the bank .08*91/360 = $2 .062 . N=21. In August a fun d man ager has $10 million invested in a portfolio of government bonds with a duration o f 6. A US corporate bo nd (30/360 days convention with 10% coupon pays semiannually on Jan 1 and July 1. dSt= tStdt + σ tStdz St=asset price dSt=infinitesimally small price changes µt=constant instantaneous drift term σt=constant instantaneous volatility dz=normally distributed random variable Q. Term to maturity – lon g maturity bond s have greater price volatility than short maturity bonds Size of coupon – low coupon bonds have greater price volatility than high coupon bonds Clean price : Bon d price without accrued interest Dirty price : Includ es accrued interest.8%. and Exotic option 2 n −1 ω =Weighted long run variance= γVL VL=Long run avg.87 CP = DP . deviation estimate was 1. d) I. The n-year zero coupon rate is the rate of in terest earned on an investment that starts to day and lasts for n years.064 % FC Q. LIBOR is a daily reference rate based o n th e in terest rates at which banks offer to len d unsecured funds to other banks in the London wholesale money market. The bank sells a forward contract: GBP 16. Yield to maturity: (YTM) is the discount rate which returns the market price of the bo nd.5%.953 = 0. The rest of th e portfolio is lent to UK clients at 7%. respectively.108.000005 + 0. The manager decides to use Dec T-bond futures.0824 ) = 0.Financial markets Volatility Estimation Time value of money Fixed Income Securities Forward and Future Commodities Options Foreign Currency Risk Swap EWMA Distribution of Possible Future Values A Repo rate is the rate at which the ban ks can bo rrow money fro m the central bank of th e country in order to avoid scarcity of fun ds. Equivalent annual yield = [1+r/n] n .A Fixed income instrument offers annual payment of $90 fo r 10 years.225.81% Nominal Interest Rate Decomposition Ri = rri +iei Nominal Interest Rate = Real Interest Rate + Inflation Rate Contango: A market where the future prices are trading above spot prices. CPT = 1/Y = 9.25) = $1.000.85σ t2 Day count conventions On a particular day ‘t’. Greeks. Calculate YTM on this security.017% Ans. Fin d the dollar amount of interest paid over the 91 day period and the corresponding rate of interest.80 years and wan ts to h edge against interest rate mo ves between August and December. B Where r is the effective annual interest rate A poison put is an option given in a bond’s indenture to redeem the bond at par in case of a corporate restructuring. deviation) estimate for t+1 = sqrt(0. Ans.5 = 4.$25 = 1.80 × = 79 93 .1 Bond portfolio structure Bond Price C = coupon p ayment T = Time to maturity r = in terest rate/required yield F = value at maturity. The lease rate is less than the risk free rate. The forward rate is USD1. Q.20 FA Financial Markets and Products A B PV of a CF is: CT/(1+y)T FV of the bond is: PV*(1+y)T If th e Rate is Semi-annual the PV is : CT/(1+y s /2) 2T. When Bond sells at a premium: co upon yield > YTM.167 and the manager wants to reduce the Beta from 1. iii.87 Q. Convexity increases with the square of a bond's duration. comp ounded quarterly is equivalent to the EAY of a loan with a continuously compounded quoted rate of: Ans. American option. which weight will be applied to the return that is 4 days old? Ans.λ= Reactive factor 2 2 σn =λσ2−1 +(1−λ)un−1 n 2 σ n2 = ω + α u n −1 + βσ Drift C It involves option pricing. 1. 2005 an d the bond matures on July. PV = -950. The futures price is 93-02 or 93.022/(1002.85 = 3% is weight given to long-term average Volatility.824 For continuously compounded loan.000 6 . At the same time.12*(-1%)^2 + 0. it recalculates the simulated model many times and brings out the probability of the output.CPT = PV = 1. the barbell portfolio h ave greater convexity and is related to the square of maturity P0 = ∑ t =1 T C F + (1 + r )t (1 + r) T Bond Yields: Coupon yield: Coupon p ayment (C) divided by the face value = C / F Current yield: Coupon payment (C) divided by the bon d price = C / P0. Use Calculator .96% Weighted average return = 6.0625 and th e duration of the cheapest to deliver bond is 9.017%.000)/ 25.85. d) All of the above are correct.λ)*rt-1. Ans. FD Pages 2 of 6 . B E A net long (short) currency position means a bank faces the risk that th e FX rate will fall(rise) versus the domestic currency. a) I an d II. p ut-call-parity.95 to develop a forecast of the conditional variance. MP = ∑ t =1 T (1 + YTM )t C + F (1 + YTM) T Ways to Terminate a Commodity Swap 1) Physical Delivery 2) 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.1 = 0. σF is the σ of futures price changes. The duration of a zero-coupon bond will be greater than the duration of a coupon bond of the same maturity.29. N=10.0792 Barbell : man ager uses bonds with short and long maturity Bullet : man ager buys bonds concentrated in the intermediate maturity range If a bullet and a barbell have the same duration. The current exchange rate is USD1. 85% is the weight given to latest variance estimate (persistence factor). Short 6 con tracts Q. FA FB FC FD Net Exposurei = (FX Assetsi – FX Liabilitiesi) + (FX Bought i – FX Sold i) = Net Foreign Assetsi + Net FX Bought i Where.25%. a) On-Balance Sheet hedging: matched maturity an d currency foreign assetliability book. European op tion. Calculate the volatility estimate for next day (t+1) and long-term average volatility. A bank has a USD50mn portfolio available fo r investing. Ans: Dollar in terest is $100*0. the stock index futures position taken is : Ans.642/GBP.162. $25.642 = GBP 15. 2015. Actual/360: T-bills & other market instrument Basis Risk: Arises out of two reasons a) The p roperties of th e underlying under the contract and the asset to be h edged are different b) The maturity date of the future contract is different than th e date at which asset is to be sold or bought Basis = Spot price to asset to be hedged Futures p rice of the contract Strengthening of Basis = Basis increase is good for short h edge and bad for long hedge Weakening of basis = Basis declines is good for long hedge and bad fo r sh ort hedge Optimal hedge Ratio: h* = Cov ( S . facto r affecting op tion pricing.000. C-STRIPS: Zero Coupon STRIPs derived from the coupon cash flows in a bond P-STRIPS: Zero Coupon STRIPs derived from the principal cash flow in a bond Clean and dirty price Duration & convexity When Bond sells at a discount: YTM > coupo n yield.739.04)^./par value Principal Only strips : receives p rincipal payments. This is called Pull to Par Value of a Bond with an Embedded Option = Option Free Bond Cost + Value of embedded Option Types of Bonds: Call feature/bond – allows the issuer to redeem /pay-off the bond prior to maturity. For t=4.58/GBP. c)III only.25%*0. The EWMA RiskMetrics model is defined as ht = λ*ht-1 + (1. b) The co rrelation of the underlying and the h edge vehicle is less than one an d their volatilities are unequal. i is the ith currency.Using a daily RiskMetrics EWMA model with a decay factor λ = 0.0317% Volatility (std. λ=Persistence factor/Decay Factor 1. Ans. Ans. PMT = 90. Fo r q uarterly compounded rate . and IV Ans. YTM: Bond prices go down wh en the YTM goes up and vice-versa.108*1.58 = USD 25. b) Off-Balance Sheet hedging: enter into a p osition in a forward contract Time Q.043 for r0 when t = 4.1 . iv.137.000/1. The Yield Curve describes the yield differential among treasury issues of differing maturities. is also called IRR. variance estimate for t+1 = .87. the DP = 1. it can use: 1) Strip Hedge Use many futures contacts each with a maturity that matches those dates 2) Stack Hedge (Stack and Roll) Use the near-by most liquid futures contract and roll over at that contact’s maturity. the ban k sells a forward contract equal to the expected receipts one year fro m n ow. Th e EAY of a loan with a quo ted rate o f 8%. positive relatio nship with interest rates.20 to .140.88%)2 = 0. and processing r0 through the equation three times produces a factor of (1-0. Under which scenario is basis risk likely to exist? a) A hed ge (which was initially matched to the maturity of the underlying) is lifted before expiration. 1/Y= 4.5 = $1. FV=1000.07 = GBP 16. Retiring of Corporate Bonds before Maturity 1) Call and refunding provision 1) Fixed Cost Call Provision 2) Make-Whole Call Provision 2) Sinking Funds Bonds are retired periodically Accelerated Sinking Funds grant the issuer to retire more bonds 3) Maintenance and Replacement Funds 4) Redemption through sale of assets 5) Tender Offers – not mentioned in the bond’s indenture Lease Rate The effect of the premium or discount of the bond prices decreases as the maturity date approaches.95)*0.61% Ans.0222 Rate o f interest = 2. Therefore.000 = 2. Q. Current S & P 500 future is 1.25%.96%*0. Value o f the p ortfo lio is $5 mn and the index multiplier is 250.12-0. Also. Bonds lying above the yield curve are cheap. F ) 2 σF = ρ* σS σF Where σS is the σ of spot price changes. on April. in verse relatio nship with interest rates Interest Only strips: receives interest payments. PMT = 50. Therefore. actual return was -1% & the std.140. ρ is the co rrelation btw Spot & future prices Hedging with Futures: (β * P)/ A where P is the value of the portfolio . 0% annual interest rate for a 3-year EUR deposit.75% an d 6. V swap to pay fixed =Bfloat-Bfiixed V swap to receive fixed =Bfixed. A bond has effective duration of 7. Zero coupon bond :The duration is equal to the bond’s term to maturity.2 *(1. 5% bond that is yielding 4. higher. F C = Future position with a contract. A European bank pays a 1. the longer (shorter) the duration.2) = 79.245 million c.563 % Comparative advantage D Interest rate swap BA Currency Swap Duration Duration hedging Characteristics of Duration Convexity Co.5 = 3% Arbitrage 1.t) FRA = Value of the FRA to receive fixed rate at time t N = Notional Amount . P<=X Lower bound European call on a non dividend paying stock c >= max(S0-Xe-rT. YTM of the bond is . higher the volatility.5%: N=10*2. Therefore.8% and 2. selling ABC stock and buying a Zero Coupon bond. How man y Dec T-bond futures should manager use.0625 and the duration of the CTD bo nd is 9.2 years = (10.04) ^3 / 1.(360/n)* (100 . Q. Assume th at the net payment is made o nly at the end of each year for the swap contract period.0% and 4. If F 0>S0erT.0481^2) / (1. Duration : First derivative of the price/yield relationship. the longest durations are found in stripped bonds or zero coupon bonds.1 on a $25 stock. Its is traded on an exchange.8% Q . deliver asset.045x 5/12]] = 0. Pages 3 of 6 .25e-0.F t) (T2 . b) Buying a call. Lower bound = 0. b) II only.%.4% higher than A on floating rate. to p ay LIBOR in return fo r a fixed 8% rate on a nominal principal o f $100 million. An American bank pays 2. selling ABC stock and buying a Zero Coupon bond.06/4)] . iii.25*( 100 . At the new swap rate. Ans.6650 x (0. PMT = 0. & the continuously compounded risk-free rate is 4. where F 21 = Forward rate b/w time T 2 and T 1 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. 12% coupon bond c. If the underlying stock exhibits an annual σ of 25%. a) I on ly.borrow loan.25% The forward exchange rate in USD per EUR for exchange three years: 1. respectively.CA T Bill: The cash Price is: (100 . collect loan buy asset under forward contract. However.1 = 4.01x 5/12)] -0.e.8%*1)/0. If 1 & 1. The value of the swap is closest to: Ans. Currency Forward Commodity F 0 = S0 e(r-δ)t . yield to maturity. t = Time period d= % of annual dividend I = the PV of dividend received. coupon rate. and n is the number of cash flows per year.948. Macaulay Duration 1+ r n ∆P ( ∆ y) 2 * Convexity = − D m * ∆y + P 2 The convexity relationships imply that a larger price increase occurs with a yield decrease than a price decrease associated with an identical yield increase P+ + P− − 2P 0 2 * P0 * ( ∆ y) 2 Effective Duration = (BV − ∆y − BV + ∆y ) 2 * BV * ∆y 0 DV01: Dollar value of basis point is the absolute change in the bond price from one basis change in yield DV01 = price at YTM0 .25 = 5.500 PV= 2.5% annual interest rate for a 1-year deposit and a 2. What is the market value of the lo ss incurred by Bank One as result o f the default? a.6880? Ans. •Yield decreases. Two years from now. 20% coupon bond .04 Q. •Coupon increases. independent of the discount rate. at this time Mervin Co.000[100 .059.200 USD /EUR. Fixed rate Coupon = $150* 0. A Q.25 million. Hedge ratio: [DV01 (per $100 of initial position)* beta]/ DV01 of hedging instrument). deliver to cover short sale. ii. the volatility decreases as the strike price increases. Value of Swap = $150 millio n . time to maturity.1 = 2.0225^2) = 1. lower. Duration Hedging: 1. the BSM value of the put is: Ans. Ans:Forward rate = [6% pa/360 (qtrly)]/ [365/90*log(1+.Kert For discrete dividend paying stock : f = S0 . the duration of a bond is positively correlated with the bond's A. % price change = [-duration *∆y*100] + [(1/2)*convexity*∆y^2 *100] =Decrease by 5.02)^3 / 1. -After inception. commodity with convenience yields Consumption Commodities F 0 <= (S0 + M) ert Where r = annual interest rate.SoN(-d 1) = $0. $1.948. Assume B & A wants to raise money in a fixed and floating rate respectively. FV=100.5%. If the current USD/AUD rate is 0.81% The 2 year forward rate in Europe = √ [(1. $992. Price at 4. Foreign Exchange F 0 = S0 e(r-rf)t . Find the forward rate when the 8-year eurodollar futures price q uote is 94. B Q.6650 USD) and the risk-free rates for the USD and AUD are 1. Company A has absolute advantage in fixed and floating rate. can be used for linear estimates of bond price changes. a CTD bond is for which the following is the least: Quoted spot price . Find the convexity adjustment and hence the forward rate. Using a semiannual compounding. Which of the following is the closest to the value of the corresponding put option Ans: p=c + D-S0 + Xe – rt d 2 = d1 − σ T Q. Therefore B has comp arative advantage in raising loan on floating rate interest and A in fixed rate.(n/360)*(100 .0) Forward Rate Agreements Q. The relevant discount rates (continuously compounded) for 1 year and 2-year obligations are currently 5.coupon bond b. for commodity with lease rentals F 0 = (S0 + M) ert.I . the potential payoff is affected. CPT = PV . if the yield rise by 82 bps. Fixed Rate Flo ating Rate A 4% L + 20 B 5% L + 60 In this example. Duration of the Zero Coupon Bond is its term to maturity.0263 and N(-d 2) = 0. Given the time to maturity. The payoff profile is th e same as a short position on a put o ption and a long position on a call. which is $2. T-Bill Futures quoted price Z = 100 .44 Bfloating = $150 million. 0% coupon bond Ans.Short sell the asset. -One party pays fixed and other p ays depending on th e floating reference rate (LIBOR is the reference rate) -At incep tion. 14% coupon bond d.03 Forward Rate Agreement (FRA) is an agreement to pay or receive a certain rate between two future dates : FRA = N(F0 . The modified duration of the bond is closest to: Ans. all of the above. which one of the following bonds has the smallest price volatility? a. d eclares bankruptcy and defaults on its swap obligation.5%) * (90/360)* $1m = $2. selling ABC stock and selling a Zero Coupon bond Ans B Q.5% respectively. C<=S0 Upper bound European/American put: p<= Xe-rT .1.5 . E C Definitions Forward Pricing Forward Valuation Interest Rate Futures Factors affecting an option price Rules for exercising American Option Put Call Parity Binomial Option pricing Black ScholesMerton Model Option trading strategies Greeks Volatility smiles Futures Contracts: Agreement to buy or sell an asset for a certain price at a certain time.T1) e -r(T2 .T 1) .9958) -0. Ans. 5-year. the Macaulay Duration is not valid anymore & Modified duration is used •Maturity increases. Bank One enters into a 5-year swap contract with Mervin Co.062.0) Lower bound European put on a non dividend paying stock p >= max(Xe-rT-S0. the investment that is expected to have the greatest convexity is a. The 2 year forward rate in US = √ [(1.225 Q.80 years and wan ts to hedge against interest rate mo ves between Aug and Dec. Callable 6% coupon bond of 10 year duration Ans. The cost of carry is the storage cost plus the interest costs less the income earned Investment Asset F 0 = S0 ert. the replacement cost o n the swap is $1 million a year discounted at 7% fo r each of the 3 years. Q. C. duration decreases .Y)* 360/n where Y is the Quoted Price(QP)in the market. B. These bonds have the greatest interest rate elasticity. 2. so modified duration = 5/(1+0. C. discrete dividend paying stock.015] . Q. D. d) I and IV. Modified duration (D*) : In case of n times compounded yield. Actual 90-day LIBOR at settlement is 6% Ans. 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. 5.4% annual interest rate on a 1-year deposit and a 4.024] .927 million b.5%. Consider a 1 year European call option with a Strike price of $27.0575 + 158. whereas the maturity of th e rate underlying the futures is 8. iv. The time to maturity is 8 yrs.5*9.Y) The T-Bill futures cash invoice price =10. duration increases.e. d) Buying a call. Royal Bank has a $25 million par position in a 5-year.K) ert For continuous yielding underlying : f = S0e-dt . buying ABC stock and buying a Zero Coupon bond.000*6. D Q.2%.25e-0.866 years. A put option with a $20 strike price that expires in 6 months is available.coupon bond c. compute DV01 for 10 yrs. Ceteris paribus. C. the value for the swap is the difference b/w the PV of the remaining fixed & floating rate payments.5/2. the value o f a swap is 0.624 million. duration increases. The 1 year risk free rate is 6%. d) Pay the floating-rate leg and receive the fixed-rate leg of a plain vanilla interest-rate swap. Higher the sensitivity of the bond to its interest rate. Evaluate.055/2) = 4. Please note: while inserting PV type it as a negative value. based on 90-day LIBOR.0% annual interest rate on a 3-year USD deposit. If a swap h as a positive value to on e counterparty th at party is exposed to credit risk.055 = $8.5 years is : Ans.5 and a convexity of 104. Contago vice versa 1st Graph. the duration of a zero coupon bond is higher when the discount rate is A.6880 x (0.6880 x [exp-(0.½*0. According to Put Call parity for European options. D.0625*2 = $147. 10 year zero. For non yielding asset : f = (F0 . 1.Q)] Cheapest-to-deliver ( CTD) The party with the short position will have an option to deliver the CTD bo nd.5. If the correlation between the risky counterparty an d the variable payment declines. the market rate o n 3-year swaps at LIBOR is 7%. buy spot. Current 1-year forward exchange rate is 1. If the quoted price for th e June 2006 Eurodollar futures contract is 96.25 years.6650 (1 AUD=0. the 6-month forward rate on an investment that matures in 1. If the correlation between the risky counterparty an d the variable payment declines. PMT = 5/2. (2. invest the proceeds at risk-free rate.Bfloat Exchange payments in 2 different curren cies. Q. The stock price is $25. Equity option: have volatility skew. Forward Rates: Forward rates are interest rate between two dates in future as implied by the spot rates F 21 = (S2T 2 . Forward rate of 5%. i. Which o f the following regarding th e payoff of a 1-period risky swap to a risk-free counterparty paying a fixed amount and receiving a variable is (are) TRUE? i. repay loan at the end 2. 5-year. PMT = 5/2. what is the lower bound of a 5-month European put option on the AUD with a strike price of 0. A.261 Q. A paymen t was just mad e. more (less) sensitive the bond’s price to change in interest rates. payment can be fixed or floating .25%. for non yielding asset F 0 = S0 e(r-d)t . comparatively B has to pay 1% higher than A on fixed rate but only 0.624 million d.S1T1) / (T2 . Ans: The futures price is 93-02 or 93. N(d 2) is the probability that a call option will be exercised. zero-coupon bond that has a market value of $19.price at YTM1| Q. B.44. Ans. the value of one co ntract is closest to Ans. 1/Y= 4. where P = portfolio value. Contract Price = 10000*[100 .49%: N=10*2.6650 x [exp-(0.89.Z)] T Bond Cash price = QP +AI T Bond Futures price = (quoted futures p rice) ×(conversion factor) +AI Eurodollar Future: is a future based on Eurodollar deposits. Ans. D Macaulay duration : Weighted average term to maturity of a bond’s cash flows.N(d2) is the probability that a put option will be exercised. N(-d1) = 0. N= 10.2%. Supp ose the σ of short rate changes is 1. PV = 19. equal to the risk free rate. 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 Maturity of on a Zero Coupon Bond that matures in t-years.0349. Portfolio with a duration of 10 yrs that contains a 5 year and a 15 year zero. $1 million notional. purchasing a put option on ABC stock will be equivalent to a) Buying a call. p = Ke-rt N(-d 2) . continuous dividend paying stock F 0 = (S0 . The payoff profile is th e same as a long position on a put o ption and a short position on a call.$147. c) II and III. FV=100. buy forward today. CPT = PV Price at 4.0122 *8 *8. Price change for larger interest rates estimated by duration and convexity are more precise since convexity can capture the curvature Convexity Approximat ion = D* = where r is the yield to maturity of the bond. FRA that settles in 30 days. Holding other factors constant. Hedge ratio: P*DP /(F C*DF).$2. rf = foreign currency domestic risk free rate M 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.0. c) Selling a call. b) Pay the fixed-rate leg and receive the floating-rate leg of a plain vanilla interest-rate swap.8) / (93. at the same yield . b. Please note: The PV is always negative value. Convexity : Second derivative of the price/yield relationship. commodity with storage costs F 0 = S0 e(r+λ-c)t . rate of interest) 2nd Graph. the potential payoff is unaffected. c) Pay the floating-rate leg and pay the fixed-rate leg of a plain vanilla interest-rate swap. ∆I = (6% . Ans. DP = Duration of Portfolio. 5 year. 5-year.$2.(Quoted futures p rice x Conversion factor) Q. The forward exchange rate in USD per EUR for exchange 3 years from today is closest to: Ans.46 Q. N(d 1) is the delta of the option and therefore S0N(d1) represents the current price of delta. Which o f 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. sell forward today. DF = Duration of future contract.Kert K is delivery price in a forward contract F 0 is forward price that would apply to the contract today CA EA Option price bounds EB P+S0=C+Xe-rT c = S 0 N(d 1 ) − Xe − rT N(d 2 ) EC p = Xe − rT [1 − N(d 2 )] − S 0 [1 − N(d 1 )]  ln  d1 =  S0  + r + σ2 * T K 2  σ* T EE ED Currency option: implied volatility is lower for ATM option than it is for away from the money option.5%.49/2. Bfixed = 8.011 millio n Ans. 6% coupon bond of 10 year duration d.000. 1/Y= 4.50 that is currently valued at $4. Ft = Forward rate between time T2 and T 1 quoted at time t F0 = Rate between time T 2 and T 1 quoted at time 0 r = Risk free rate =5 Variable S0 EA c + ? + + - p + ? + + C + + + + - P + + + + K T σ r D Q. volatility used to price a low-strikeprice option is higher than that used to price a high-strike-price option ( ) Upper bound European/American call: c <= S0 . the price of the bond will: Ans.059.9814)=0. Q.2%*1.5 years’ spot rates are 1. If F 0<S0erT.I) ert . Q.500 / (1 + (90/360)*6%) = $2. Q.A fund manager has $10 mn invested in a portfolio of government bonds with a duration of 6. Macaulay duration is equal to maturity for a zero-coupon bond.$3. False c. A delta of 0.20 = 8% Relative risk = 8%*100 =8 Q. 4%.961* 0.500 options and sell 1. Buy 1 call with low exercise price.25 $400 0. purchase no. false i.5=17.5 for at the money. Beta (βa ) = Cov (ra.500 options and buy 1. 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. Buy a put with a SP of $25.2*0. As the maturity nears. MAR is min imum accepted return.5^2)*(0. which of the following would accomplish his goals? I. false f.1 suggests that for 1% increase in volatility. of portfolio’s excess return over Benchmark index) .e. CAPM is a special case of APT with only one factor exposure: market risk premium. Bull Spread Ans.4743 . 30% * 100+ 40%*200=110. •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. but has a -5. c.000 options and sell 1.25^2 + 0. r = Rf + beta3 x ( Km .0. p eople risk.Theta is more pronounced when the option is “in the money”.2^2)+2*0.00 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. What is the portfolio's SR? 0. highest for longterm ATM options.2^2)+2*0. •Sovereign Risk: Willingness an d ability to repay.750 long delta.000 options and buy 1. An option is available that has a gamma of 2 and a delta of 0. 10% * 125+ 90% *300 = 282.2^2)+ (0.750 shares of the underlying stock. B 2 options with different expiration.SML: indicates a return an investor should earn in the market for any level of Beta risk. Fixed coupon bonds.Strangle b. largest when option is ATM. E(RP)-E(Rb)/TE Q. Gamma: rate of change in delta as underlying stock price change ( also Convexity). and if rates increase is 96.Gamma means we are short on options. Sell a put with a SP of $75. Last 4 years. capital withdrawals. lower is the risk of large losses Alpha: measure of assessing an active man ager's p erformance as it is the return in excess o f a benchmark index.025/2)] = 97 EE Delta Theta Gamma Vega Rho • • Covered call Protective Put Bull spread Bear spread Butterfly spread Calendar spread Long straddle Strangle Strips & Straps Delta: estimates the change in value of an option for a unit change in stock price. Both the option has same expiry date in bull and bear Spreads Q. and the price of the bond if rates decline is 98.5 for at the money and .2*0 =0. American put options on stocks paying large dividends a) I and IV. (Rp – MAR) / SSD.5*0. Butterfly Spread c. The following formula is applied at each node: European Option Payoff = [ p x Option up + (1-p) x Option down] x exp . The minimum acceptable return is 7%. Butterfly buyer is betting the stock will stay near the price of the written calls Bet on volatility.2*-1 =0.Call option delta range from 0 to 1. of options sold. True e.30 $500 0. if S is the current price. (0. R n = ΣXn. & subsidized the purchase with sale of a call option with a higher Strike price 3 different options. 1 for deep in the money. αi < rf: the manager h as destroyed value αi = rf: the manager h as neither created nor destroyed value αi > rf: the manager h as created value The d ifference αi − r f is called Jensen's alpha Jensen’s α excess return o f a stock.750 shares of the underlying stock Ans: A. ITM options and OTM options have low gammas. or exercise value. C b) II and IV. At the money options close to the maturity tend to have a high a. Where Ep = Rp – Rb .750 shares to be gamma and delta neutrality.5*0. 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. Assume the weights to be 50 % for A & B. Strap : when upside move is expected to be more likely . Portfolio return σp = 25% Portfolio benchmark σ B = 20% Correlation .Rho for fixed income is small.2^2)+ (0. Liquidity Risk: risk of n ot being able to quickly liquidate a p osition at a fair p rice.Y) = cov(X.PV (Before edging) . Profit is earned if stock price has a large change in either direction Similar to straddle except purchased option is out-of-the money. where Rp = portfolio return.30 $500 0. Bull Spread Ans C Foundation of Risk Management Foundation of Risk Management Sources of Risk Tools for Risk Management Risk Management & value creation Performance Measurement Risk & return Portfolio Markowitz efficient frontier Capital Market Line (CML) Beta Security market line (SML) & CAPM Business risk: Sp ecific fo r the busin ess house.Which of the following have –Delta a.r t Q: Assume that a binomial interest-rate tree indicates a 6-month period spot rate of 2. •Funding liquidity: Inability to honor margin calls.5*0. to create a gamma-neutral portfolio (5000/2) = 2.Go long 10.40 Debt $300 Bankruptcy Cost $75 PV (after h edging) Prob $200 0. Case 2 : ρAB = 0.2σ1σ2 . Credit risk: risk o f loss due to coun terparty default.Go long 10. -The equation of the SML is CAPM (return & systematic risk equilibrium relationship -CAPM: E(Ri )=RF+β i[E(Rmkt)-RF] .2^2)+2*0. 9%. 100% *300=300 Eq uity value= probability *expected payment to equity i.Risk-free asset has 0 variance in returns . at expiration of the option the option value is simply its intrinsic.2*1 =0. Efficient Frontier CML: When a risky portfolio is combined with some allocation to a risk free asset.5 2. Rho: sensitivity of the option price to changes in the Risk free rate. b. & short 2 calls with an exercise price in b/w. over its required rate o f return as determined by CAPM: α = Rp – Rc . buy two puts and one call of same strike and expiry.5^2)* (0. buy a call & a put of same exercise price & expiration date. 10%.Case 3 : ρAB = -1 Case 1. True h. American call options on stocks paying small dividends. d ev.Correlation: ρ (X. but expected financial distress / bankruptcy costs because of leverage hamper th e firm value beyon d a level. •Settlemen t Risk: Failure of counterparty to deliver its obligation •Expo sure & recovery rate: Calculated on th e hap pening of a cred it event. 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.5 Q.5 means that price of a call option will change by $. so it cheaper to implement. another with high exercise price. American call options close to maturity. Sell a short dated option & buy a long dated option. Rb = benchmark return –Lower the tracking error lesser the risk d ifferential between portfolio and the benchmark index TE Volatility(TEV) = ω = √ σA2 . 0.Bear Spread d. True or False a. •Asset Liquidity: Large positions affecting asset prices.02 Case 3 : (0. Largest for ITM option “ITM” calls and Puts are most sensitive to changes in the rates than “OTM” Q. Market Risk: risk of value decrease due to change in prices of assets in th e market. and no one can earn excess returns CML Efficient Frontier . Positive gamma is beneficial.Investors will only be compensated systematic risk since Unsystematic risk can be diversified. More relevant when the distribution is more skewed to th e left. also called portfolio insurance Purchases call option with low Strike price. •VaR: maximum lo ss at given confidence level.45. E(RA ) = 10%. buy two calls and one put of same strike and expiry.750 shares of the underlying stock.e. where U=e√σ and D=1/U.25* 0. . create positive gamma Vega: sensitivity of the option price to changes in the volatility of the underlying stock.45 + (1-p)*96] / [1+ (0. Sell a call with Strike price (SP) 50 II. 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. Vega decreases with maturity. However this will change the position from delta-neutral portfolio to 2. Strip : when downside move is expected to be more likely.Options are most sensitive to changes to volatility. Calendar Spread b.Go long 2. Operation risk: risk due to inadequate mon itoring. Buy a call with a SP of 25. where Rp = p ortfolio return.Go short 2. c) III only. Put option delta : .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 Q. By Reducing The Probability Of Debt Overhang: Debt Overhang refers to situation where the amt of d ebt the firm is carrying p revents the shareholders fro m investing in +ive NPV p ro jects By Reducing The Pro blem Of Information Asymmetry: In formation Asymmetry results in two problems: •Investors have to rely on mgmt estimates fo r profitability of new projects •Exten t to which the performance is due to man agement decisions o r external factors Treynor Ratio: Is th e excess return divided by per un it of market risk( Beta) in an investment asset [E(RP )-RF]/βp Sharpe Ratio: Is th e excess return d ivided by per unit of to tal risk in an investment asset: [E(RP )-RF]/σp. Ex: Increase in the prices of cement fo r a construction company Financial Risk: result of a firm’s financial market activities. then in the next period the price will either be : Sup = S*u and Sdown = S *d. III. the option price will increase by $.1 deep in the money.Theta is the most negative for OTM options. iv.5*0.Probability $200 0.5^2)* (0.Variance for 2 asset portfolio σ2=w 12σ12+w22σ22+2w1w2 ρ1. D Purchase call with high strike price.. human error.30%*100+ 45%* 200 =120. Long on a Call and short on a put on the same stock with higher strike price and same maturity is called a. it implies that value of the asset drops more slowly and increases more quickly. True Q. If the price of the underlying asset is $46. d) III and IV. Delta of a option = δ c /δ s Delta of a forward position is equal to 1 Delta of Future= ert Theta: Time decay. σA = 20%.Theta affects the value of a call and a put in similar way. •Notional Limit: Maximum amount to be invested in a asset. Calculate portfolio returns when :Case 1 : ρAB = 1 . Bear spread with puts involves buying put with high exercise price and selling put with exercise price . True b.It is possible for a European put option that is “in the money” to have a positive theta value. Thus EV=392. Early exercise of an option is more likely for: i.2*0. At each final node of the tree i.500*.[E(Rmkt)-RF] is the risk premium SML Risk free return Rf The Arbitrage Pricing Theory (APT) : APT model points towards the relationship between factors and expected returns. but downside is limited Ans. Ex: Lehman. leg al an d compliance risk Efficient frontier: The optimal portfolios plotted along the curve have the highest expected return possible for a given amount of risk. Straddle c. b) II and III. . Other tools include: •Stop-loss limit: Limit on th e amount of losses in a p osition. R n = Excess return for stock n Xn = Exposure of Stock n to factor k . Value o f portfolio =100. Reducing WACC: Also we can reduce th e tax outgo by in creasing interest outgo. Rf = risk free return Sortino Ratio (SR): Excess return divided by Semi standard deviation(SSD) which considers only d ata points that represent a loss. c) II and IV.2*0. Firms can use risk management to move their income across time horizon and reduce tax burden .5^2)*(0. Thus EV=420–10=410 3.Y)/(σX *σY) . 4%.2* ρAB* σA* σB+ σB2 Relative Risk W= ω *P Information ratio: is d efined as excess return divided by TE.Rf ) + bs x SMB + bv x HML + alpha Pages 4 of 6 . Long positions in options. close to 0 when option is deep ITM or OTM. Vega Q.10 $300 0.7 = 1.e. have positive convexity. European calls options on stocks paying large dividends.e.Theta usually decreases in absolute terms as expiration approaches. An existing option short position is delta neutral.5^2)*(0. Investor profits if stock price stays in a narrow range. False d. man agement failure. which indicates that option price changes very fast as Stock price changes.20 $400 0. Bear Spread d. The return s of the benchmark were 7%. th e returns on a po rtfolio were 6%.5%. ( calls or puts) . when they are “At the money”. 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. & 10%.7. Ans. Stock price have to move more to be profitable Delta Neutral Hedging: •To completely hedge a short call position . iii. Rc = return predicted by CAPM Tracking error(TE) : TE = σ Ep (Std . a) I only.5^2)* (0.k * bk + un . of shares of stock=delta*no. bk = Factor return for factor k .00 $300 0. Q. If Hed gin g cost is 10 & after h edging PV are also sh own as abo ve Debt value =probability * expected p ayment to d ebt i.5*0. Incremental benefit=410-392.Expected return : E(RP)=∑ni=1W iE(R i) . Q. σB = 20%. volatility in various market related in struments Ex: The recent market crash. Gamma c. So sell 1. d.0 for deep out of money.Lower the correlation greater the benefits from diversification . Long stock plus short call Long stock plus long put. All points along the CML have superior risk-return profiles to any portfolio on the Efficient Frontier Efficient-market hypothesis: it is impossible to consistently outperform the market by using any information that the market already knows The 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.2^2)+ (0.5 Eq uity value = probability * expected payment to equity i. it is assumed that the underlying instrument will move up or down by a specific factor (u or d) per step of the tree.750 shares of the underlying stock.un = Stock n’s specific return` Ans Debt value=probability*expected payment to debt i. & 12%.5*0. IV. but downside is limited The upside is limited. NPV Ans B d. ED EC Binomial Pricing: At each step. Types of Financial Risk Derivatives is the mo st popular tool used by Risk Managers for RM.it is also uncorrelated with any other asset -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. .Rho b. Ans B Q.000 gamma exposure. ρPB =0. systems failure. Higher the SR. d) III and IV.500 long option.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 . the option gamma increases.5 for $1 change in value of the stock.961 Calculate TEV Ans: ω = √0.Rp = portfolio return. ii. True g. •This model considers the fact that value and small cap stocks outperform markets on a regular basis.A Vega of . In a butterfly spread Both the upside & downside are unlimited Both the upside and downside are limited The upside is unlimited.e. the resulting riskreturn combinations will lie on a straight CML.04 Case 2 : (0. So. [p*98.45 By handling bankruptcy costs :∆ (Expected Value of firm) = ∆ (Present Value of firm) + ∆ (PV of bankruptcy costs) – Risk management cost. Call options delta : 0 for deep out of money. what is the Profit/Loss on the position? Ans. 0. •Model risk. E(R B) = 10%. Q.10. •Exp osure limit: Exposure to risk factors like d uration for debt instruments & Beta fo r Equity Investments. What actions should be taken to create a gamma neutral position that will remain delta neutral? a.20^22*0. 6 Q. If the current price of XYZ is $50. . short call with low strike price. Osaka.4 in a portfolio. you can explicitly calculate the dollar cost of the increase in CFAR and include it as an additional cost of the project.41.000.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. •It had various risk exposures . European and Japanese Govt. h t = 0.08*.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-day s = (VAR(X%)1-day s )*√n Value at Risk (VaR) has become the standard measure that financial analysts use to quantify this risk.73 mn Q.04r 2(t-1) + 0.4 mn The additional project cost due to increased cash flow volatility is: ($113.02 + 0. Let h t be the variance at t and r 2(t-1) the squared return at t . The current estimate of daily volatility is 1. VARstock (5%) = 1. Unknown risk.89 results in a 6. it was expected to losses more than ~500 million USD in once in 20 months. Employee/People risk Q: Which of the following is false? a.0231* $5. Ans. Project VAR: when considering a new project. •MG bought futures on NYMEX to offset its forward commitments exposure with hedge ratio of one (every barrel was hedged). 3. A trader has an allocation equal to 8% of the firm’s capital. Using the EWMA model with λ = 0. The MVAR are 0. Correlation A & B is 10% Ans. If the ann ual volatility is 0.70 Manager B 5% .0316 VAR = 1.060606*1. A spike in correlations 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.4 STDEV(an nual) =0.000 or more. Z5%*σ = 1. the updated estimate of volatility is? Ans.65 * 0. EWMA approach of Risk Metrics is a particular case of a GARCH process.9*120 million = 8.999.32. VAR = $10.e. •LTCM used proprietary mathematical models to engage in arbitrage trading in U.000. Expected return on the surplus. Swaps.Volatility of security A & B are 7% & 3% respectively. it indicates that there is only 5% chance that on any given day.430 The area under the normal curve for confidence value is: D Risk Budgeting involves choosing and managing exposure to risk. trader at Baring PLC. calculate the 1 -day 5% VAR for this call option.94.96 Optio ns Value = 20 units VAR fo r option = 0. Therefore portfolio weight in Y should increase to move the portfolio towards the optimal portfolio.96h t-1 Ans. He took arbitrage positions on Nikkei derivatives on different exchanges viz.6 & 0. In 1998. Q.08 CVAR = 99.05 + 0.25 Days = 50 daily STDEV= 0. An increase in interest rates on on-therun Trasuries •Nick Lesson. What should manager do to move towards the optimal portfolio? Ans. c. A Fund has $200 mn in assets and $180mn in liabilities. •Fall in copper prices in June 1996 after revelation of Hamanaka’s unfair dealings led to ~2. •Lesson was solely responsible for back & front office operations of Singapore.Weight of asset A & B are 0.06 and 0.6* [√10/ (√252)]* 2.5 respectively.90 MVAR = 99. b.000* 0. The daily volatility is = 1.copper trader at Sumitomo manipulated copper prices on London Metal Exchange. Ans. scales by assets is 4%. 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). Funding Liquidity Risk.b ) VAR of uncorrelated positions: VAR portfolio = √ (VAR12 + VAR22 ) Q. with modified duration of 3. The closing price of an asset yesterday & today are $30 and $30. He took double long exposure on the same index from different exchanges.7 mn)*.90*.90*.VAR represents maximum potential loss in value of a portfolio of financial instruments with a given probability over a certain horizon.6 yrs and annualized yield of 2%. Liquidity Funding Risk. the beta of trader’s return with the return of the firm is . Example: The daily 5% VAR is $10. Determine the optimal weight ratio TE vol Ratio IR Manager A 5% . New project with a cost of $30 mn and cash flow volatility of $20 mn. D 1. futures.If the value of stock is 100 and the value of the put option at 110 is 20.such as Operational Risk. 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.94h t-1 c. h t = 0. Credit Derivatives.7 mn CFAR (at 5%) existing = 1. B=37% and remaining 12% in benchmark Pages 5 of 6 . Post Russian default on its ruble denominated debt. Tokyo & SIMEX.$68.50 Benchmark 0% 0 Portfolio 3% .060606.28) = 5 mn Risk Budgeting with Active Mangers: is done using Tracking error ( Active Returns . Investment in security A & B are USD 1. It assumes that all prices will move in the worst direction simultaneously.93h t-1 b.06%. Market Risk. assuming there are 252 days in a year.01r 2(t-1) + 0. h t = 0.500. An increase in interest rates on on-the-run Treasuries Ans: D. Ex: MRM & LTCM Incorrect measurement of known risks. Next is the optimal allocation of assets for that risk exposure. δ projects = sqrt (602 + 302 + 2*(. Which of the following GARCH models will take the shortest time to revert to its mean? a. which is unrealistic. Calculate VAR at 95% confidence level. 5. as measured by VAR.000 = 99.65.3 and 0. Employee/ People Risk. Ex: Barrings & Sumitomo Ineffective risk communication Ignorance of significant known risks. •It had its bet on convergence of Russian & American G-sec yield.65*60 = $99 mn CFAR (at 5%) with new project = 1.01 = 6. Market Risk σn2= λ*σ2n-1+(1-λ)r2n-1 Where λ = weight on previous volatility and (1.65 *1.015811 Z at 97..60 only •It had various risk exposures such as Operational Risk. The volatility of surplus is 10%. Funding liquidity risk. expected to grow by $8 mn to a value of $28 mn.35*2.6mn c) $9. $7.186 Q. Delta of the call = 0. 1st step is to determine the total amount of risk. Russian.300. gamma expected change in the delta of an option( or convexity in bonds). Market Risk. VaR for Linear and Non Linear Derivatives Linear Assets: When the value of the delta is constant for all changes in the underlying.λ) weight on squared return Q. Ex: MRM & LTCM. Danish. what is the deficit with the loss associated with the VAR. h t = 0. Ans. Ineffective risk monitoring. Risk metrics failure.7 = $113. 6. calculate the 10 day holding period VaR of the position with 99% confidence interval.000.65*68. Taylor Approximation: large changes can be explained by the 2nd derivative i. d.1975 and the risk free rate of return is 5%.1975 /√250). Ans.000 = 234.06% = 13. A B Value At Risk Types of VAR How to measure VAR •VaR (daily VaR) (in %) = ZX% * σ .b) VaRport (daily VaR) (in %): =√ (wa2 (%VaRa)2 + wb2 (%VaRb )2+2wawb *(VaRa)*(%VaRb)* σab) $ VAR portfolio = √($ VARa 2 + $ VARb 2 +2$ VARa *$ VARb* σ a. Delta (1st derivative or duration in bonds) can be used to estimate the VAR for linear derivatives.999. Which of the following reasons does not help explain the problems of LTCM in August and September 1998? 1.8mn Ans:. th e portfolio will experience a loss of $10.05r 2(t-1) + 0. The Expected excess Return ratio for X and Y are 1. Based on its models.6 mn d) $10. A 1% decrease in the stock to $10. 4. DVARP = z* std dev* portfolio value =√(VAR12+VAR22) Undiversified VAR: sum of the individual VARs for each risk factor.6 respectively.8 respectively.5mn and 3 mn respectively.4% = 2. it offered fixed price contract for supplying gasoline for 5 to 10 years.5% =1.247923 units Risk Budgeting Q.35 .such as Basis Risk.0635/.31%. VAR Linear Derivative = Delta * VAR Underlying risk factor 2. which is lowest for d.999.4 = $32. Examples: Options. Example: Forwards. A portfolio is composed of 2 securities. Sovereign Risk. GARCH allows for time-varying volatility.ZX% : the normal distribution value fo r the given probability (x%) (normal distribution has mean as 0 and standard deviation as 1) . Ans. 1.5 %. A drop in liquidity 4. A bond of $10 mn.. Use Z =1.000 = $122.25% (0. and 0. Mean μ=0 Approximately Normal Curve Representing VAR Q. Q.1 = $33 mn The deficit is: ( 33 . Market risk. What is the MVAR of Asset B and CVAR of Asset B at a 95%. the market price of the underlying stock is $11. •Yasuo Hamanaka . 1. however after Jan’95 earthquake. •Company was exposed on rising spot prices.such as Model Risk.8/1. Before failing in 1998.02*3. LTCM lost more than 4bn USD in 4 months. Ans: Surplus = (200 .000. A=51%.82 Ans. VARP =√(VAR12+VAR22+2VAR1VAR2) =VAR1+VAR2 Marginal VAR is the change in VaR of the portfolio with one unit change in the components = DVAR *β A /portfolio value Incremental VAR : The change in VAR from the addition of a new position in a portfolio.075 respectively. Ex: MRM & LTCM.4% and asset has a current value of $5. The correlation between two cash flows is 0.12=$1.90.000. LTCM’s positions were highly leveraged (1:28) with ~ USD 5: 130 billion of equity and assets. 10 units change in the underlying brings in change of 4 units change in the option premium. bonds.3)* 60*20) = $68. Calculate d aily VaR at 97.3.95*0.12.03 + 0. Component VAR is the Amount a portfolio VAR would change by deleting either of the assets from a portfolio = DVAR *β A * weight of asset A. •It used Stack and roll hedging strategy •In 1991. took concentrated positions Nikkei 225 derivatives for bank in Singapore International Monetary Exchange (SIMEX). it had given spectacular returns in 1995-97 periods (upto 40% post-fees).3 mn.95h t-1 d.25. σ port = √ wa2 σ a2 + wb2 σb2+2wawb* σa* σb* correlation (a. calculate the VAR ( 5%) on both percentage & dollar basis. 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.e.S. Ans: DVAR = 1.65* 1.65* 20* .8 mn b) $8. •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. It eventually lost more than USD 1. Funding Risk: is the risk that the value of the assets will not be sufficient to cover the liabilities of the fund Q. Its failure led to a huge bailout by large commercial & merchant banks under the guidance of Federal Reserve •It had various risk exposures .25%= 2. it is .1%. If the annual volatility of the stock is s = 0.64 million.A firm with existing projects have expected cash flow of $100 mn and cash flow volatility of $60 mn. •He kept on building his positions even after Nikkei kept on falling. i. VaR = 1.04 + 0. (D) The speed of mean reversion is defined by α1 + β. The expected excess return of X is 9% and that of Y is 12%. surplus is expected to grow by $8 mn over 1st year. σ portfolio = √(1/3)2 (7%) 2 + (2/3) 2 (3%) 2 + 2*(1/3)*(2/3)*10%* 7%*3% = 0. If beta (β) of an asset is β X with the portfolio then the cash flow at risk (CFAR) = βX * CFAR of portfolio.180 ) = $20 mn.1.Taylor approximation is ineffective for callable bonds & mortgage backed securities.8 *0.000 = $0.5 and 1.06r 2(t-1) + 0.4 mn . If the assets has a daily σ of returns equal to 1. •LTCM’s model assumed maximum volatility of 20% annually. he could not sustain his positions & failed to honor the margin calls •It eventually led to the collapse of Barings bank. The contribution of the trader to the Firm’s VAR of $120 million is: a. when it was sold to ING for mere $1.5% assuming 250 days? Delta = 0.35% decrease in the call option with a value of $1. the respectively. Q.97 Q.0316 * 4.5096 Q. GARCH can produce fat tails in the return distribution. if the betas of asset A and asset B are 1.33 = $334.5bn in 1993. VARcall = ∆ VARstock = 6. A 6 month call option with a strike price of $10 is currently trading for $1. if the cost of cash flow volatility is $0.Value at Risk Case Studies Value at risk EWMA Model Factors affecting portfolio risk GARCH Estimation Model Linear vs Nonlinear derivatives Types of VaR Methods of VAR calculation Project & CF VAR Risk budgeting D Types of Risk Management Failure A LTCM Metallgesellschaft (MRM) Baring Sumitomo Assets concentration Assets volatility Assets correlation Systematic risk σn2=ω+αr 2n-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 Relationship b/w an underlying factor and the derivative’s value are linear in nature B C Cash Flow at Risk: It is a measure of the expected cash flow at loss beyond a confidence level. •LTCM was a hedge fund using highly leveraged arbitrage trading activities in fixed income in addition to pairs trading.630 Q. An increase in stock index volatilities 3. A portfolio has an equal amount invested in X and Y. VAR Benefits: •Aggregates all the risks in a portfolio into a sin gle number Provides an approach to arrive at economical capital. •Relates capital with th e exp ected losses •Scaled to time VAR Diversified VAR: accounts for diversification effects. 2. The deltanormal approach (generally) does not work for portfolios of nonlinear securities. he never imagined that he could be susceptible to steep decline of copper prices •It had various risk exposures . The value of the total portfolio is USD1 million and its σ is 0. GARCH imposes a positive conditional mean return. Non Linear Assets: When the value of the delta keeps on changing with the change in the underlying asset. which however diverged after Russian default. •EVT stresses on the extreme value. MCVAR will approach the delta-normal VAR as the number of replications increases.e.. depending if the position is net long or short options. Pitfalls: Time variation in risk.5%. & MCS are methods available to compute VAR. •EVT is calculation of the tail events & captures expected value of the fat-tail. Q. d. 5. It can sometimes overstate the VaR. It depends on the correlations. What is the VAR of the position at 99%.3%. putting them in order from worst to best. The current price is 935 and the multiplier is 250.9%. Ans: B.6%.-5. b. expected loss is then determined from distribution. Stress testing complements VAR by providing information about the magnitude of losses that may occur in extreme market conditions Q. d. It performs most poorly for a portfolio of deep ITM options. The historical price data for the previous 300 days. Marked to market d. The delta-VAR could understate or overstate the true VAR. b. historical simulation. Therefore (935)*250* (0. b. Full revaluation c. they converge when the returns are ND. parametric VAR at high confidence levels will generally underestimate VAR Q: If you use delta-VAR for a portfolio of options.7%.. Parametric VaR will be lower. the relationship bw parametric delta-normal VAR and historical VAR will tend to be a.9%. The number of scenarios increases greatly with additional risk factors.Full valuation: to take into account nonlinear relationships C Local valuation: for Linear derivatives Historical Simulation: simply reorganizes actual historical returns. c. as the sample size increases.1%. MCVAR will be identical to the HS VAR. which of the following is always correct? a. Historical correlations mix normal and hectic periods. In finite samples.. It performs most poorly for a portfolio of deepOTM options. Delta-normal method VAR will be identical to the MCVAR.. c.059) = $13. Ans: B. Worst Case Scenario (EVT): focuses on distribution of worst possible outcomes given in an unfavourable event. 5%.. 5. •Back Testing: process of testing a trading strategy on prior time periods. 5.9% Ans: The 99% return among 300 observations would be the 3rd worst observation i. VAR(X%)=zx%*σ •Pitfall: non linearity .. None of the above are correct. -5. Ans: C. Returns: -6.-5. The only difference is that the risk factor returns are measured as logarithms of the price ratios.. Ans: C Q: Which of the following methods would be most appropriate for stress testing your portfolio? a. Delta-normal. the HS VAR will be in different from the delta-normal method. so the deltaVAR would predict zero risk. It identifies important factors not observed in historical data. then a. Parametric VaR will be higher. unusual events 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. Delta-normal method VAR will be identical to the HS VAR. Q. Q: Under usually accepted rules of market behaviour. If underlying returns are normally distributed (ND). because these have low gamma and for OTM options. c. b. •The expected shortfall is mean of the observations exceeding VaR value. Calculate VAR for an S&P 500 futures contract using the HS approach. delta is close to zero.9%. fat tails underestimate the occurrence of large observations because of its reliance on a ND RiskMetrics approach is similar to the delta-normal approach. c. d..791. d. 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. Delta-normal VAR Ans: B Pages 6 of 6 . 5. 5. It necessarily understates the VaR because it uses a linear approximation. Assumes history repeats itself. instead of rates of returns..-6%. Calculated losses may be extremely high relative to the 99% VAR significance level.1%. it is generally better for ITM options. Pitfalls: Model risk Delta-Normal or Variance-Covariance Method: assumes that the portfolio exposures are linear and that the risk factors are jointly normally distributed (ND). Which of the following is NOT a drawback to stress testing? a. Delta-gamma valuation b.4.