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## Financial Risk Management

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**Financial Risk Management**Zvi Wiener Following P. Jorion,Financial Risk Manager Handbook FRM**FRM 2000**• Capital Markets Risk Management 20 • Legal, Accounting and Tax 6 • Credit Risk Management 36 • Operational Risk Management 8 • Market Risk Management 35 • Quantitative Analysis 23 • Regulation and Compliance 12 Zvi Wiener**Quantitative Analysis**Jorion, Value-at-Risk. Jorion, Financial Risk Manager Handbook Hull, Options, Futures and Other Derivatives. Fabozzi F., Bond Markets: Analysis and Strategies. Fabozzi F., Fixed Income Mathematics. Golub B., Risk Management. Crouchy, Galai, Mark, Risk Management. Zvi Wiener**Quantitative Analysis**• Bond fundamentals • Fundamentals of probability • Fundamentals of Statistics • Pricing Techniques Zvi Wiener**Chapter 1Quantitative AnalysisBond Fundamentals**Following P. Jorion 2001 Financial Risk Manager Handbook FRM**Bond Fundamentals**Discounting, Present Value Future Value Zvi Wiener**Compounding**US Treasuries market uses semi-annual compounding. Continuous compounding Zvi Wiener**A bond pays $100 in ten years and its price is $55.9126.**This corresponds to an annually compounded rate of 6% using PV=CT/(1+y)10, or (1+y) = (CT/PV)0.1. This rate can be transformed into semiannual compounded rate, using (1+ys/2)2 = (1+y), or ys = ((1+0.06)0.5-1)*2 = 5.91%. It can be transformed into a continuously compounded rate exp(yc) = 1+y, or yc = ln(1+0.06) = 5.83%. Zvi Wiener**Note that as we increase the frequency of the compounding**the resulting rate decreases. Intuitively, since our money works harder with more frequent compounding, a lower rate will achieve the same payoff. Key concept: For a fixed present and final values, increasing the frequency of the compounding will decrease the associated yield. Zvi Wiener**FRM-99, Question 17**Assume a semi-annual compounded rate of 8% per annum. What is the equivalent annually compounded rate? A. 9.2% B. 8.16%C. 7.45% D. 8% Zvi Wiener**FRM-99, Question 17**(1 + ys/2)2 = 1 + y (1 + 0.08/2)2 = 1.0816 ==> 8.16% Zvi Wiener**FRM-99, Question 28**Assume a continuously compounded interest rate is 10% per annum. What is the equivalent semi-annual compounded rate? A. 10.25% per annum. B. 9.88% per annum. C. 9.76% per annum. D. 10.52% per annum. Zvi Wiener**FRM-99, Question 28**(1 + ys/2)2 = ey (1 + ys/2)2 = e0.1 1 + ys/2 = e0.05 ys = 2 (e0.05 - 1) = 10.25% Zvi Wiener**Price-Yield Relationship**Here Ct is the cashflow t - number of periods to each payment T number of periods to maturity y - the discount factor. Zvi Wiener**Face value, nominal.**Bond that sells at face value is called par bond. A bond has a 8% annual coupon and IRR of 8%. What is the price of the bond? Is this always true? Zvi Wiener**Price-yield Relationship**$ Price of a straight bond as a function of yield y Zvi Wiener**FRM-98, Question 12**A fixed rate bond, currently priced at 102.9, has one year remaining to maturity and is paying an 8% coupon. Assuming that the coupon is paid semiannually, what is the yield of the bond? A. 8% B. 7% C. 6% D. 5% Zvi Wiener**FRM-98, Question 12**ys = 5% Zvi Wiener**Taylor Expansion**To measure the price response to a small change in risk factor we use the Taylor expansion. Initial value y0, new value y1, change y: Zvi Wiener**Derivatives**F(x) x Zvi Wiener**Properties of derivatives**Zvi Wiener**Bond Price Derivatives**D* - modified duration, dollar duration is the negative of the first derivative: Dollar convexity = the second derivative, C - convexity. Zvi Wiener**Duration of a portfolio**Zvi Wiener**Macaulay Duration**Modified duration Zvi Wiener**Bond Price Change**Zvi Wiener**Example**10 year zero coupon bond with a semiannual yield of 6% The duration is 10 years, the modified duration is: The convexity is Zvi Wiener**Example**If the yield changes to 7% the price change is Zvi Wiener**Duration-Convexity**Price of a straight bond as a function of yield $ y Zvi Wiener**Effective duration**Effective convexity Zvi Wiener**Effective Duration and Convexity**Consider a 30-year zero-coupon bond with a yield of 6%. With semi-annual compounding its price is $16.9733. We can revalue this bond at 5% and 7%. Zvi Wiener**5% 6% 7%**Zvi Wiener**Coupon Curve Duration**If IR decrease by 100bp, the market price of a 6% 30 year bond will go up close to the price of a 30 years 7% coupon bond. Thus we associate a higher coupon with a drop in yield equal to the difference in coupons. This approach is useful for mortgages. Zvi Wiener**FRM-98, Question 20**Coupon curve duration is a useful method to estimate duration from market prices of MBS. Assume that the coupon curve of prices for Ginnie Maes is as follows: 6% at 92, 7% at 94, 8% at 96.5. What is the estimated duration of the 7s? A. 2.45 B. 2.4 C. 2.33 D. 2.25 Zvi Wiener**FRM-98, Question 20**Zvi Wiener**FRM-98, Question 21**Coupon curve duration is a useful method to estimate duration from market prices of MBS. Assume that the coupon curve of prices for Ginnie Maes is as follows: 6% at 92, 7% at 94, 8% at 96.5. What is the estimated convexity of the 7s? A. 53 B. 26 C. 13 D. -53 Zvi Wiener**FRM-98, Question 21**Zvi Wiener**Duration of a coupon bond**Zvi Wiener**Exercise**Find the duration and convexity of a consol (perpetual bond). Answer: (1+y)/y. Zvi Wiener**Convexity**Exercise: compute duration and convexity of a 2-year, 6% semi-annual bond when IR=6%. Zvi Wiener**FRM-99, Question 9**A number of terms in finance are related to the derivative of the price of a security with respect to some other variable. Which pair of terms is defined using second derivatives? A. Modified duration and volatility B. Vega and delta C. Convexity and gamma D. PV01 and yield to maturity Zvi Wiener**FRM-98, Question 17**A bond is trading at a price of 100 with a yield of 8%. If the yield increases by 1 bp, the price of the bond will decrease to 99.95. If the yield decreases by 1 bp, the price will increase to 100.04. What is the modified duration of this bond? A. 5.0 B. -5.0 C. 4.5 D. -4.5 Zvi Wiener**FRM-98, Question 17**Zvi Wiener**FRM-98, Question 22**What is the price of a 10 bp increase in yield on a 10-year par bond with a modified duration of 7 and convexity of 50? A. -0.705 B. -0.700 C. -0.698 D. -0.690 Zvi Wiener**FRM-98, Question 22**Zvi Wiener**FRM-98, Question 29**A and B are perpetual bonds. A has 4% coupon, and B has 8% coupon. Assume that both bonds are trading at the same yield, what can be said about duration of these bonds? A. The duration of A is greater than of B B. The duration of A is less than of B C. They have the same duration D. None of the above Zvi Wiener**FRM-97, Question 24**Which of the following is NOT a property of bond duration? A. For zero-coupon bonds Macaulay duration of the bond equals to time to maturity. B. Duration is usually inversely related to the coupon of a bond. C. Duration is usually higher for higher yields to maturity. D. Duration is higher as the number of years to maturity for a bond selling at par or above increases. Zvi Wiener**FRM-99, Question 75**You have a large short position in two bonds with similar credit risk. Bond A is priced at par yielding 6% with 20 years to maturity. Bond B has 20 years to maturity, coupon 6.5% and yield of 6%. Which bond contributes more to the risk of the portfolio? A. Bond A B. Bond B C. A and B have similar risk D. None of the above Zvi Wiener**Portfolio Duration and Convexity**Portfolio weights Zvi Wiener**Example**Construct a portfolio of two bonds: A and B to match the value and duration of a 10-years, 6% bond with value $100 and modified duration of 7.44 years. A. 1 year zero bond - price $94.26 B. 30 year zero - price $16.97 Zvi Wiener