1. Advanced Risk Management I Lecture 2

2. Cash flow analysis and mapping Securities in a portfolio are collected and analyzed one by one. Bonds are decomposed in their cash flows. Then, cash flows and other securities are distributed in a limited set of reprsentative exposures, for a more synthetic reporting of risk.

3. Coupon bond cash flows Let us define P(t,T;c) the price of a bond paying coupon c on a schedule {t 1, t 2, …,t m =T}, with trepayment of capital in one sum at maturity T. The cash flows of this bond can be replicated by a basket of ZCB with nominal value equal to c corresponding to maturities t i for i = 1, 2, …, m – 1 and a ZCB with a nominal value 1 + c iat maturity T. The arbitrage operation consisting in the purchase/sale of coupons of principal is called coupon stripping.

4. Indexed (floating) coupons An indexed coupon is determined based on a reference index, typically an interest rates, observed at time , called the reset date. The typical case (known as natural time lag) is a coupon with – reference period from  to T – reset date  and payment date T – reference interest rate for determination of the coupon i( ,T) (T –  ) = 1/v ( ,T) – 1

5. Replicating portfolio What is the replicating portfolio of an floating coupon, indexed to a linear compounded interest rate for one unit of nominal? Notice that at the reset date  the value of the coupon, determined at time  and paid at time T, will be given by v ( ,T) i( ,T) (T –  ) = 1 – v ( ,T) The replicating portfolio is then given by –A long position (investment) of one unit of nominal available at time  –A short position (financing) for one unit of nominal available at time T

6. Cash flows of a floating coupon Notice that a floating coupon on a nominal amount C corresponds to a position of debt (leverage)

7. No arbitrage price: indexed coupons The replicating portfolio enables to evaluate the coupon at time t as: indexed coupons = v(t,  ) – v(t,T) At time  we know that the value of the position is: 1 – v( ,T) = v( ,T) [1/ v( ,T) – 1] = v( ,T) i( ,T)(T –  ) = discount factor X indexed coupon At time t the coupon value can be written v(t,  ) – v(t,T) = v(t,T)[v(t,  ) / v(t,T) – 1] = v(t,T) f(t, ,T)(T –  ) = discount factor X forward rate

8. Indexed cash flows Let us consider the time schedule  t,t 1,t 2,…t m  where t i, i = 1,2,…,m – 1 are coupon reset times, and each of them is paid at t i+1. t is the valuation date. It is easy to verify that the value the series of flows corresponds to –A long position (investment) for one unit of nominal at the reset date of the first coupon (t 1 ) –A short position (financing) for one unit of nominal at the payment date of the last coupon (t m )

9. Floater A floater is a bond characterized by a schedule  t,t 1,t 2,…t m  –at t 1 the current coupon c is paid (value cv(t,t 1 )) –t i, i = 1,2,…,m – 1 are the reset dates of the floating coupons are paid at time t i+1 (value v(t,t 1 ) – v(t,t m )) –principal is repaid in one sum t m. Value of coupons: cv(t,t 1 ) + v(t,t 1 ) – v(t,t m ) Value of principal: v(t,t m ) Value of the bond Value of bond = Value of Coupons + Value of Principal = [cv(t,t 1 ) + v(t,t 1 ) – v(t,t m )] + v(t,t m ) =(1 + c) v(t,t 1 ) A floater is financially equivalent to a short term note.

10. From portfolios to exposures The securities collected and evaluated are transformed into exposures to risk factors. The process to transform cash flows into securities is called mapping. Aim of the mapping is to give a synthetic, but informative, representation of the risks to which a portfolio is exposed, and to provide a guideline for risk management.

11. Fixed income exposures Fixed income exposure mapping requires the definition of a set of reference maturities on which to collect the cash flows. These reference maturities are called bucket. Each and every cash flow of the replicating portfolio of a bond that does not coincide exactly with a bucket is split in two flows on the closest buckets. The splitting is designed in such a way as to preserve the financial features of the cash flow, as closely as possible.

12. The art of reporting How many bucket? The number of buckets should reflect the number of risk factors characterizing the yield curve. –A number of buckets too small can lead to ignore important movements of the yield curve –Too many buckets may induce too much “noise” in the movements of the yield curve In most markets, three or four factors are sufficient to represent the a huge percentage of the movements of the yield curve, but in reality ten or twelve buckets are used.

13. Why more buckets? Typically, more buckets than needed are used for two reasons A limited number of buckets increase the distance, in financial terms, between the product and its mapped version. A less coarse representation in terms of buckets provides a better guideline for hedging strategies.

14. Cash flow-mapping Cash-flow mapping is based on the following requirements –The sign of the positions is preserved –The value of the positions is preserved –The risk of the positions is preserved

15. Cash flow mapping: two options Fisher & Weil: cash-flows are split preserving i) sign ; ii) market to market value; iii) duration Opzione RiskMetrics™ : cash-flows are split preserving i) sign ; ii) market to market value; iii) volatility

16. Fisher Weil option Let c  be a nominal flow (positive or negative) corrisponding to maturity . The cash-flow must be decomposed in two flows of the same sign c i-1 and c i on the closes maturities t i-1 and t i. The solution requires that c i-1 v(t, t i-1 ) + c i v(t, t i ) = c  v(t,  ) (t i-1 - t)c i-1 v(t, t i-1 ) + (t i - t) c i v(t, t i ) = (  - t) c  v(t,  ) We used Macaulay duration, but we could do the same with modified duration.

17. Fisher e Weil: solution Solving the system we get …and the value of the item is split in proportion to the ratio of duration differences.

18. RiskMetrics™ options Define:  i-1,   and  i the interest rate volatility of maturities , t i-1 and t i respectively and  i,i-1 their correlation. We want back out , the share of value c  v(t,  ) that will be allocated to vertex t i-1 v(t, t i-1 )c i-1 =  c  v(t,  ) v(t, t i )c i = (1-  ) c  v(t,  ) The sign and value conditions are then verified for 0    1 and the duration condition is substituted by  2 D i-1 2  i-1 2 + (1 -  ) 2 D i 2  i 2 + + 2 (1 -  )  D i-1 D i  i,i-1  i  i-1 = D  2   2

19. RiskMetrics™: solution The  value is obtained by solving the second degree equation a  2 + b  + c = 0 with a = D i-1 2  i-1 2 + D i 2  i 2 -  i,i-1 D i-1 D i  i  i-1 b = 2(  i,i-1 D i-1 D i  i  i-1 - D i 2  i 2 ) c = D i 2  i 2 – D  2   2 We choose the solution between 0 and 1, assuming it exists!

20. Cash flow mapping: pros and cons Opzione Fisher & Weil: pros –Independent of volatility estimation –It is possible to estimate the degree of approximation Opzione RiskMetrics™: pros –If the volatility estimate is accurate, it gives a better approximation. –Consistent with many changes of the term structure

21. A particular case Assume the volatility to be constant for all buckets and perfect correlation i)  i-1 =   =  i =  ii)  i,i-1 = 1 The RiskMetrics™ condition is  2 D i-1 2  2 + (1 -  ) 2 D i 2  2 + 2 (1 -  )  D i-1 D i  2 = D  2  2 from which [  D i-1 + (1 -  ) D i ] 2 = D  2 and the condition corresponds to Fisher and Weil.

23. New RiskMetrics options RiskMetrics launched another proposal based on linear inerpolation of interest rates r(t,  ) =  r(t, t i-1 ) + (1 –  ) r(t, t i ) Mapping uses three funds instead of two: the closest buckets and the cash bucket The mapping is built in such a way that the sensitivity of the value with respect to the rate of each bucket be the same as the sensitivity of the mapped cash flow. The idea is to allow for the two rates to move independently of each other.

24. The three equations

25. The solutions

26. Reporting for equity risk The positions in the equity markets are reported in appropriated buckets. Reporting choices: –Market exposure = value of the position –Market exposure = systematic risk –Stock exposure = overall risk

27. Reporting for FX exposures For exposures denominated in foreign currency j, with exchange rates e j with respect to the reference currency, the exposure to exchange rate risk is added to the market risk of the product. The percentage variation of every exposure i, from the point of view of the reference currency, can be decomposed as r i = r* i + r e where r* i and r e are respectively the percentage change of the position in domestic currency and that of the exchange rate.

28. Exchange rate exposure From the decomposition of the percentage change of value, the variance of the position is  i 2 =  * i 2 +  e 2 +2  ie  * i  e Every exposure denominated in foreign currrency then implies an exposure to exchange risk for the same mark to market amount. Notice that in the extreme case  ie = 1  i =  * i +  e