4. Greek Letters, Value-at-Risk

Transcription

1 4 Greek Letters, Value-at-Rsk 4 Value-at-Rsk (Hull s, Chapter 8) Math443 W08, HM Zhu Outlne (Hull, Chap 8) What s Value at Rsk (VaR)? Hstorcal smulatons Monte Carlo smulatons Model based approach Varance-covarance method Comparsons of methods Testng The Queston Beng Asked n VaR What loss level s such that we are X% confdent t wll not be exceeded n next N busness days? 99% VaR 3

2 VaR and Regulatory Captal (Hull s Busness Snapshot 8, page 436) We are X% certan that we wll not lose more than V dollars n the next N days VaR s the loss level V that wll not be exceeded wth a specfed probablty Regulators use VaR n determnng the captal a bank s requred to to keep to reflect the market rsks t s bearng The market-rsk captal s k tmes the 0-day 99% VaR where k s at least 30 4 Advantages of VaR It captures an mportant aspect of rsk n a sngle number It s easy to understand It asks the smple queston: How bad can thngs get? 5 Tme Horzon N Instead of calculatng the N-day, X% VaR drectly analysts usually calculate a -day X% VaR and assume N-day VaR = N -day VaR Ths s exactly true when portfolo changes on successve days come from ndependent dentcally dstrbuted normal dstrbutons 6

3 Methods of Estmatng VaR Hstorcal smulatons Monte Carlo smulatons Model buldng approachs 7 Hstorcal Smulaton (Hull, See Tables 8 and 8, page )) Create a database of the daly movements n all market varables The frst smulaton tral assumes that the percentage changes n all market varables are as on the frst day The second smulaton tral assumes that the percentage changes n all market varables are as on the second day and so on 8 Hstorcal Smulaton contnued Suppose we use m days of hstorcal data Let v be the value of a varable on day There are m- smulaton trals The th tral assumes that the value of the market varable tomorrow (e, on day m+) s v vm+ = vm v 9 3

4 Example A (Hull, page 438) Day Market Varable Market Varable n Portfolo value ($ mllons) Example A (Hull, page 439) Scenaros generated for tomorrow (Day 50) Scenaro Number Market Varabl e Market Varable n Portfolo value ($ mllons) 37 3 Change n value ($ mllons) Example B A portfolo consst of 9 dfferent stocks The number of each stock n the portfolo s , respectvely We have 3 year hstorcal data of each stock Estmate -Day 95% VaR usng hstorcal smulatons 4

5 Example B 787 smulatons; VaR = 97388e+004 (the 39 th ) 3 Monte Carlo Smulaton (page ) To calculate VaR usng MC smulaton we Value portfolo today Sample once from the multvarate dstrbutons of the x Use the x to determne market varables at end of one day Revalue the portfolo at the end of day 4 Monte Carlo Smulaton Calculate P Repeat many tmes to buld up a probablty dstrbuton for P VaR s the approprate fractle of the dstrbuton tmes square root of N For example, wth,000 trals the percentle s the 0th worst case 5 5

6 Example B A portfolo consst of 9 dfferent stocks The number of each stock n the portfolo s , respectvely We have 3 year hstorcal data of each stock Estmate -Day 95% VaR usng hstorcal smulatons 6 Example B 000 smulatons; VaR = 656e+005 (the 50 th ) 7 The Model-Buldng Approach Another alternatve approach s to make assumptons about the probablty dstrbutons of return on the market varables and calculate the probablty dstrbuton of the change n the value of the portfolo analytcally Ths s known as the model buldng approach or the varance-covarance approach 8 6

7 Daly Volatltes In opton prcng we measure volatlty per year In VaR calculatons we measure volatlty per day σ day σ year = 6% σ 5 year 9 Daly Volatlty contnued Strctly speakng we should defne σ day as the standard devaton of the contnuously compounded return n one day In practce we assume that t s the standard devaton of the percentage change n one day 0 Mcrosoft Example (page 440) We have a poston worth $0 mllon n Mcrosoft shares The volatlty of Mcrosoft s % per day (about 3% per year) We use N=0 and X=99 7

8 Mcrosoft Example contnued We assume that the expected change n the value of the portfolo s zero (Ths s OK for short tme perods) The standard devaton of the change n the portfolo n day s $00,000 We assume that the change n the value of the portfolo s normally dstrbuted Snce N( 33)=00, the -day 99%VaR s 33 00, 000 = $ 466, 000 Mcrosoft Example contnued The 0-day 99% VaR 466, = $, 473, 6 3 AT&T Example (page 44) Consder a poston of $5 mllon n AT&T The daly volatlty of AT&T s % (approx 6% per year) The SD of change n the value of the portfolo n -day s 5, 000, = $ 50, 000 The 0-day 99% VaR s 50, = $ 368,

9 Portfolo Now consder a portfolo consstng of both Mcrosoft and AT&T Suppose that the correlaton between the returns s 03 X: change n the value of the poston n Mcrosoft over -day perod Y: change n the value of the poston n AT&T over -day perod 5 SD of Portfolo A standard result n statstcs states that σ + = σ + σ + ρσ X Y X In ths case σ X = 00,000 and σ Y = 50,000 and r = 03 The standard devaton of the change n the portfolo value n one day s therefore 0,7 Y X σ Y 6 VaR for Portfolo The 0-day 99% VaR for the portfolo s 0, = $,6,657 The benefts of dversfcaton are (,473,6+368,405),6,657=$9,369 What s the ncremental effect of the AT&T holdng on VaR? 7 9

10 The Lnear Model We assume The daly change n the value of a portfolo s lnearly related to the daly returns from market varables The returns from the market varables are normally dstrbuted 8 The General Lnear Model contnued (equatons 8 and 8) n P= α x σ = n n P = αα jσσ jρj = j= n P = + j j j = < j σ α σ αα σσ ρ where x: the return on asset n day α : amount beng nvested on asset σ : the volatlty of varable σ P : the portfolo's standard devaton 9 The Lnear Model and Optons Consder a portfolo of optons dependent on a sngle stock prce, S Defne P δ = S and S x = S 30 0

11 Lnear Model and Optons contnued (equatons 83 and 84) As an approxmaton Smlarly when there are many underlyng market varables where δ s the delta of the portfolo wth respect to the th asset P = δ S = Sδ x P = S δ x 3 Quadratc Model For a portfolo dependent on a sngle stock prce t s approxmately true that P = δ S + γ( S ) ths becomes P = Sδ x + S γ ( x) 3 Quadratc Model contnued Wth many market varables we get an expresson of the form n n P = S δ x + S S γ x x where Ths s not as easy to work wth as the lnear model = = P δ = S j j P γj = S S j j 33

12 Comparson of Approaches Hstorcal smulaton lets hstorcal data determne dstrbutons, but s computatonally slower Monte Carlo smulaton can handle any dstrbuton and s easy to ncorporate back and stress tests It allows to acheve any accuracy (f the tme s not an ssue) It s computatonally ntensve Model buldng approach assumes normal dstrbutons for market varables It tends to gve poor results for low delta portfolos 34 Stress Testng Ths nvolves testng how well a portfolo performs under some of the most extreme market moves seen n the last 0 to 0 years 35 Back-Testng Tests how well VaR estmates would have performed n the past We could ask the queston: How often was the actual 0-day loss greater than the 99%/0 day VaR? 36