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1、Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Introduction to Value-at-Risk; the Historical Simulation Method,Neil D. Pearson,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Value-at-Risk (VaR),Single, summary measure of possible portfolio losses A measure of loss due to
2、“normal” market movements in the sense that losses greater than the value-at-risk are suffered with only a specified small probability Value-at-risk is a probabilistic measure of portfolio losses,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Why a portfolio-level measure?,To many diff
3、erent underlying “assets” or market factors is J.P. Morgan exposed? How many deltas does J.P. Morgan compute every day? Can J.P. Morgans risk management committee/senior management understand and meaningfully review tables of hundreds of deltas, gammas, etc.? These considerations push one toward usi
4、ng a portfolio-level measure,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Why a probabilistic measure?,What is the worst possible outcome for J.P. Morgan? That is, what is the largest possible loss?,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Value-at-Risk,VaR measur
5、es risk of loss from equally likely market movements across instrument classes VaR is the amount that an organization can expect to lose less than x percent of the time (if positions are left unchanged for t days) Example: probability of 5 percent, horizon of 1 day VaR of $1 million means the organi
6、zation expects to suffer daily losses of $1 million or more on only 5 percent of business days,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,What (really) is value at risk?,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Or, if you like Normal distribution,Notes: (1) a cu
7、toff point 1.65 standard deviations below the mean leaves 5% of probability in left hand tail (for Normal distribution); (2) In this example mean is assumed to be zero,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Back to the definition of VaR,Value at risk is the amount that an entit
8、y can expect to lose less than x percent of the time (if positions are left unchanged for t days) Example: probability of 5 percent, horizon of 1 day value at risk of $86,625 means the entity expects to suffer daily losses of $86,625 or more on only 5 percent of business days,Copyright 1996-2002 by
9、Neil D. Pearson. All rights reserved.,VaR Estimation Methods,Historical simulation Delta-Normal (or Variance-Covariance or parametric) approach Monte Carlo (stochastic) simulation,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Historical simulation: A naive approach,Profit/loss on $10
10、million of par value of 30-yr. US Treasury bond, held during first 100 days of 1994,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Historical simulation: a naive approach,What is wrong with this approach?,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Historical simulatio
11、n: A better approach,Example: 1-year THB-denominated note with face value of THB 5 billion, paying semi-annual interest at the rate of 22.5 percent per year. Interest payment of THB 0.5 0.25 5 billion = THB 562.5 billion in one-half year. Interest payment of THB 562.5 million and also return the pri
12、ncipal of THB 5 billion at end of year USD-denominated loan with a principal amount of USD 50 million paying semi-annual interest at rate of 6 percent per year interest payment of USD 0.5 0.06 50 million = USD 1.5 million in one-half year interest payment of USD 1.5 million and repay the principal o
13、f USD 50 million in one year Current date is Friday, 30 January we want to measure risk from Friday to Monday, 2 February,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Historical simulation: A better approach,The current value of the Thai baht note is THB 5,047,516, 023 0.01860465 USD
14、/THB = USD 93,907,275 Present value of the liability on the dollar-denominated loan is USD 50,201,942. Thus, the current value of the portfolio is USD 93,907,275 USD 50,201,942 = USD 43,705,333.,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Determinants of portfolio value,THB bond is
15、equivalent to 6-month zcb with a face value of THB 562.5 million; and 1-year zero-coupon bond with a face value of THB 5.5625 billion USD loan is equivalent to 6-month zero-coupon bond with face value USD 1.5 million; and 1-year zero-coupon bond with a face value of USD 51.5 million This decompositi
16、on of the portfolio gives us,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Determinants of portfolio value,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Determinants of portfolio value,Current USD mark-to-market value is determined by the basic “market factors”: 6-month
17、 USD interest rate 1-year USD interest rate 6-month THB interest rate 1-year THB interest rate USD/THB exchange rate We need data on these 5 market factors,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Obtain data on market factors,Copyright 1996-2002 by Neil D. Pearson. All rights re
18、served.,Compute hypothetical values of market factors,Start with 30 Jan. 1998 USD interest rate: 5.625% Compute % change using first day in sample, 6 to 7 Feb. 1997: Apply this % change to 30 Jan. 1998 rate:,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Compute hypothetical values of
19、market factors,Do same for 1-year USD rate, THB rates, USD/THB exchange rate Obtain the following hypothetical values: 6-month USD interest rate: 5.63281% 1-year USD interest rate: 5.65625% 6-month THB interest rate: 21.58044% 1-year THB interest rate: 22.01408% USD/THB exchange rate: 0.01859392 USD
20、/THB,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Compute the hypothetical mark-to-market value for 2 Feb. 1998,Compute hypothetical mark-to-market value: Using the values of the market factors from previous slide, this is USD 44,140,027 Change in value = USD 44,140,027 - 43,705,333
21、= USD 434,694,Some time has passed from Friday 30 Jan 1998 to Monday 2 Feb 1998,Here we are cheating a bit, and assuming that the rate for 362 days is the same as the 1-year rate. We really should interpolate the rate for 362 days from the 6-month and 1-year rates.,Copyright 1996-2002 by Neil D. Pea
22、rson. All rights reserved.,Computation of hypothetical change in mark-to-market value,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Repeat 249 more times (total of 250*),*In the “real world,” financial institutions typically use many more than 250 observations. I use 250 only to keep
23、the example a manageable size.,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Order from largest profit to largest loss,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Order from largest profit to largest loss,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Va
24、lue-at-Risk,95 percent confidence value-at-risk is 0.5(2,738,866) + 0.5(2,777,376) = 2,758,121,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Advantages/disadvantages,What dont you like about this methodology? What do you like about it?,Copyright 1996-2002 by Neil D. Pearson. All right
25、s reserved.,Determinants of portfolio value,Current USD mark-to-market value is determined by the basic “market factors”: 6-month USD interest rate 1-year USD interest rate 6-month THB interest rate 1-year THB interest rate USD/THB exchange rate How do we handle a cash flow to be received in say 8 m
26、onths?,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Value of 8-month cash flow,Market value of cash flow to be received in 8 months might be computed as where is estimate of the 8-month rate obtained by interpolating from the 6 and 12-month rates.*,*Of course, one might use interpola
27、tion schemes more sophisticated than linear interpolation.,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,Value of 8-month cash flow,Does this introduce another market factor, the 8-month rate? No. The point of the interpolation was to avoid introducing another market factor,Copyright
28、1996-2002 by Neil D. Pearson. All rights reserved.,Key ideas,Chose the basic market factors: important (and conflicting) considerations are: convenience richness/ability to capture relevant risks Formulas expressing instruments values in terms of market factors Collect data past market factors Use data and formulas to estimate distribution of possible changes in value,Copyright 1996-2002 by Neil D. Pearson. All rights reserved.,End of Lecture,Next time we begin discussion of “Delta-Normal” method,