Marin Karaga AN EXAMPLE OF QUANT’S TASK IN CROATIAN BANKING INDUSTRY Introduction... A person has all of hers/his available money invested in one equity (stock) At the same time, she/he needs certain amount of money (for spending, other investments etc.) What can this person do in order to get the money? First option... First option is to close the position in equity (sell the equity) and use the proceeds from that transaction Money is obtained in simple and relatively quick way However, there is no longer the position in equity, so the person is no longer in a position to profit from potential increase of equity price Second option... Person strongly believes that the equity price will rise in the near future What to do? Person needs the money and yet is reluctant to sell the position in equity Second option could solve this problem: ask the bank for an equity margin loan! What is equity margin loan? Margin loan Equity margin loan is a business transaction between bank and its client in which client deposits certain amount of equity in the bank as collateral and receives the loan. If the client doesn’t meet hers/his obligations on a loan (i.e. doesn’t repay the loan) bank has the right to sell the collateral and use the proceeds from that transaction to cover its loss from the loan. Second option... Person strongly believes that equity margin loan is the best solution and approaches the bank with hers/his equity and asks for a equity margin loan. What are the main questions for the bank? What amount of loan can we issue to the client for a given amount of equity which is deposited as a collateral by the client? What are the risks associated with this loan? Risks... In every moment during the life of loan, bank has to be able to quickly sell the collateral and receive enough money from that transaction to cover its loss, should the client default on a loan (if the loan isn’t fully repaid) So, there are two main sources of risks associated with the equity... Risks... 1. Uncertainty about movements of equity price Equity price could fall significantly and bank might not be able to receive enough money from closure of equity position... 2. Uncertainty about equity liquidity The more time it takes you to close the position in equity, the more time its price has to fall below acceptable levels... Risks... How to quantify these risks? A task for bank’s quants! What we need to do? We need to quantify equity price risk and somehow take liquidity of equity into account. Equity price risk St - equity price at the end of day t St 1 Let’s look at the ratio ln St Let’s assume that for every t, these ratios are independent and identically distributed random variables with following distribution St 1 ln ~ N(0, 2 ) St Equity price risk (EWMA) Exponentially Weighted Moving Average Estimate volatility of random variable by looking at its observations (realizations) in the past How it works? Let’s define random variable r and assume the following r ~ N(0, r ) 2 EWMA Let’s look at N past observations of this random variable r1 , r2 , r3 , ..., rN Possible estimate of variance (or its square root – standard deviation, volatility) 2 r 1 N 2 rt N t 1 EWMA 2 r 1 N 2 rt N t 1 We treat each squared observation equally, they all have the same contribution toward the estimate of variance 1 N Can we improve this reasoning? EWMA Yesterday’s equity price is more indicative for tomorrow’s equity price that the price from, for example, 9 months ago is So, let’s assign different weights to observations of our random variable, putting more weight on more recent observations EWMA Let’s choose the value of factor w, 0 < w < 1, and use it to transform the series r1 , r2 , r3 , ..., rN to w 0 w1 w2 w N 1 r1 , r2 , r3 , ..., rN F F F F where we set N 1 w F w i 1 1w i 1 N EWMA We have changed the weight assigned to i-th observation 1 N w i 1 N k 1 w 1 w i 1 w N 1w k 1 Let’s see how the series of weights depends on the choice of factor w EWMA N=250 0,018 0,016 0,014 0,012 0,010 0,008 0,006 0,004 0,002 0,000 1 21 41 1/N 61 81 101 w=0,995 121 141 w=0,990 161 181 201 221 w=0,985 One can understand why factor w is commonly called decay factor 241 Equity price risk (cont.) Using the same formula for variance estimation, now applied to the EWMA weighted series, we get 2 r 2 1 N 2 rt N t 1 r If we apply this to our ratio ln 1w 1 wN N St 1 we get St w t 1 1w 1 wN t 1 St 1 ln St 2 N w t 1 t 1 2 t r Equity price risk Let X be a random variable, X ~ N( , 2 ) Let’s define random variable Z, Z X Obviously, Z ~ N(0,1) Hence, for some α, 0 < α < 1, we have P Z where (.) represents cumulative distribution function of random variable that has standard normal distribution Equity price risk We have P Z 1 X 1 P P X 1 P X 1 Equity price risk If we apply the previous formula to our random variable St 1 ln ~ N(0, 2 ) St we get St 1 1 P ln St What this actually tells us? Equity price risk St 1 1 P ln St St 1 1 P exp St St 1 1 P exp St St 1 1 P 1 1 exp St St St 1 1 P 1 exp St Equity price risk St St 1 - equity price decrease over one day horizon St St St 1 1 P 1 exp St For α close to zero, we can say that there is only 100 percent chance that the equity price over one day horizon will fall by more than 1 100 1 exp percent Now we have some measure of equity risk that comes from the uncertainty about movements of its price Equity price risk + liquidity Let’s assume that it takes us H days to close the position in equity Since it takes us H days to close the position so we are exposed to movements of equity price for H days Using previous notation, we need to examine following random variable St H ln St What is its distribution? Equity price risk + liquidity St H St H 1 St H St 2 St 1 ln ln St St 1 St St H 1 St H 2 St H St H 1 St 1 ln ln ln St H 1 St H 2 St St 1 Since for each t we have ln ~ N(0, 2 ) and they are St all independent, we have the following Equity price risk + liquidity St H Var ln St St H St H 1 St 1 Var ln ln ln St H 2 St St H 1 H 1 St 1 i H 1 2 Var ln i 0 St i i 0 that is, we have St H ln ~ N(0,H 2 ) St Equity price risk + liquidity Applying the same procedure as before, we get St H 1 P ln H St and finally St St H 1 P 1 exp H St All that remains is to figure out how to determine variable H Equity liquidity There are numerous ways to estimate equity liquidity We’ll again look at the past observations of equity liquidity and try to estimate how long it would take us to close our position in collateral The main factor determining how many days it could take us to close the position is, obviously, the size of position Let’s denote the size of equity position with C (expressed as market value of equity position; number of equities we have times its current market price) Equity liquidity Let’s now look at the daily volumes that were traded with this equity on the equity market during last M days (daily volume – size of trades with equity during one day, market value of position that exchanged hands that day) Let’s denote the following: VM – volume that was traded during the first day (the oldest day) in our M day long history VM-1 – volume that was traded during the second day (second oldest day) in our M day long history etc. Equity liquidity Now, let’s see how many days we would have needed in order to close the equity position if we had started to close it on day M After first day we have C VM of our position left, after second day we have C VM VM 1 of our position left, etc. Let’s define TM J TM min J N : C VM 1k 0 k 1 Equity liquidity J TM min J N : C VM 1k 0 k 1 TM is the number of days we would have needed in order to close the equity position if we started to close it on day M In a similar way we can define TM-1 J TM 1 min J N : C V(M 1)1k 0 k 1 as number of days we would have needed in order to close the equity position if we started to close it on day M-1 Equity liquidity If we continue with these definitions, we will get the series of numbers TM , TM 1 , TM 2 , all representing number of days we would have needed in order to close our position if we started to close it on certain days in the past We need to determine our variable H based on the previous series of numbers, let’s be conservative and set H max TM , TM 1 , TM 2 , Equity price risk + liquidity risk Now we have everything we need: - estimate of equity price volatility H - estimate of equity liquidity Combined measure of risk St St H 1 P 1 exp H St Practical use Remember what our question was: What amount of loan can the bank issue to its client for a given amount of equity which is deposited as a collateral by the client? Let’s assume that the bank wants that in 99% of cases value of collateral doesn’t fall below the value of the loan during the selling of collateral Expressed in language of our model: α = 0,01 Next, let’s assume that the bank finds appropriate to set the decay factor w to be equal to 0,99 Practical use – loan approval Let C denote the initial value of position in equity Bank calculates H and Then bank looks at the following St St H 1 P 1 exp 0,01 H 0,01 St St H 1 P exp 0,01 H 0,01 St 1 P St H St exp 0,01 H 0,01 Practical use – loan approval 1 P St H St exp 0,01 H 0,01 In 99% cases, St H 1 St exp 0,01 H In other words, in 99% of cases, during the selling of collateral, price of collateral won’t fall below 1 St exp 0,01 H where St is the value of collateral at the start of closure of equity position Practical use – loan approval So, the bank sets the value of loan 1 L C exp 0,01 H 1 equivalent ly, L C C 1 exp 0,01 H “haircut” We have solved our problem! Important note: once the loan has been issued, L is constant and C varies, so the client is obliged to maintain appropriate size of collateral – above relation has to be true during the entire life of loan Examples C = HRK 10 million Using the data from last 250 days (1 year) we get (α = 0,01, w = 0,99): HT: = 0,0127 (1,27%), H = 19 1 exp 0,01 H 87,92% L HRK 8.792.000 haircut 100% 87,92% 12,08% INGRA: = 0,0339 (3,39%), H = 82 1 exp 0,01 H 48,92% haircut 100% 48,92% 51,08% L HRK 4.892.000 Summary We have seen: “Real life” case from Croatian banking industry Identified risks associated with margin loan Used EWMA to model equity volatility Enhanced EWMA results in order to take equity liquidity risk into account Transformed analytical result into straightforward figure (haircut) that can be quoted to potential clients Two examples of haircut calculation Final remarks Every model is nothing more than just a model Check the model assumptions, try to improve it, confirm its results by comparing them with the results form different models etc. In “historical” model one needs to constantly update the underlying historical data in order to feed the model with the most recent information Compare the actual losses with the level of losses predicted by the model – test the soundness of model Questions Thank you for your attention!