Presentation on theme: "An Example of Quant’s Task in Croatian Banking Industry"— Presentation transcript:
1An Example of Quant’s Task in Croatian Banking Industry Marin KaragaAn Example of Quant’s Task in Croatian Banking Industry
2Introduction...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?
3First option...First option is to close the position in equity (sell the equity) and use the proceeds from that transactionMoney is obtained in simple and relatively quick wayHowever, 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
4Second option...Person strongly believes that the equity price will rise in the near futureWhat to do? Person needs the money and yet is reluctant to sell the position in equitySecond option could solve this problem: ask the bank for an equity margin loan!What is equity margin loan?
5Margin loanEquity 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.
6Second 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?
7Risks...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...
8Risks...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 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...
9Risks... 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.
10St - equity price at the end of day t Equity price riskSt - equity price at the end of day tLet’s look at the ratioLet’s assume that for every t, these ratios are independent and identically distributed random variables with following distribution
11Equity price risk (EWMA) Exponentially Weighted Moving AverageEstimate volatility of random variable by looking at its observations (realizations) in the pastHow it works?Let’s define random variable r and assume the following
12EWMA Let’s look at N past observations of this random variable Possible estimate of variance (or its square root – standard deviation, volatility)
13EWMAWe treat each squared observation equally, they all have the same contribution toward the estimate of varianceCan we improve this reasoning?
14EWMAYesterday’s equity price is more indicative for tomorrow’s equity price that the price from, for example, 9 months ago isSo, let’s assign different weights to observations of our random variable, putting more weight on more recent observations
15EWMALet’s choose the value of factor w, 0 < w < 1, and use it to transform the seriesto where we set
16EWMA We have changed the weight assigned to i-th observation Let’s see how the series of weights depends on the choice of factor w
17EWMAOne can understand why factor w is commonly called decay factor
18Equity price risk (cont.) Using the same formula for variance estimation, now applied to the EWMA weighted series, we getIf we apply this to our ratio we get
19Equity price risk Let X be a random variable, Let’s define random variable Z,Obviously,Hence, for some α, 0 < α < 1, we have where represents cumulative distribution function of random variable that has standard normal distribution
23Equity price risk - equity price decrease over one day horizon For α close to zero, we can say that there is only percent chance that the equity price over one day horizon will fall by more than percentNow we have some measure of equity risk that comes from the uncertainty about movements of its price
24Equity price risk + liquidity Let’s assume that it takes us H days to close the position in equitySince it takes us H days to close the position so we are exposed to movements of equity price for H daysUsing previous notation, we need to examine following random variableWhat is its distribution?
25Equity price risk + liquidity Since for each t we have and they are all independent, we have the following
27Equity price risk + liquidity Applying the same procedure as before, we get and finallyAll that remains is to figure out how to determine variable H
28Equity 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 collateralThe main factor determining how many days it could take us to close the position is, obviously, the size of positionLet’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)
29Equity liquidityLet’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.
30Equity liquidityNow, 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 MAfter first day we have of our position left, after second day we have of our position left, etc.Let’s define TM
31Equity liquidityTM is the number of days we would have needed in order to close the equity position if we started to close it on day MIn a similar way we can define TM 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
32Equity liquidityIf we continue with these definitions, we will get the series of numbers 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 pastWe need to determine our variable H based on the previous series of numbers, let’s be conservative and set
33Equity price risk + liquidity risk Now we have everything we need: estimate of equity price volatility H - estimate of equity liquidityCombined measure of risk
34Practical useRemember 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 collateralExpressed in language of our model: α = 0,01Next, let’s assume that the bank finds appropriate to set the decay factor w to be equal to 0,99
35Practical use – loan approval Let C denote the initial value of position in equityBank calculates H andThen bank looks at the following
36Practical use – loan approval In 99% cases,In other words, in 99% of cases, during the selling of collateral, price of collateral won’t fall belowwhere is the value of collateral at the start of closure of equity position
37Practical use – loan approval So, the bank sets the value of loanWe 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“haircut”
38Examples 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 = 19INGRA: = 0,0339 (3,39%), H = 82
39Summary We have seen: “Real life” case from Croatian banking industry Identified risks associated with margin loanUsed EWMA to model equity volatilityEnhanced EWMA results in order to take equity liquidity risk into accountTransformed analytical result into straightforward figure (haircut) that can be quoted to potential clientsTwo examples of haircut calculation
40Final 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 informationCompare the actual losses with the level of losses predicted by the model – test the soundness of model