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Marin Karaga

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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?

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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

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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?

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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 doesnt meet hers/his obligations on a loan (i.e. doesnt 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.

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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?

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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 isnt fully repaid) So, there are two main sources of risks associated with the equity...

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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...

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Risks... How to quantify these risks? A task for banks quants! What we need to do? We need to quantify equity price risk and somehow take liquidity of equity into account.

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Equity price risk S t - equity price at the end of day t Lets look at the ratio Lets assume that for every t, these ratios are independent and identically distributed random variables with following distribution

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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? Lets define random variable r and assume the following

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EWMA Lets look at N past observations of this random variable Possible estimate of variance (or its square root – standard deviation, volatility)

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EWMA We treat each squared observation equally, they all have the same contribution toward the estimate of variance Can we improve this reasoning?

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EWMA Yesterdays equity price is more indicative for tomorrows equity price that the price from, for example, 9 months ago is So, lets assign different weights to observations of our random variable, putting more weight on more recent observations

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EWMA Lets choose the value of factor w, 0 < w < 1, and use it to transform the series to where we set

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EWMA We have changed the weight assigned to i-th observation Lets see how the series of weights depends on the choice of factor w

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EWMA One can understand why factor w is commonly called decay factor

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Equity price risk (cont.) Using the same formula for variance estimation, now applied to the EWMA weighted series, we get If we apply this to our ratio we get

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Equity price risk Let X be a random variable, Lets 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

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Equity price risk We have

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Equity price risk If we apply the previous formula to our random variable we get What this actually tells us?

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Equity price risk

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- 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 percent Now we have some measure of equity risk that comes from the uncertainty about movements of its price

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Equity price risk + liquidity Lets 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 What is its distribution?

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Equity price risk + liquidity Since for eac h t we have and they are all independent, we have the following

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Equity price risk + liquidity that is, we have

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Equity price risk + liquid ity Applying the same procedure as before, we get and finally All that remains is to figure out how to determine variable H

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Equity liquidity There are numerous ways to estimate equity liquidity Well 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 Lets 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)

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Equity liquidity Lets 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) Lets denote the following: V M – volume that was traded during the first day (the oldest day) in our M day long history V M-1 – volume that was traded during the second day (second oldest day) in our M day long history etc.

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Equity liquidity Now, lets 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 of our position left, after second day we have of our position left, etc. Lets define T M

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Equity liquidity T M 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 T M-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

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Equity liquidity If 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 past We need to determine our variable H based on the previous series of numbers, lets be conservative and set

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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

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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? Lets assume that the bank wants that in 99% of cases value of collateral doesnt fall below the value of the loan during the selling of collateral Expressed in language of our model: α = 0,01 Next, lets assume that the bank finds appropriate to set the decay factor w to be equal to 0,99

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Practical use – loan approval Let C denote the initial value of position in equity Bank calculates H and Th e n bank looks at the following

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Practical use – loan approval In 99% cases, In other words, in 99% of cases, during the selling of collateral, price of collateral wont fall below where is the value of collateral at the start of closure of equity position

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Practical use – loan approval So, the bank sets the value of loan 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 haircut

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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 INGRA: = 0,0339 (3,39%), H = 82

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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

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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

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Questions

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Thank you for your attention!

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