Variance, covariance method / parametric method

Calculating VaR on percentage basis and VaR on dollar basis:

VaR on percentage basis.

In order to calculate VaR on %age basis, we would use a critical z-value of -1.65 and multiply by the standard deviation of percent returns. The resulting VaR estimate would be the percentage loss in asset value that would only be exceeded 5% of the time.

Formula: VaR_(%) = - zσ (where value of z at 5% is 1.65 and at 1% is 2.31) VaR on dollar basis.

VaR can also be estimated on a dollar rather than a percentage basis. To calculate we simply multiply the percent VaR by the asset value. It is interpreted as; the dollar loss in asset value that will only be exceeded 5% of the time.

Formula: VaR_($) = - zσ (p) (where p is the value of asset / portfolio) Example 30:

Calculating percentage and dollar VaR.

A risk management officer at a bank is interested in calculating the VaR of an asset that he is considering adding to the bank‘s portfolio. If the asset has a daily standard deviation of returns equal to 1.4% and the asset has a current value of $5.3 million . calculate the VaR (5%) on both a percentage and dollar basis.

Solution:

σ = 1.4% p= 5.3 million

VaR_(%) = - zσ = -1.65(.014) = -.0231 = -2.31%

VaR_($) = - zσ (p) = -1.65(.014)(5.3) = -.0122430

Thus, there is a 5% chance that, on a given day, the loss in the value of asset will exceed from 2.31% or $122,430.

Time Conversion of VaR

VaR, as calculated above measures the risk of a loss over a period of one day. We can calculate it for longer periods such as a week, month, quarter or year. It can be converted simply by multiplying the daily VaR to the required period.

HINT: (always consider 5 days in a week, 20 days in a month and 240 days in a year) To calculate multiply the daily value with √n

Example 31:

VaR(5%)daily = - zσ = -1.65(.014) = -.0231

VaR(5%)weekly = VaR(5%)daily √n = -.0231 x √5 = -.052 VaR(5%)month = VaR(5%)daily √n = -.0231 x √20 = -.10 VaR(5%)quarter = VaR(5%)daily √n = -.0231 x √60 = -.18

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VaR_(5%)year = VaR(5%)daily √n = -.0231 x √240 = -.358

Or we can calculate it from monthly or quarterly values of VaR.

VaR(5%)quarter = VaR(5%)monthly √n = -.10 x √3 = -.18 VaR_(5%)year = VaR(5%)monthly √n = -.10 x √3 = -.358

HINT: in the same way VaR_($) can be calculated by simply multiplying the value of VaR%

to asset value.

Assumptions of VaR 1. Stationarity

Stationary assumes that the probability of experiencing a fluctuation is the same in all periods. It also assumes that mean and variance do not change over the time.

Stationarity can be defined in precise mathematical terms, but here, we mean a flat looking series, without trend, constant variance over time, a constant autocorrelation structure over time and no periodic fluctuation.

2. Random walk

It means, a deviation in one period I independent form a deviation in another period.

Price follow a random behavior.

3. Non-negativity

Limited liability asset values are never negative. This assumption is violated by derivatives such as futures, forwards and swaps, which can have negative value.

4. Time consistency

All assumptions that apply in one period also apply in a multiple-period scenario. The assumptions for a 1-day period and a 1-week period are same.

5. Normal distribution

One period fluctuation in return is assumed to be normally distributed with a mean of zero and standard deviation of one.

Continuously Compounded Rates of Returns

Normally, we calculate VaR on daily basis. A risk manager may be interested in calculating VaR on a multi-period basis by extending the daily VaR using the square root rule.

In doing so, however, it is important that the distributional assumptions of the single-period VaR are preserved after transforming the measure into a multiple-period VaR. using continuously compounded rates of return allows the preservation of the single-period assumptions. We calculate compounded returns by natural log of returns.

R_t = ln(P_t / P_t-1)

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Thus, it preserves the assumption that VaR is normally distributed no matter what time period is used. In addition, continuously compounded returns also preserve the assumptions of

Stationarity and time consistency.

Exception: although it is generally preferable to use continuously compounded rates of return to calculate VaR, there is one exception. For interest rate related variables (i.e. yield to maturity, credit spread etc), compounded rates are inappropriate since the focus is on the absolute yield change in basis points. In this situation, VaR must be adjusted to account for duration and convexity.

Portfolio VaR

Portfolio risk, as measured by standard deviation, decreases as the correlation among assets within the portfolio decreases. In a similar manner, VaR is affected by the diversification effect that assets with low correlation bring to the portfolio. For a two-asset portfolio, VaR(%) is calculated as follows:

VaR(%)portfolio = - z σp =

VaR($)portfolio = - z σp(a,b) = (a,b, is the value of portfolio assets) Whereas:-

𝜎𝑝 = ω1²σ1² + ω2²σ2² + 2ω1 ω2 σ1σ2 r1,2 or σp2 = ω1²σ1² + ω2²σ2² + 2ω1 ω2 σ1σ2 r1,2 Example 32:

A fund manager manages a portfolio of two investments: A and B. Of the portfolio‘s current value of $6 million, A make up 4 million and B, 2 million. The standard deviation for A is .06 and for B 0.14. Correlation (r_1,2) is -0.5. Calculate the VaR for this portfolio.

Solution:

A = 4 m B= 2 m σ1 = 0.06 σ2 = 0.14 ω1 = 4/6 = 0.67 ω2 = 2/6 = 0.33 r_1,2 = -.5 VaR(%)portfolio = - z σp and VaR($)portfolio = - z σp(a,b)

𝜎𝑝 = ω1²σ1² + ω2²σ2² + 2ω1 ω2 σ1σ2 r1,2

𝜎𝑝 = . 67 ² . 06 ² + . 33 ²(.14)² + 2(.67)(.33)(.06)(.14)(−.5) = .044 VaR(%)portfolio = - z σp = -1.65(.044) = .0726

VaR($)portfolio = - z σp(a,b) = -1.659.044(6) = 0.43 million

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Factors affecting the portfolio VaR

There are four primary factors that affect the risk of a portfolio. These factors Include asset concentration, asset volatility, asset correlation and systematic risk.

1. Asset concentration: Asset concentration means weight of a particular asset in portfolio.

As the portfolio becomes heavily weighted towards one asset, portfolio risk increases.

2. Asset volatility: As the variance of assets within the portfolio increases, portfolio risk increases.

3. Asset correlation: As the asset correlation between assets increases towards +1, portfolio risk increases.

4. Systematic risk: As the number of assets within the portfolio becomes large, systematic becomes the more relevant factor for assessing additional assets.

Individual VaR

Individual VaR is the VaR of an individual position in isolation. Let the position weight be ωi, the portfolio vale be P, the position volatility be σi, then the individual VaR is;

VaR = -zσi ǀωiǀ p

We use the absolute value of the weight because both long and short positions pose risk.

Example:

VaR1 = 2.4, VaR2 = 1.6, what is VaR when r_1,2 is 0 ? Solution:

𝑉𝑎𝑅𝑝 = Var² + VaR2² = 2.4 ² + (1.6)² = 8.32

VaR, when assets are uncorrelated, positively correlated or negatively correlated.

1. No correlation r_1,2 = 0 𝑉𝑎𝑅𝑝 = Var² + VaR2² or

(𝑉𝑎𝑅𝑝)² = W1²(Var1)² + W2²(Var2)² 2. Positively correlated r_1,2 = + 1

𝑉𝑎𝑅𝑝 = W1(Var1) + W2(Var2)

3. Negatively correlated r_1,2 = - 1 𝑉𝑎𝑅𝑝 = W1 Var1 + W2 Var2

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VaR for portfolio with more than two assets (n, number of assets) a. Equally weighted;

1. Equal investment in all assets.

2. Same risk for each asset class.

Incremental VaR is the change in VaR from the addition of a new position in a portfolio. It can be calculated precisely from a total revaluation of the portfolio.

Component VaR

Component VaR for position I, denoted CVaR_i is the amount a portfolio VaR would change from deleting that position in a portfolio. In a large portfolio with many positions, the approximation is simply the marginal VaR multiplied by the dollar weight in position.

CVaR_i = MVaR_i (ω_i x p)

Credit risk

Credit risk refers to the risk that a borrower will default on any type of debt by failing to make required payments.^[1] The risk is primarily that of the lender and includes lost principal and interest, disruption to cash flows, and increased collection costs. The loss may be complete or partial and can arise in a number of circumstances. For example:

 Bond. A business or government bond issuer does not make a payment on a coupon or principal payment when due

 Loan. A consumer may fail to make a payment due on a mortgage loan, credit card, line of credit, or other loan

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 Firm. A company is unable to repay asset-secured fixed or floating charge debt

 Individual. A consumer does not pay mortgage loan, credit card, line of credit, or other loan

Assessing Default Probabilities

There may be two aspects of assessing the probability of default. Qualitative and Quantitative A. Qualitative Model

Following are some factors that must be evaluated under qualitative model before making decision about issuance of loan.

1. Collateral. Since collateral backed loan give the debt holder a first claim against specific assets of the borrower, collateral decreases the probability of default.

2. Leverage. Above certain debt-to-equity levels, the probability of default increase dramatically. A large debt burden imposes a drain on cash flows that are used to make interest and principal payments.

3. Volatility of earnings. As earning‘s volatility increases, the probability that a borrower will not be able to make debt payments also increases.

4. Reputation. The borrower‘s past credit performance is assumed to continue into the future. A borrower who has consistently made payments on time is more likely to receive favorable terms from the lender.

5. Business performance. The phase of business cycle has a significant effect on default probabilities. The phases under which, the business is currently running will show its ability to pay or default.

6. Other debts. The other debts a company is currently paying or requiring to pay off also decide the credit paying worth of the borrower.