 # PORTFOLIO DELTA-NORMAL VAR

The VaR - value-at-risk - is the value of potential losses, over a given horizon, which will not be exceeded in more than a given fraction of all possible events. The delta-normal VaR applies to linear instruments in that it relies on the constant sensitivity to risk factor assumptions. Furthermore, it assumes a normal distribution of the portfolio value. Such assumptions should be relaxed with non-linear portfolios, such as those with optional instruments.

## Steps for Determining VaR

The process for detraining portfolio VaR requires several steps:

1 The first step is to map the positions to the risk factors influencing their values.

2 Mapping of each position to risk factors allows defining their sensitivities with respect to each risk factor. Sensitivities are the "deltas" of the delta-VaR.

3 The variations of the portfolio value become a linear function of the variations of each risk factor, if we assume that sensitivities are constant.

4 The volatility of the portfolio becomes the volatility of a linear function of random variables. Volatility is defined for a given horizon, for example daily.

5 Moving from volatility to VaR implies an assumption about the distribution of the changes of value. The delta-normal VaR relies on the normal distribution assumption.

6 Since confidence levels and multiples of volatility are related in normal distributions, the VaR at given confidence level follows.

As a preliminary to a more complex example, we use a simple example with two simple assets.

## A Simple Portfolio of Two Zero-coupon Bonds

It is easier to illustrate the process and highlight dependencies effect on a simple two-asset portfolio in a first step (Table 35.1). The portfolio is made of two zero-coupon bonds with the following characteristics.

The four steps for deriving delta-VaR are:

TABLE 35.1 Portfolio of two zero bonds

 Portfolio Bond A Bond B Value 1000 500 Duration 2 5 Interest rate volatility (annually) 2% 1%

• mapping to market parameters

• deriving sensitivities to these risk factors

• deriving the volatility of the portfolio value

• deriving the VaR under the normal distribution assumption.

##### 35.1.2.1 Mapping to Risk Factors

A zero-coupon bond has only one flow at maturity, hence it depends on a single interest rate, the one observed on the yield curve for this maturity. The values of each of the two zero-coupon bonds B and Bb depend on interest rates i and ib. These two interest rates are the risk factors.

##### 35.1.2.2 Sensitivities to Selected Market Parameters

The portfolio value of two bonds is B + Bb. The random changes of value of the portfolio depend on the sensitivities, or durations, of each bond: If sensitivities and initial values of each bond are constant, the random changes of value of the portfolio are a linear function of the changes of the random market parameters The volatility of the portfolio value is the volatility of a linear function of two random variables, Ai and Ai., as long as we can consider current values of bonds and durations constant. The a b' ° initial data on the portfolio is in the table below. The portfolio value is B^ + Bh = 1500. The random changes of the portfolio value are: The volatility of the portfolio value is the volatility of a linear function of two random variables as long as we can consider current values and durations constant.

##### 35.1.2.3 Volatility of the Portfolio Value

The volatility of the portfolio value is the volatility of a linear function of two random variables, Ai and Ai., as long as we can consider current values of bonds and durations constant. The a b' ° two interest rates, to which each position is mapped, do not deviate by the same amount at the same time, which implies inputting their correlations.

The standard formula for the volatility of two variables uses the volatilities of each random variable and their correlation. Volatilities and correlations are derived from historical time series. The standard formula starts from variance "a2" and derives volatility "a" (standard deviation) as the square root of variance: In this general equation, a and b are constants, X and Y are random, is their correlation (between -1 to +1), O is the standard deviation. Using this formula with the random variations of portfolio value, we have: Using annual volatilities in this example: o(A/a) — 2% and o(A/b) 1%, we need to input the correlation pab, for example 30% for finding the portfolio value variance and volatility.

##### 35.1.2.4 VaR under the Delta-normal Assumptions

Finally, we use the normal distribution multiples for determining the variance of the portfolio value, then its volatility and its VaR as a loss percentile.

The example in Table 35.2 shows that using simple assumptions, the VaR is easily calculated when there are dependencies. The VaR is proportional to the horizon used for volatilities. For varying the horizon, we use the scale root of time rule for adjusting volatilities and calculating the VaR. If we need a daily volatility, we scale down the annual volatilities by the factor 1/V250. The portfolio daily volatility is 53.15/^250= 53.15/15.811 = 3.362 and the daily VaR becomes 26.33.

A more elaborate example, calculating the VaR for a single forward contract on foreign exchange, illustrates more comprehensively the VaR calculation process. Before that, we address some general issues.