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Sensitivities measure the response to a given shock of the underlying random parameter, or risk factor referring to a given change of risk factors (such as a 1% shift of interest rates).

Sensitivities are not constant and vary when market conditions change. They are "local" measures and considering them as constant is only a proxy. The zero-coupon bond depends on a single risk factor, the 2-year interest rate applicable as of current date t, or i(t, T), where t is the current date and T the maturity date equal to 2 years. The interest rate is, for example, a risk-free rate: i(t, 2) 5%. The value of the risk-free zero-coupon bond with maturity T— 2 years is the discounted value of the final cash flow F(T):

Dropping subscripts, t being current date and T being equal to 2 years, and using a final cash flow of 100, the risk-free zero-coupon bond value is:

The response of value S to a change of / by Ai is the derivative of B with respect to /.

The derivative can be calculated from the closed-form theoretical formula. In the case of a bond, it is the duration. The duration of a zero-coupon bond equals its maturity T—2. The random value variations of the zero-coupon bond are a linear function of the random deviations of the interest rate. Using D as duration, with D = 2:

With S 90.703, and Ai = 1%, the monetary value of the bond sensitivity fora 100 basis points change of/ is €1.81 or 1.81% of current value.

From Sensitivity to Volatility

The volatility of a position depends on both its sensitivity and the volatility of the underlying risk factors. When there is a single risk factor, the relation is quite simple. When there are several risk factors, all sensitivities to all factors and their dependencies have to be considered. Dependencies are dealt with a separate set of chapters (see Section 9).

The volatility measures the magnitude of the variations of a random variable, such as returns or market parameters. Measuring volatility is discussed in the previous chapter (Chapter 16). In this example, we rely on the common simple measure of historical volatility. The historical volatility is the standard deviation of a time series of observations. The daily historical volatility of interest rate i, Oj(i), is calculated as the standard deviation of the interest rate observed over 250 working days in a year.

There is a linear relation between random value variations and random interest rate variations: AS — -D B Ai, considering the initial value of the bond as known as well as its duration, and assuming that the duration is approximately constant. Both AS and Ai are random variables. From the relationship, the magnitude of the changes of the bond value depends on both its sensitivity and the magnitude of the variations of interest rates. The standard deviation of a random variable multiplied by a constant is the product of the constant and the volatility of the variable. From above relation, we find the relationship between the volatilities of the two random variables is straightforward.

For proceeding, the daily volatility of / is required. It can be measured in percentage terms or in absolute terms. Let us use 0.1% as the daily volatility of the 2-year rate. The volatility is in monetary units, like the value of the bond. The monetary value of the sensitivity of the zero-coupon is:

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