Risk Factor: Full Yield

We consider a historical simulation, where the risk factors are given by the full yield curve, Ri(t) for i = 1, . . . , n. The yield R(t, T −t) at time t0 ≤ t ≤ t1 for the remaining time to maturity T −t is determined by means of linear interpolation from the adjacent values Ri(t) = R(t, Ti) and Ri+1(t) = R(t, Ti+1) withTi≤T−t < Ti+1(for reasons of simplicity we do not consider

remaining times to maturityT −t < T1andT−t > Tn): R(t, T−t) =[Ti+1−(T −t)]Ri(t) + [(T −t)−Ti]Ri+1(t)

Ti+1−Ti

. (3.1) The present value of the bondP V(t) at timetcan be obtained by discounting,

P V(t) = 1

1 +R(t, T−t)T−t, t0≤t≤t1. (3.2)

In the historical simulation the relative risk factor changes

∆(_ik)(t) = Ri t−k/N −Ri t−(k+h)/N Ri t−(k+h)/N , 0≤k≤N−1, (3.3)

are calculated fort0≤t≤t1and each 1≤i≤n. Thus, for each scenariokwe obtain a new fictive yield curve at timet+h, which can be determined from the observed yields and the risk factor changes,

R(_ik)(t+h) =Ri(t)

1 + ∆(_ik)(t), 1≤i≤n, (3.4) by means of linear interpolation. This procedure implies that the distribution of risk factor changes is stationary betweent₋(N₋1+h)/Nandt. Each scenario corresponds to a drawing from an identical and independent distribution, which can be related to an i.i.d. random variableεi(t) with variance one via

∆i(t) =σiεi(t). (3.5)

This assumption implies homoscedasticity of the volatility of the risk factors, i.e., a constant volatility level within the observation period. If this were not the case, different drawings would originate from different underlying distributions. Consequently, a sequence of historically observed risk factor changes could not be used for estimating the future loss distribution.

In analogy to (3.1) for timet+hand remaining time to maturity T −t one obtains R(k)(t+h, T−t) =[Ti+1−(T −t)]R (k) i (t) + [(T−t)−Ti]R (k) i+1(t) Ti+1−Ti

for the yield. With (3.2) we obtain a new fictive present value at timet+h:

P V(k)(t+h) = 1

1 +R(k)_{(t+h, T−t)}T−t. (3.6)

In this equation we neglected the effect of the shortening of the time to maturity in the transition fromt tot+hon the present value. Such an approximation should be refined for financial instruments whose time to maturity/time to expiration is of the order of h, which is not relevant for the constellations investigated in the following.

Now the fictive present valueP V(k)_{(t+h) is compared with the present value}

for unchanged yieldR(t+h, T−t) =R(t, T −t) for each scenariok(here the remaining time to maturity is not changed, either).

P V(t+h) = 1

1 +R(t+h, T−t) T−t

The loss occurring is

L(k)(t+h) =P V(t+h)−P V(k)(t+h) 0≤k≤N−1, (3.8) i.e., losses in the economic sense are positive while profits are negative. The VaR is the loss which is not exceeded with a probabilityαand is estimated as the [(1−α)N+ 1]-th-largest value in the set

{L(k)(t+h)|0≤k≤N−1}.

This is the (1−α)-quantile of the corresponding empirical distribution.

2. Mean Adjustment:

A refined historical simulation includes an adjustment for the average of those relative changes in the observation period which are used for generating the scenarios according to (3.3). If for fixed 1 ≤ i ≤ n the average of relative changes ∆(_ik)(t) is different from 0, a trend is projected from the past to the future in the generation of fictive yields in (3.4). Thus the relative changes are corrected for the mean by replacing the relative change ∆(_ik)(t) with ∆(_ik)(t)−

∆i(t) for 1≤i≤nin (3.4): ∆i(t) = 1 N N−1 X k=0 ∆(_ik)(t), (3.9) This mean correction is presented in Hull (1998).

3. Volatility Updating:

An important variant of historical simulation uses volatility updating Hull (1998). At each point in time t the exponentially weighted volatility of rel- ative historical changes is estimated fort0≤t≤t1by

σ2_i(t) = (1₋γ) N−1 X k=0 γk ∆(_ik)(t) 2, 1_≤i_≤n. (3.10) The parameterγ∈[0,1] is a decay factor, which must be calibrated to generate a best fit to empirical data. The recursion formula

σ2i(t) = (1−γ)σ 2

i(t−1/N) +γ

∆(0)_i (t) 2, 1≤i≤n, (3.11) is valid fort0≤t≤t1. The idea of volatility updating consists in adjusting the historical risk factor changes to the present volatility level. This is achieved by

a renormalization of the relative risk factor changes from (3.3) with the corre- sponding estimation of volatility for the observation day and a multiplication with the estimate for the volatility valid at time t. Thus, we calculate the quantity

δ(_ik)(t) =σi(t)·

∆(_ik)(t)

σi(t−(k+h)/N)

, 0≤k≤N−1. (3.12) In a situation, where risk factor volatility is heteroscedastic and, thus, the process of risk factor changes is not stationary, volatility updating cures this violation of the assumptions made in basic historical simulation, because the process of re-scaled risk factor changes ∆i(t)/σi(t)) is stationary. For each k

these renormalized relative changes are used in analogy to (3.4) for the deter- mination of fictive scenarios:

R(_ik)(t+h) =Ri(t)1 +δ (k)

i (t) , 1≤i≤n, (3.13)

The other considerations concerning the VaR calculation in historical simula- tion remain unchanged.

4. Volatility Updating and Mean Adjustment:

Within the volatility updating framework, we can also apply a correction for the average change according to 3.4.1(2). For this purpose, we calculate the average δi(t) = 1 N N₋1 X k=0 δ_i(k)(t), (3.14) and use the adjusted relative risk factor changeδ_i(k)(t)−δi(t) instead ofδ

(k) i (t)

in (3.13).