BBA FN415 (18) Lect 09 VAR Method.pptx

VaR Methods

VaR Methods

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VaR: Definition



_{Value at Risk is a summary measure}

of downside risk expressed in dollars,

or in the reference currency.



_{A general definition is}

“VaR is the maximum loss over a

target horizon such that there is a

Local vs. Full Valuation



_{Local valuation}

_{methods make use of}

the valuation of the instruments at

the current point, along with the first

and perhaps the second partial

derivatives.



_{Full valuation}

_{methods, in contrast,}

Local Valuation



_{Local valuation methods, also known}

as analytical methods.



_{These include linear models and}

nonlinear models.



_{Linear models are based on the}

covariance matrix approach.



_{Nonlinear models take into account}

the first and second partial

Local Valuation

 _{VAR was born from the recognition that we need}

an estimate that accounts for various sources of risk and expresses loss in terms of probability.

 _{Extending the duration equation to the worst}

change in yield at some confidence level dy,

 _{For a long position in the bond, the worst}

Local Valuation

 _{This approach is called local valuation, because it}

uses information about the initial price and the exposure at the initial point. As a result, the VAR for the bond is given by

 _Advantage  simplicity: The distribution of the price is the same as that of the change in yield.

 _{Convenient for portfolios with numerous sources}

0.02 0.04 0.06 0.08 0.1 y 20 40 60 80 100 120 140 160 P

Example with a simple Bond

 _{The price of a coupon bond is given below.}

 _{If we set c = 5 and F = 100, plot the bond price as a}

function of y.

BondPrice





i 1 n

c



1



y





100

Using Duration to approximate the change in bond price



_{The change in the price of the bond is}

linked to the modified duration by the

expression



_{If we look at the bond with c = 5 and F =}

100, and y = 6% with price 92.64. Then

the change in price can be

approximated.



P





D



P



y

where

D





1

P





P

0.02 0.04 0.06 0.08 0.1 y 20

40 60 80 100 120 140

P

Using Duration to approximate the change in bond price

Linear approximation vs true price

y 20

40 60 80 100 120 140 160

P

Approximation bad when Δy is large

0.02 0.04 0.06 0.08 0.1 y 80

100 120 140 160

P

Linear approximation vs exact price

VaR(99%) Δy = +1.59%

= (-7.57*92.64)x(1.59%) = -11.1481

Linear approximation vs exact price



_If

_{y = 7.59 (= 6.00 + 1.59), c = 5}

_{. Then}

the price of bond is

82.2949



_{The approximation gives the price of}

the bond of



_{P + ΔP =}

_{92.6399 -11.1481}

_{= 81.4948}

Price of a coupon bond P 



i 1 n

c





100

Practice questions

(Q1) Using linear

approximation, if the interest

rate VaR(99%) is +1.5%. Find

the 99% VaR for a bond with a

modified duration of 25 years

and a current price of

Practice questions

(Q2) Using linear approximation, if

the interest rate VaR(95%) is

+0.7%. Find the 95% VaR for a

Full Valuation



_{More generally, to take into account}

nonlinear relationships, one would have to

reprice the bond under different scenarios

for the yield.



Defining

y

as the initial yield



_{We call this approach full valuation}

Full Valuation

dPworst  P









 P













i 1

10

5





100









10

5





100







 82.2949  92.6399

  10.3451

Note that the VaR from the full valuation and the linear

Example of the Full Valuation

Consider a zero coupon bond with maturity 20 years, face value 100. If the interest rate VaR at 98% is +3.4%. The current price of the bond is 35. Calculate the VaR at 98% for this bond.

(1) Calculate the yield to maturity.

(2) Find the bond price with yield (y₀+ dy)

Delta-Gamma Method

 _{Ideally, we would like to keep the simplicity of the local}

valuation while accounting for nonlinearities in the payoffs patterns.

 _{Using the Taylor expansion,}

How well does the delta-gamma model perform.

0.02 0.04 0.06 0.08 0.1 y

80 100 120 140

P

This one uses both the first and the second derivative to approximate the price of the bond, note that we start seeing some curvature coming in.

This is the approximation

made using the first derivative only, namely the modified

How well does the delta-gamma model perform.

y 80

100 120 140 160

P

Performance of Gamma-Delta VaR



_{The improvement brought about by this}

method depends on the size of the

second-order coefficient, as well as the

size of the worst move in the risk factor.



_{For forward contracts, for instance, Γ = 0,}

and there is no point in adding second

order terms. Similarly for most

Example: Delta-Gamma Model

Calculate the 95% VaR on the bond whose current price is 50, current yield is 6%, the modified duration is 4.2 and the