11/5/2010
1
MGMT 4370 / MGMT 7760
Risk Management
Aparna Gupta
Lally School of Management and Technology
Office: PITTS 2104
Email: [email protected]
Phone: x2757
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Interest Rate Risk
Assessing Risks in FixedIncome Instruments 
Sensitivities
•
A popular risk measure among traders is “
DV01
”, meaning change
in value of a bond due to change in yield of
1 basis point or 0.01%
.
•
A more rigorous definition of measuring sensitivity of price of a
Bond w.r.t. interest rates (YTM) is defined by
Duration
.
23
1.
2.
3.
.(
)
.....
1
(
1)(
1)
(
n
cF
cF
cF
n cF
F
yy
y
y
D
P
+
+++
+
++
+
+
=
•
Duration (or Macaulay Duration)
is the weighted average of the
dates (in years) of each cash flow, where weights are the present
value of the cash payment divided by the sum of the weights, i.e.,
the price of the bond.
•
Duration of a coupon paying Bond is less than the maturity of the
bond
– It is exactly equal to the maturity of the bond if there are no
coupons.
– Longer the maturity of a bond, higher its duration.
Assessing Risks in FixedIncome Instruments 
Sensitivities
•
Modified Duration, D* = D/(1+y), and modified duration has a better
interpretation:
•
Higher the duration, greater the sensitivity of the Bond price to
change in yield.
•
But the priceyield relationship for a Bond is
Nonlinear
! Duration is
1
*
P
D
Py
Δ
−≅
Δ
only the
firstorder approximation of impact of change in yield
on the
price of a bond, so works for sensitivity of price to yield only in small
variations of the yield.
•
Convexity of the Bond price
: A second order adjustment to more
accurately predict change in value of a bond due to larger changes
in yield comes.
2
2
1
P
C
Δ
≅
Δ
Assessing Risks in Portfolio of FixedIncome
Instruments
•
Price sensitivity of instruments from the same yield curve can be
aggregated from individual sensitivities by calculating weighted
average duration of the instruments held in the portfolio.
•
But this is restrictive for a general portfolio.
**
12
11
2 2
()
PP
PD
yyy
ΔΔ
ΔΠ
=+≅
− +
ΔΔΔ
•
For a general portfolio of fixedincome instruments the robust
method is the
VaR methodology
– This requires models for future termstructures of interest rate
– Determine price of the instruments in the portfolio under this
modified termstructure
– Compute the change in value of the portfolio, and
– Determine the 1day or 10day VaR.
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 Fall '10
 Gupta
 Management, Derivatives, Derivative, Interest Rates, Forward contract, Interest rate swap

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