QFIP-158: Guide to DV01 and Risk Transformations

Question 1

Describe DV01

Answer 1

DV01 is the derivative of price of a fixed income instrument with respect to its yield:
DV01
dPV
dy

y is typically the yield-to-maturity

For bonds/swaps with $100 notional, DV01 will be measured as dollars per 100 bp
change in yields

For actual portfolios, though, DV01 is more often measured as dollars per 1 bp
change in yields

Question 2

Describe Modified Duration

Answer 2

Modified duration is the derivative of the percentage change in bond price with
respect to its yield:
Modified Duration
100
V
dPV
dy 100
DV01
V

Question 3

Compare and contrast DV01 and modified duration

Answer 3

DV01 measures sensitivity in dollar terms (risk per unit notional), while modified
duration measures risk in percentage terms (per $100 invested)

Question 4

Suppose we have DpDv, the partial DV01s of a portfolio of instruments with respect to
the curve variables. Describe how to transform the partial DV01s of the portfolio in
terms of an alternative set of variables x

Answer 4

The partial DV01 of the financial instruments with respect to the alternate variables is
given by: DpDx DpDv DvDx:

Question 5

Suppose we have DpDv, the partial DV01s of a portfolio of instruments with respect to
the curve variables. Describe how to transform the partial DV01s of the portfolio in
terms of an alternative set of yields y through the prices of the alternate instruments x

Answer 5

DpDy


BP1
By1
. . . BP1
Byk …
. . .
…
BPn
By1
. . . BPn
Byk


DpDvDvDy


BP1
Bv1
. . . BP1
Bvk …
. . .
…
BPn
Bv1
. . . BPn
Bvk




p
dx1
dy1
q1 Bx1
Bv1
. . . p
dx1
dy1
q1 Bx1
Bvk …
. . .
…
p
dxk
dyk
q1 Bxk
Bv1
. . . p
dxk
dyk
q1 Bxk

Question 6

Is the DV01 of a bond/swap with respect to the par rate equal to the DV01 with respect
to the par yield?

Answer 6

In general, the DV01 of a bond/swap using the par yield is not equal to the DV01 using
the par rate

This is because when the market yield curve is not flat, the derivative of the
yield-to-maturity and the par rate with respect to the curve variables (i.e. forward
rates) are different:
Bpar coupon
Bvi

Bpar yield
Bvi

Question 7

Is a 10 bp parallel shift in the par swap rates equivalent to a 10 bp parallel shift in
continuously compounded zero yields?

Answer 7

No, this is not true. As a result, the total DV01s between the two metrics are slightly
different.