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Subject: DURATION, SENSITIVITY AND PLA IN BONDS
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I would like to help some of you with a general explanation on how to
calculate sensitivity and PLA in bonds. Many of you may know these
issues,
but I prefered to send a general message. Please disregard this CM if
this
is your case.
The market factor (what generates the risk) in a bond, is the yield
(the interest rate embedded in the investment). This means that the
Position Sensitivity should relate to changes in yields. This
sensitivities,
then, multiplied by the volatility of the yields, would give us the PLA
associated with the bond positions (expected portential loss if the
yield
moves agains us).
To calculate the Position Sensitivity, first of all, you should know the
modified duration of the bonds that you are holding.
Duration is defined as the equivalent tenor in a bond, expressed in
terms
of a zero coupon bond (a bond that has only one payment at maturity and
it is traded at discount).
This means that for example, an investor should be completely indiferent
to invest in a zero coupon bond of 2.25 years than in a 4 years bond
(let's
say with annual principal and interest payment) with also a 2.25 years
duration.
How to calculate this duration (also known as Macaulay duration):
Let's suppose this bond's cash flow:
($100 bond with 4 equal annual principal payment and 10% interest rate
on
outstandings).
Let's also assume that we bought at $96 (at discount), equivalent to a
12% yield.
Coupons Disc at 12% % on price coupon tenor (1) * (2)
Ppal+ Interest in years (1) (in years)(2)
--------------------------------------------------------------------
1 25+10 = 35 31.25 33% 1 0.33
2 25+ 7.5= 32.5 25.91 27% 2 0.54
3 25+ 5 = 30 21.35 22% 3 0.66
4 25+ 2.5= 27.5 17.49 18% 4 0.72
------- -------- -------
96 100% 2.25
The duration of this bond is 2.25 years, even though the final maturity
is 4 years, because there are some coupons that are received before the
4 years. As you see, duration is related with the current level of yiels
How to calculate the modified duration:
Just by dividing the Macaulay duration by (1+the yield in one discount
period).
In the example above, the discount period is 1 year (it was done on an
annual basis, so we should discount the annual yield. However, if the
discount would have been done, for example, in a semi-annual basis, the
discount period would have been 6 months, and we should divide by the
semi-annual yield).
Modified duration = macaulay duration divided by (1+yield)
Modified duration = 2.25 / (1.12) = 2.01
How to calculate Position Sensitivity:
PS = Volume of position * 0.01 * modified duration (unit shift =
1%)
PS = Volume of position * 0.0001 * modified duration (unit shift =
1bp)
How to calculate PLA:
PLA = PS * yield volatility * square root of days in the defeasance
period
Note that yield volatility should be expressed in terms of 1% if the
unit
shift is 1% or in terms of 1 bp, if the unit shift is 1bp.
General examples:
1) Let's assume we have the bond of the example above ($96.000
position),
the unit shift considered is 1bp, the O/N volatility of the yield is 60
bps
and the defeasance period is 4 days
PS = 96.000 * 2.01 * 0.0001 = $19.3 (each time the yield changes 1bp,
the
position changes $19.3)
PLA = 19.3 * 60 * square root of 4
PLA = 19.3 * 120 = $2316 (if the yield moves 120 bps in the wrong
direction,
the potential loss would be $2316)
1) Let's assume we have the bond of the example above ($96.000
position),
the unit shift considered is 1%, the O/N volatility of the yield is 60
bps
(0.6%) and the defeasance period is 4 days
PS = 96.000 * 2.01 * 0.01 = $1930 (each time the yield changes 1%, the
position changes $1930)
PLA = 1930 * 0.6 * square root of 4
PLA = 1930 * 1.2 = $2316 (if the yield moves 1,20 % in the wrong
direction,
the potential loss would be $2316)
As you see, the PLA for both examples is the same. By changing the unit
shift, we only change the way we report sensitivity, but the risk of the
whole transaction (PLA) should be the same.
Word Count: 736
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