DURATION, SENSITIVITY AND PLA IN BONDS

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.