Duration.pptx

8/10/2019 Duration.pptx

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Interest Rate Risk Management


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Interest Rate Risk

Interest rates change substantially over time and

their variation poses large risks to financial

institutions, portfolio managers, corporations and

the governments.

A systematic methodology is required to assess the

riskiness of a bond portfolio to movements in

interest rates.

A methodology is also required to effectively manage

such risk.


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Duration

Duration of a security with price P is the (negative of the)

percent sensitivity of the price P to a small parallel shift in the

level of interest rates.

Let r(t, T) be the continuously compounded term structure of

interest rates at time t.

Due to uniform shift of size dr across rates, the price of the

security moves by dP

The duration of the asset is then defined as:

Duration =1 dP

DP dr


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Duration

Given the duration DP of a security with price P, a uniformchange in the level of interest rates brings about a change inthe value of:

Change in portfolio value= dP = - DP *P*dr

For example, if a Rs.100 million portfolio has a duration of 10,a one basis point increase in the level of interest rates willresult in a loss of Rs.100,000 in the portfolio value as shown

below:

Change in portfolio value= dP = - 10*Rs.100 m*0.0001

= - Rs.100,000


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Duration of a Zero Coupon Bond

For a Zero Coupon Bond we get:

Pz(t,T) =100 * Z(t,T) = 100* e-r(T-t)

= -(T-t)*Pz(t,T)

= T-t (Time to maturity)

( )100 ( ) r T tZdP T t ed r

1 d PD

P d r


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Example

The duration of a 5-year STRIPS is 5.

This implies that a one basis point increase in interest

rates decreases the value of the portfolio by 5 basis

points. A portfolio of Rs.100 million will experience a decline of

approximately Rs.50,000.

dP =-DP *P*dr = -5*Rs.100 million*0.0001 = -Rs.50,000


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Duration of a Portfolio

Duration of a portfolio is a weighted average of the duration

of assets, where the weights correspond to the percentage of

the portfolio invested in a given security

The duration of a portfolio of nsecurities is given by

where wiis the fraction of the portfolio in security i, and Diis

the duration of security i

1

n

W i i

i

D w D


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An example:

A bond portfolio manager has Rs.100 millioninvested in 5-year STRIPS and Rs.200 millioninvested in 10-year STRIPS

The impact of a one basis point parallel shift ofthe term structure on the value of the portfoliocan be found by computing the duration of theportfolio

The 5-year and 10-year strips have duration of 5and 10, respectively

The total portfolio value is Rs.300 million


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Duration of the portfolio:

Therefore, a one basis point increase in interestrates generates a portfolio loss of Rs.249,000

Loss in portfolio value = Rs.300 million 8.3 0.01%= Rs.249,000

100 2005 10 8.3

300 300


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Duration of a Coupon Bond

A coupon bond can be considered a portfolio of the coupons

and the principal, so applying the same principal we have that

the duration of a coupon bond is:

Where for i= 1,,n-1;

,

1 1i

n n

W i z T i i

i i

D w D w T

/ 2 0,

0,

z i

i

c n

c P Tw

P T

1 / 2 0,

0,

z n

n

c n

c P Tw

P T


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Duration of a 10-year 6% Coupon Bond


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Duration of a Floating Rate Bond

The general formula for a semi-annual floating rate bond withzero spread sis:

The duration of a floating rate bond is simply equal to the timeleft to the next coupon payment date Ti+1t

In particular, if today is coupon date (but the coupon has notbeen paid yet), the duration is zero

1

2

1 1 2

1

1

,

,1100 1 / 2

,

1, 100 1 / 2

,

FR

FR

FR

i

i

FR

i i i

FR

i

dPD

P t T dr

dZ t T r T

P t T dr

T t Z t T r T P t T

T t

1 2, , 100 1 / 2FR i iP t T Z t T r T


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Dollar Duration

Duration implicitly assumes that the security, or the portfolio,

has non-zero value.

In many cases involving no arbitrage (Long-Short) strategies,

the security or the portfolio may have a value exactly equal to

zero.

In such cases and for certain swaps, we calculate the dollar

duration

Dollar Duration of a security is defined by

For a non-zero valued security: D$P= P DP

For a portfolio of nsecurities (D$W) with Niunits of security i

$

P

dPD

dr

$ $

1

n

W i i

i

D N D


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Price Value of a Basis Point

The dollar losses due to a basis point increase in the level of

interest rates is a common measure of interest rate risk and is

called price value of a basis point

The price value of a basis point PV01 (or PVBP) of a security

with price P is defined as:

Price value of a basis point = PV 01 (or PVBP) =$

P

D dr


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Dollar Duration and PVBP

Dollar duration and PVBP are also useful when comparing

investments with unequal dollar amounts.

A 30-year 5% coupon bond with a yield of 4.646% has a

duration of 15.9 years . It sells at a price of Rs.105.58.

A 30-year zero trading at a yield of 4.6775 has a duration of 30

years. The price of the zero coupon bond is Rs.24.98.


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Dollar Duration and PVBP

As the duration of the zero is much greater than the coupon

bond, the zero coupon bond might be considered much

riskier.

However, as the zero sells at a considerable discount to its par

value, the price risk of the zero for the same par value as the

coupon bond is much lower .

The dollar duration of the zero is 749.7 (Rs.24.98 x 30) and

that of the coupon bond is Rs.1,678.72 (Rs.105.58 x 15.9)

The PVBP of the zero is 0.0749 and that of the coupon bond is

0.1678.

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