risk management topic 3: interest rate risk & interest rate risk measures

Risk Management

Topic 3:Interest Rate Risk & Interest Rate Risk Measures

Topic 3 - IR Risk 2

Interest Rate Risk

Interest rate risk is an important factor when analysing any security or transaction whose return depends on interest rates.

The risk arises from potential unexpected moves up or down in interest rates. Interest rate securities’ prices move inversely to yields, which results in:

Interest rates Up = capital loss for investor

Interest rates Down = capital gain but increased reinvestment risk.

Topic 3 - IR Risk 3

Interest Rate RiskTwo main types of risk for bond

investors: credit risk, and interest rate risk.

Interest rate risk cannot be reduced by “diversification” between issuers

I/R risk can only be managed by: investing in less sensitive bonds hedging

Topic 3 - IR Risk 4

Measures of Interest rate Risk

All are measures of sensitivity of a security’s value to changes in interest rates: Volatility PVBP/DVBP Duration Convexity

Topic 3 - IR Risk 5

Volatility (Bonds)Calculating volatility

Volatility is percentage change in bond price in response to a change in market yields, and is calculated by:

100%xprice Original

yieldin change 0.01% afor pricein Change

(note: Some texts and portfolio software take an average of the change in price for a 0.01 increase in yield and the change in price for a 0.01 decrease in yield and divide the result by 2. Whilst this is a more accurate result, for a small change in interest rates, the above formula is acceptable.)

Topic 3 - IR Risk 6

Example:

A 10yr bond is paying a coupon of 10%, current yield is 10%.What is the volatility of this bond?

Price per $100 at 10.00% = 100.000Price per $100 at 10.01% = 99.938

0.062%1

100x

100.00

99.938100.00

Volatility

Interpretation:For a 0.01% change in yield, the price of this security will change by 0.062%.

Topic 3 - IR Risk 7

PVBP

Price Value per Basis Point (aka DVBP) Change in value of a security in response

to a 1 basis point change in yield

Consider 2 securities: A 3-yr bond with face value $100,000 and

11% semi-annual coupon; 10-yr bond with FV $100,000 and 11% semi-

annual coupon

Topic 3 - IR Risk 8

PVBP

At 12% yield, 3yr bond has a value of:

34.97541$6)06.01(

000,10006.0

6)06.01(15500

P

At 12.01% yield, 3 yr bond has a value of:

14.97517$6)06005.01(

000,10006005.0

6)06005.01(15500

P

Therefore:

PVBP = 97541.34 - 97517.14 = $24.20

Topic 3 - IR Risk 9

PVBP

At 12% yield, 10 yr bond has a value of:

At 12.01% yield, 10 yr bond has a value of:

Therefore:

PVBP = 94265.04 – 94210.04 = $55.00

04.94265$20)06.01(

000,10006.0

20)06.01(15500

P

04.94210$20)06005.01(

000,10006005.0

20)06005.01(15500

P

Topic 3 - IR Risk 10

PVBP

The 10-year bond is more sensitive to interest rate movements than the 3-year bond. It will usually be true that longer term securities will

be more sensitive to interest rates

BUTSensitivity does not vary proportionally with

the term of the security. (Note that the PVBP10yr is not 3.33 times PVBP3yr)


Duration

Definition:

The Duration of a security is the weighted average time to receipt of all of the cash flows generated by the security.

The weights are the market value weights of each cash flow (present values of each cash flow divided by the price of the security)


Duration

CF

(1)

PV

(2)

Weight

(3)

Time

(4)

Duration (periods)

(3)x(4) 5500 *5188.68 **0.053195 1 0.053195

5500 4894.98 0.050184 2 0.100368

5500 4617.91 0.047343 3 0.142029

5500 4356.52 0.044663 4 0.178652

5500 4109.92 0.042135 5 0.210675

105500 74373.34 0.762480 6 4.574880

97541.35 1.000000 5.259799

Consider the cash flows on our 3-year bond:

68.5188

06.01

55001

PV

35.97541

68.5188Weight D = 5.26 Periods


Duration

Note that this approach to duration calculates duration in periods. In our example, the bond pays semi-annual coupons, therefore each period is one half year. The duration of our 3-year bond will be 5.26/2 = 2.63 years.

This is known as Macauley duration.

For securities that pay cash flows during their life, the duration will always be less than the term.


Duration – Partial PeriodsDealing with Partial Periods:

ALL Australian Commonwealth Government Bonds pay semi-annual coupons and that coupons are paid and the bond matures on the 15th day of the coupon month.

The approach used in the previous example assumes that we are trading the bond and/or estimating its duration on a coupon payment date, BUT, bonds are traded every day, so how do we estimate duration on other trading days?


Duration – Partial PeriodsEg: A 7/13 bond would mature on the 15th

July 2013, and would pay its coupons on 15th July and 15th January each year until it matures on 15th July 2013.

What if we want to estimate duration on a day that is NOT 15th January or 15th July? We need to take into account the partial coupon period remaining until the next coupon payment date, as a proportion of the whole of the current coupon period.


Duration – Partial PeriodsConsider a bond similar to the bond in our

previous worked example – it matures on 15th July 2008, and we want to estimate its duration on 10th April 2005. 6 full coupon periods remaining, plus the period from

10th April 2005 to 15th July 2005. Last coupon paid on 15/1/05, next is due on 15/7/05,

#days between these dates is 181. #days between 10/4/05 and 15/7/05 is 96, so

proportion of current coupon period remaining is 96/181 = 0.5304.

We adjust time weighting in our duration calculation and the PV calculation for each cash flow as follows


Duration – Partial Periods

CF PV Wt t Dcont5500 5332.62 0.053377 0.530

40.028311

5500 *5030.77 0.050355 1.5304

0.077064

5500 4746.01 0.047505 2.5304

0.120207

5500 4477.37 0.044816 3.5304

0.158219

5500 4223.93 0.042279 4.5304

0.191542

5500 3984.84 0.039886 5.5304

0.220586

105500 72109.92 0.721782 6.5304

4.713522

99,905.47 1.000000

5.509451

77.5030

06.01

55005304.1

PV D= 5.51/2 = 2.75


Duration Assumptions

Yield curve is flat and term structure is assumed to be flat.Changes in yield curve are constant across all maturities, i.e. yield curve moves in parallel shifts, which implies that all yields on all bonds are perfectly correlated.Only small yield curve changes occur at small intervals.


Duration Characteristics Duration generally increases with term to

maturity

For par and premium bonds, duration increases as maturity increases

Higher coupon bonds with the same maturity and yield will have a shorter duration than lower coupon bonds

Duration is higher for lower yielding bonds.


Duration Characteristics

Further points to note:

For Bank Bills and Zero-coupon Bonds, Duration will be equal to the term of the security.

Duration gives us a sensitivity measure in terms of time, which is of limited usefulness

Our interpretation of duration needs to be modified to give us a measure of price sensitivity to changes in interest rates.


Modified Duration

r

DD

1`

is the Macauley Duration discounted one period. This gives us a measure of interest rate sensitivity in percentage terms.

The Modified Duration of our 3-year bond will be:

%481.206.01

63.2`

D


Mod.D and Price

The relationship between price and modified duration is:

This can be rewritten as:

rDP

P

`

PrDP .̀

If we take r to equal 1 basis point, P gives us a representation of PVBP:

20.24$35.541,97$0001.0481.2 P


Mod.D Approximation

One method of approximating Mod. Duration uses the annuity factor:

i

iaD

n

in

11

`

Where n = #periods, and i = interest per periodFollowing our 3-yr bond example:

%458.2""9173.4

06.0

06.011'

6

06.06

periodsaD

Where a bond is trading at par, this estimate will be exact. Most of the time it will be unacceptably inaccurate.


Portfolio Duration

Security(1)

MV(2)

Duration (3)

Weight (4)

Dcont(3)*(4)=(5)

100,000 FV 3yr CGS

97,541.35 2.63 0.1349 0.3548

300,000FV 10yr CGS

282,795.12 6.19 0.3911 2.4209

200,000FV 6mth BAB

198,765.45 0.5 0.2749 0.1374

200,000FV 3mth BAB

143,987.65 0.25 0.1991 0.0498

723,089.57 1.0000 Dp = 2.96


Duration Gap

We have so far seen duration used as a measure of interest rate exposure for a single asset. We can use duration to measure the interest rate exposure in a portfolio of assets and liabilities.

DGAP = DA – K.DL

where:

Assets

sLiabilitieK


Duration Gap

Consider the following portfolio:

Assets $ Liabilities $Mortgages 1,000,000 Equity 500,00090 day Bills 1,000,000 Zeros (2.5yr) 1,500,000

2,000,000 2,000,000

(Assume mortgages have duration of 3.95 years)

DA = ½ x 3.95 + ½ x 0.25 = 2.1 years

DL = 2.5 years

22.088.11.25.20.2

5.11.2 DGAP


Duration Gap & Volatility

Once DGAP is determined, we can treat the portfolio as a single security.

We can use DGAP to estimate gains or losses on our portfolio in response to interest rate changes, as before:

rr

DGAP

P

P

)1(

For a 1% change in interest rates, and assuming a 13.5% expected return on all balance sheet components, we would expect:

0019.001.0)135.01(

22.0

P

P

or –0.19% change in the value of capital


Duration Gap

Rearranging the previous expression:

Prr

DGAPP

1

If we set r =1basis point, we can determine the PVBP for the share capital:

61.9$000,5000001.0135.01

22.0

P

So for a 1bp increase in interest rates, the value of the owner’s equity will decrease by $9.61


DGAP Characteristics

DGAP allows us to treat a portfolio of interest rate sensitive securities as a single security

If DGAP > 0, portfolio behaves as an asset as i/r , portfolio

If DGAP < 0, portfolio behaves as a liability as i/r , portfolio


Bond Vol. CharacteristicsProperty 1:

although directional P in response to y is same for all bonds, %P is different

Property 2: |%P| for a given bond is practically the same

regardless of direction, when |y| is very smallProperty 3:

Property 2 does not hold when |y| is largeProperty 4:

For a given large |y|, P > P


Bond Characteristics & Volatility

Characteristic 1: for given T and initial Y, C Volatility

Characteristic 2: for a given C and initial Y, T Volatility

Implications: If we want to increase a portfolio’s

volatility, switch to bonds with longer maturities or lower coupons (and vice versa)


PVBP

PVBP also varies with yield. Let’s look again at the 3-year security: If market yield goes from 11% to 12% (100

basis points), Price falls by $2458.66,

BUT PVBP at 12% yield is $24.20, suggesting a

price change of $2420, and PVBP at 11% is $24.97, suggesting a price

change of $2497.


Duration

Let’s look again at the 3-year security: If market yield goes from 12% to 11.5% (50

basis points) Recalculating the bond’s price directly we get

a new price of $98,760.95, a price change of $98,760.95 - $97,541,35 =

$1219.60

BUT Using Modified Duration to estimate the price

change we get:

210,1$35.541,97$005.0481.2 P


Price-Yield RelationshipWe know that bond price changes inversely with the change in yields. However, this change is not linear i.e. if interest rates changes 5%, prices will not change 5%. The diagram below helps illustrate the convex relationship:

1

r

P

Price

Yield

2

r

P

1

r

P

2

r

P

P when yields are low

P when yields are high

Note:

for a given absolute r1 = r2,

Plow yield > Phigh yield


Convexity•Duration serves well for small changes in yield.•Duration assumes a linear relationship between changes in prices and changes in yield•As changes in yield become larger, the relationship between change in price and change in yield becomes convex:

Large changes in yield require a more complex measure – sensitivity to large changes in yield is measured by “convexity”.

Price

Yield


Convexity

Duration measures the ‘slope’ of present value profile, whereas convexity measures the curvature of the present value profile.

Convexity: increases with maturity decreases with rising coupon decreases with rising interest rates.

– fin

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