Duration Convexity Bond Portfolio Management

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DURATION ANDCONVEXITY



Duration

In 1938, Frederick Macaulay suggested a method for

determining price volatility of bonds.

He gave the name duration to the measure

now called Macaulay duration

Due to regulation there was very little volatility of interest

rates, so Duration was not popular till the 1970s

1970s onwards interest rates started to rise dramatically Investors and traders became interested



Measures how long, in years, it takes for the price of

a bond to be repaid by its internal cash flows.

Or Weighted average life of a bond

Which considers the size and timing of each cash flow

The weight assigned to each time period is the PV of the

cash flow paid at that time as a proportion of the price of

the bond

Macaulay Duration



Features of Duration

Duration for Zero coupon bonds will be same as

maturity

The Duration has a value between 0 and maturity

period



Duration of a Zero-Coupon Bond(maturing in 4 years)

The red lever above represents the four-year time period it takes for a zero-coupon

bond to mature.The money bag balancing on the far right represents the future value of the bond.

The fulcrum, represents duration, which must be positioned where the red lever is

balanced.

The entire cash flow of a zero-coupon bond occurs at maturity, so the fulcrum is located

directly below this one payment.



Duration of a Coupon Bond(coupons paid annually and maturing in five years)

The moneybags represent the cash flows you will receive over the five-year period.

To balance the red lever at the point where total cash flows equal the

amount paid for the bond, the fulcrum must be farther to the left, at a

point before maturity.

It pays coupon payments throughout its life and therefore repays

the full amount paid for the bond sooner



Bonds with high coupon rates and, in turn, high yields will tend

to have lower durations than bonds that pay low coupon rates

or offer low yields.



Macaulay Duration

The Formula

where:t = time period in which the coupon or principal payment occurs

Ct = interest or principal payment that occurs in period t

i = yield to maturity on the bond

price

)(

)1(

)1(

)(

1

1

1

n

t

t

n

t

t

t

n

t

t

t C PV t

i

C

i

t C

D



t wt Duration

T

t

1

CF Cash Flow for period t t

Or



Coupon 8% paid annually;

Time to maturity 8 years

Discount rate 10% pa

Face and maturity value 1000

Calculate the Duration of the Bond



Time Cash Flow PV at 10% Time x PV

1 80 72.73 72.732 80 66.12 132.23

3 80 60.11 180.32

4 80 54.64 218.56

5 80 49.67 248.37

6 80 45.16 270.95

7 80 41.05 287.37

8 1080 503.83 4030.62

Sum 5441.15

Price 893.30

Duration =Sum/Price 5441.15/893.3 6.09105

years



Time Cash Flow PV at 10%

PV’s Proportion

of Price (Wt)

Time X PV’s

Proportion of Price

1 80 72.73 0.08 0.081

2 80 66.12 0.07 0.148

3 80 60.11 0.07 0.202

4 80 54.64 0.06 0.245

5 80 49.67 0.06 0.278

6 80 45.16 0.05 0.303

7 80 41.05 0.05 0.322

8 1080 503.83 0.56 4.512

Price 893.3 1.00

Duration 6.091

(in years)

Duration Calculation



Betty holds a five-year bond with a par value of $1,000 and coupon rate of

5%. For simplicity, let's assume that the coupon is paid annually and that

interest rates are 5%. What is the Macaulay duration of the bond?

= 4.55 years



Another example

Coupon 12% annually

Time to maturity 8 years

Discount rate 10% pa

FV and MV 1000



Time Cash Flow PV at 10% Time x PV

1 120 109.09 109.09

2 120 99.17 198.35

3 120 90.16 270.47

4 120 81.96 327.85

5 120 74.51 372.55

6 120 67.74 406.42

7 120 61.58 431.05

8 1120 522.49 4179.91Sum 6295.69

Price 1,106.70

Duration = 6295.691/1106.7

= 5.689



Time Cash Flow PV at 10%

Proportion of

Total value

Time X

Proportion of

Total Value

1 120 109.09 0.10 0.099

2 120 99.17 0.09 0.179

3 120 90.16 0.08 0.244

4 120 81.96 0.07 0.296

5 120 74.51 0.07 0.337

6 120 67.74 0.06 0.367

7 120 61.58 0.06 0.389

8 1120 522.49 0.47 3.777

Price 1,106.70

Duration 5.689



Importance of Duration

Measures bond price sensitivity to interest rate movements,which is very important in any bond analysis If two bonds have the same coupon rate and yield, then the bond

with the greater maturity has the greater duration.

If two bonds have the same yield and maturity, then the one withthe lower coupon rate has the greater duration.

Bonds with higher durations carry more risk and have higherprice volatility than bonds with lower durations.

Allows comparison of effective lives of bonds that differ inmaturity, coupon

Used in bond management strategies particularlyimmunization



But Duration does not tell investors exactly how

much a bond's price changes given a change in

yield.

There is a relationship between Macaulay durationand the first derivative of the price/yield function.

This relationship lead to the definition of modified

duration



Modified Duration and Bond Price Volatility

Modified Duration provides a good approximation,

particularly when interest-rate changes are small, for how

much the security price changes for a given change in

interest rates Modified Duration Formula (D mod)

mYTM1

DurationMacaulay

mod

D

where:

m = number of payments a year

YTM = nominal YTM



Betty holds a five-year bond with a par value of $1,000 and

coupon rate of 5%. The coupon is paid annually and that interest

rates are 5%. Macaulay duration is 4.55

= 4.33 years for Betty’s problem



where:

Dmod is the modified duration;

D is the Macaulay duration;

i is the periodic yield;

P(i) is the price of the bond at yield i .

This formula can be used to estimate the change in price for a small change

in the periodic yield:



Modified Duration and Bond Price Volatility

As A Measure of Bond Price Volatility

Bond price movements will vary proportionally with

modified duration for small changes in yields

% change in price = i D P

P

mod

where:

P = change in price for the bond P = beginning price for the bond

-Dmod = the modified duration of the bond

i = yield change in %



A Bond with Mac D of 8 years and YTM to be 10% with semi annualcompounding

Modified Duration = 8/(1+.05) = 7.62

Assume the YTM to decline by 75 basis points from 10 to 9.25%

The estimate change in the price of the bond

= - 7.62 x (-0.75%) = 5.72%

This means the bond price should increase approximately 5.72% inresponse to the 75 basis points decline in YTM

If price was Rs 900 before change, then after the drop in interestrates, the price would be 951.48

M difi d D ti d B d P i V l tilit



Modified Duration and Bond Price Volatility:

Trading Strategies Using Modified Duration

To maximize returns fund managers constantly adjust theduration of the bond portfolio

Longest-duration security provides the maximum price

variation

If you expect a decline in interest rates, increase the

average modified duration of your bond portfolio to

experience maximum price volatility i.e., buy long bonds

If you expect an increase in interest rates, reduce theaverage modified duration to minimize your price decline

i.e. sell long bonds and buy short bonds or come into

cash.



Bond Convexity

Modified duration is a linear approximation of bond pricechange for small changes in market yields

But, price changes are not linear, but a curvilinear (convex)function of bond yields

Convexity refers to the degree to which duration changes as theyield to maturity changes

The estimate using only modified duration will underestimate theactual price increase caused by a yield decline and overestimatethe actual price decline caused by an increase in yields

Modified Duration is to be combined with the convexity to get abetter approximation of price







Some relationships

There is an inverse relationship between coupon and

convexity (yield and maturity constant) — that is, lower

coupon, higher convexity.

There is a direct relationship between maturity andconvexity (yield and coupon constant) — that is, longer

maturity, higher convexity.

There is an inverse relationship between yield and

convexity (coupon and maturity constant). This means

that the price – yield curve is more convex at its lower-

yield (upper left) segment.



18-29

Exhibit 18.21



Bond Convexity

The Formula

P

di

P d

2

2

Convexity



3 year Bond 9% YTM

12% Coupon FV 1000

Years CF PV t2+t Product

1 120 110.09 2 220.18

2 120 101.00 6 606.01

3 1120 864.85 12 10378.15

Price 1075.94 Sum 11204.34

Sum/(1.09^2) 9430.47

Convexity 8.76



Price change due to convexity

= ½ x convexity x (change in yield^2)



Example

Consider an 8% 10-year bond at a price of 100 and a modifiedduration of 6.80.

If the yield increases to10%, duration estimates the price change asfollows:

Price change = -[Duration] x [Yield Change]

Price change = -[6.80] x [2]

Price change = -13.60

The duration estimated price is 86.40 (100 – 13.60).

Similarly, for a 2% decrease in yields, the estimated price is 113.60(100 + 13.60).

The actual prices should be 87.71 and 114.72, and therefore,duration tends to underestimate the price during both rising andfalling rates.



We can get a better approximation of the new price byincluding convexity :

Price Change = (- Duration x Price Yield) + (0.5 x Convexityx (Yield Change)^2))

If the 8% 10-year bond has a 0.60 convexity, the newestimated price change is :

Price Change = (-6.80 x 2) + (0.5 x 0.60 x 4)

Price Change = -12.40

The estimated price using convexity is now 87.60 (100 – 12.40).

The convexity estimate of 87.60 is much closer to the actualprice of 87.71 than the duration estimate.



Approaches for measuring interest rate risk:

Full Valuation Approach

Simplest yet comprehensive way to measure interest rate risk in abond.

We start with the current market yield and price of the bond.

Then we fix on the different scenarios (interest rate changes) atwhich we want to value the bond, say a 0.5% increase in interest

rates. We then re-value the bond for each interest rate scenario.

The new value is then compared to the current value to determinethe gain/loss due to changes in interest rates.

This method is also sometimes referred to as scenario analysis.

While performing scenario analysis on a portfolio of bonds, eachbond is re-valued at different interest rates and the portfolio valueis recalculated.



Approaches for measuring interest rate risk:

The Duration Convexity Approach

Full Valuation Approach is recommended and mostaccurate approach to measuring interest rate risk

But it is not always practical especially when it’s a large

portfolio.

The full valuation approach is also very time consuming.

Managers may prefer a simpler approach which couldquickly give them an idea of how bond prices willchange with changes in interest rates.

This can be achieved by using the duration/convexityapproach.



BOND PORTFOLIO MANAGEMENT

STRATEGIES



Bond Portfolio Strategies

Passive Portfolio Strategies

Active Management Strategies

Hybrid: Immunization



Passive Portfolio Strategies

Buy and hold

A manager selects a portfolio of bonds based on theobjectives and constraints of the client with the intent ofholding these bonds to maturity

Indexing

The objective is to construct a portfolio of bonds that

will track the performance of a bond index




Active management strategies attempt to beatthe market

Mostly the success or failure is going to come

from the ability to accurately forecast futureinterest rates




Forecasting Interest-rate changes Riskiest as it involves relying on uncertain forecasts of

interest rates

Preserve capital if interest rates are expected to increase

Achieve high capital gains if interest rates are expectedto decline

Valuation Analysis Select bonds based on their intrinsic value

Credit Analysis: detailed analysis of the bond issuer todetermine expected changes in its default risk

Exploiting Mispricing among securities



Hybrid Techniques

Immunization Strategies

The process is intended to eliminate interest rate risk that

includes:

Price Risk

Coupon Reinvestment Risk

A portfolio manager (after client consultation) may

decide that the optimal strategy is to immunize the

portfolio from interest rate changes

The immunization techniques attempt to derive a

specified rate of return during a given investment horizon

regardless of what happens to market interest rates



Classical Immunization

Immunize a portfolio from interest rate risk by

keeping the portfolio duration equal to the

investment horizon



Consider a 12.5% bond redeemable on

1/July/2020 at a premium of 5%. If the interest

rates prevailing is 15% on 1/July/2015, what will

be price of the bond?



Face Value 100 Interest rate is 15% Coupon

12.50% Redeemable at 5% premium

Date 1-Jul-16 1-Jul-17 1-Jul-18 1-Jul-19 1-Jul-20 Total

No. of

years 1 2 3 4 5

Cash Flow 12.5 12.5 12.5 12.5 117.5

Present

Value 10.87 9.45 8.22 7.15 58.42 94.11 ie the Price



Calculate the Duration


No. of years 1 2 3 4 5

Cash Flow 12.5 12.5 12.5 12.5 117.5

Present

Value 10.87 9.45 8.22 7.15 58.42 94.11

Year * PV 10.87 18.90 24.66 28.59 292.09 375.11

DURATION 375.11/94.11

3.99 or 4 years

An investor buys the 5 year bond on 1/July/2015 Sells the



An investor buys the 5 year bond on 1/July/2015 Sells the

bond on 1/July/2019; Holding period 4 years same as the

Duration ; Reinvest the interest amounts at 15%


No. of

years 1 2 3 4 4

Cash

Flow 12.5 12.5 12.5 12.5 102.17(is the PV of 117.5 to be

received after one year)

Terminal

Value 19.01 16.53 14.38 12.50 102.17 164.59 at 15%



If the interest rates rise to 16%

Date 1-Jul-16 1-Jul-17 1-Jul-18 1-Jul-19 1-Jul-19Total

No. of

years 1 2 3 4 4

Cash Flow 12.5 12.5 12.5 12.5 101.29

Terminal

Value 19.51 16.82 14.50 12.50 101.29 164.62



If the interest rate falls to 14%

Date 1-Jul-16 1-Jul-17 1-Jul-18 1-Jul-19 1-Jul-19Total

No. of

years 1 2 3 4 4

Cash Flow 12.5 12.5 12.5 12.5 103.07

Terminal

Value 18.52 16.25 14.25 12.50 103.07 164.58

For the 4 year holding period (equal to Duration),

there is no interest rate risk at all as we see no change in the terminal value

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Duration Convexity Bond Portfolio Management

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