fnce 30001 week 9 managing fixed income portfolios

8/22/2019 FNCE 30001 Week 9 Managing Fixed Income Portfolios

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FNCE 30001 Investments 9.0

FNCE 30001 InvestmentsSemester 2, 2011

5 & 7 October 2011Week 9: Managing Fixed Income

PortfoliosProfessor Rob Brown


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Week 9: Managing Fixed Income

Portfolios

Overview of Lecture1. Overview of Bond Portfolio Risks

2. Some Mathematical Properties of Bond Prices

3. Duration and Convexity

4. More Properties of Duration

5. How Duration is Used in Portfolio Management

6. Measuring Portfolio Yield

Reading: Bodie et al, Chapter 16.How Duration Changes Over Time (available on the LMS)


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1. Overview of Bond Portfolio Risks


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Overview of Bond Portfolio Risks

There are four main risks in investing in bonds:

1. Interest rate risk If interest rates decrease, bond prices rise, but

If interest rates increase, bond prices fall capital loss.

2. Inflation risk Most bonds promise payments in nominal (rather than

real) terms.

If inflation is higher than expected

lower real returnthan expected.

In real terms, the return may even be negative.


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Overview of Bond Portfolio Risks

3. Credit (or, default) risk

Applies only to non-government debt. Occurs if the promised nominal payments are not made

in full and/or on time.

The loss may be small or large, depending on the case.4. Exchange rate risk

Applies only if you invest in a bond that pays interestand/or par value in a foreign currency.

If the home currency appreciates lower return inhome currency terms.

In this lecture our focus is on interest rate risk.


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2. Some Mathematical Properties ofBond Prices


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Some Mathematical Properties of Bond Prices

Interest rate risk arises because interest rate changes affect thebond price.

Hence, we first need to understand the detail of how interestrates and bond prices are related.

This relationship has the following five features:

1. The bond price is inversely related to the required yield.2. This relationship is convex.

3. All other things equal, the longer the term to maturity, the greater is thevolatility of the bond price.

4. All other things equal, the higher the coupon rate, the lower is thevolatility of the bond price.

5. All other things equal, the greater the convexity, the more attractive isthe bond to investors.


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1, 2 and 3: Price change vsYield for different terms to maturity

T= 30 years

T= 15 years

T= 5 years



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4. Price change vsYield for different coupons

c= 5%

c= 15%

c= 25%



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Recall the pricing formula for a coupon bond:

To simplify the notation a little, I will use iinstead ofytmand drop thesubscript 0 from P.The price, P, is a function ofi.

We are interested in howPresponds to changes in i.

That is, if ichanges to i+ i, then Pchanges to P+ Pand we want to

know how the change in the price (P) is related to the change in theyield (i).

We are also interested in the proportionate change in price P/P,because this tells us the percentage capital gain or loss on the bond.

2 3( ) ...1 1 1 1TC C C C Par P i i i i i

0 2 3 ...1 1 1 1 TC C C C Par

P ytm ytm ytm ytm


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We now use a Taylors series expansion and get:

22

2

22

2

22

2

1

higher-order terms2

Therefore:

1

21

2

dP d P

P i i P i i i di di

dP d P P P i i P i i i

di diP dP i d P

iP di P P di

How can this term be

interpreted?


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By definition, the yield elasticity () of the bond price is:

That is, the proportionate change in yield multiplied by the bonds yieldelasticity.

dP idi P

So,dP i i dP i i

di P i di P i


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From the last slide:

2

2212

P dP i d P iP di P P di

The %capital gain

or loss

Proportionatechange in

yield x bondsyield elasticity

2

nd

derivativetells us about

convexity

Usually quitesmall, butmeasurable.


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Recall:

2 3

( ) ...1 1 1 1

TC C C C Par P i

i i i i

2 3 4 1

2 3

Therefore:

2 3 ...1 1 1 1

1 2 3...

1 1 1 1 1

T

T

T C Par dP C C C di i i i i

T C Par C C C

i i i i i


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2 3 4 1

2

2 3 4 5 2

2 2 3

Because:

2 3

... ,1 1 1 1

therefore:

12 6 12

...1 1 1 1

11 2 6 12...

11 1 1 1

T

T

T

T C Par dP C C C

di i i i i

T T C Par d P C C C

di i i i i

T T C Par C C C

ii i i i


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Lets go back to the term

We now know that:

The big term in square brackets looks like a big mess!!

But actually it has a very simple and intuitive meaning.

It is called duration (D).

2 3

2 3

/ 1 2 / 1 3 / 1 ... / 1So

1

/ 1 / 1 / 1 / 12 3 ...

1

T

T

C i C i C i T C Par i dP i i

di P i P

C i C i C i C Par i iT

i P P P P

2 3 1

1 2 3...

1 1 1 1 1T

T C Par dP C C C

di i i i i i

.

d P i

d i P


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Duration ( D) is defined to be:

Note it also follows that

2 3/ 1 / 1 / 1 / 12 3 ...TC i C i C i C Par i D T

P P P P

1

1So

dP i i Ddi P i

i dPD

P di


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3. Duration and Convexity


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Duration and Convexity

Consider the following bond:

Today is a coupon payment date (coupon has been paid) Term = 5 years

Coupon rate = 8.5% pa

Par value = $100 Yield = 10% pa

What is its duration?


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Time

(t)

Cash

flow ($)

PV of cash

flow ($)

PV of cash flow

/Price

(PV of cash

flow/Price) Time

1 $8.50 $7.727272 0.081931 0.081931 years

2 $8.50 $7.024793 0.074483 0.148966 years

3 $8.50 $6.386176 0.067712 0.203136 years

4 $8.50 $5.805614 0.061556 0.246224 years

5 $108.50 $67.369963 0.714318 3.571590 years

Total $94.313818 1.000000 4.251847 years

2 3/ 1 / 1 / 1 / 1

1 2 3 ...

TC i C i C i C Par i

D TP P P P


Duration (D)


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Duration is the bondsweighted average term to maturity.

The formula takes the dates on which cash flows will occur (1, 2, 3, 4 and5), then weights each of these dates by the percentage contribution of thepresent value of that cash flow to the price of the bond.

In the example, the first coupon ($8.50) occurs at t= 1. The present valueof this coupon is $7.727272, which represents 8.1931% of the price of the

bond ($94.313818).

So, although this is a 5-year bond, in a sense the average wait for the cashflows is 4.251847 years.

This measure (4.251847) is called the Macaulay duration of the bond, or

just duration for short.


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Some simple algebra shows that the duration formula can also be writtenthis way:

And also this way:

1 1

where means the cash flow at time .

Tt

tt

t

t CF

iD

P

CF t

1

1

1

1

Tt

tt

Tt

tt

t CF

iD

CF

i


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Duration and Convexity Duration recognises that term to maturity is an inadequate

measure of the time period of an investment

we also need to take into account the pattern of cash flowswithinthe term to maturity.

Extreme example:

Suppose we invented a new security called a 30001 special

bond. A 50-year 30001 special bond pays $100 after 1 year and $1

after 50 years.

The term to maturity of this 30001 special bond is indeed50 years, but would you really think of it as a 50-yearinvestment?

The duration of this 30001 special bond would be slightly

more than 1 year.


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Duration and ConvexityWe now go back to our approximation for bond price sensitivity:

2

22

12

P d P i d P iP d i P P d i

The %

capital gain

or loss

Proportionate

change in

yield x bondsyield elasticity

2nd derivative

tells us about

convexity

Usually quite

small, but

measurable.

1

iD

i

We will call this term the

convexity adjustment,

denoted byX.


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So, to a first approximation, the percentage capital gain or loss on a bond,when the yield changes, is:

where Dis the Macaulay duration of the bond.

To a second (more accurate) approximation, we add in the convexityadjustment (X).

see next slide.

1

P iD

P i


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If we wanted to be more accurate, we could add in the convexityadjustment,X:

2

2

2

2 2 3

,1

1where 2

11 2 6 12...

12 1 1 1 1T

P iD X i

P i

d PX P di

T T C Par C C C

iP i i i i


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Example

Consider the 5-year 8.5% coupon bond (annual coupons) priced to yield10% pa, with a par value of $100. We found that D= 4.251847 years.

If the yield were to change to 10.5% pa, to a first approximation, what is theestimated percentage capital loss?

AnswerWe know to a first approximation that

Here, i = 10% and i= +0.5%.

.

1

P iD

P i

0.005

So 4.2518471.1

1.93266%

P

PP

ieP


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Duration and ConvexityExample (contd.)If we wanted to be more accurate we could use the convexity adjustment:

2 2 3

11 2 6 12

...12 1 1 1 1T

T T C Par C C C

X iP i i i i

Time

(t)

Cash

flow ($)

Cash flow

multiplier

Cash flow cash

flow multiplier

PV of (cash flow cash

flow multiplier)

1 $8.50 1 2 = 2 $17 $15.4545452 $8.50 2 3 = 6 $51 $42.148760

3 $8.50 3 4 = 12 $102 $76.634110

4 $8.50 4 5 = 20 $170 $116.112287

5 $108.50 5 6 = 30 $3255 $2021.098907

Total $2271.448609*

*This is the value of the term in square brackets.


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Duration and ConvexityExample (contd.)

So the adjustment to be made is:

2

2 2 3

2

2

11 2 6 12...

12 1 1 1 1

1

$2271.448609 0.0052 $94.313818 1.1

0.0002488

0.02488%

T

T T C Par C C Ci

iP i i i i


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Duration and ConvexityExample (contd.)

So the more accurate approximation for the percentage capital

loss is:

1.93266% 0.02488% 1.90778%

P

P


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Example (contd.)

How good are these approximations?

Using the bond price formula we can work out the exactanswer.

When the yield increases from 10.0% to 10.5%, the bond pricefalls from $94.313818 to $92.514284.

Thus the exact capital loss is $1.799534 and the exact percentage capital loss is 1.90803%.

Our approximations were:

first approximation: a loss of 1.93266% second approximation: a loss of 1.90778%


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4. More Properties of Duration


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More Properties of Duration

How Duration Behaves

All other things being equal, duration is higher if: The coupon rate is lower (why?)

The yield is lower (why?)

(Usually) the term to maturity is longer (why?) The duration of a zero-coupon bond is equal to the bonds

term to maturity.

The duration of a coupon bond is always less than the bondsterm to maturity.

1

The duration of a perpetuity is .i

i


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Between coupon dates, the duration of a bond decreases inline with the change in maturity;

ieduration decreases by one day, every day.

But on a coupon payment date, the duration of a bondincreases discretely.

In the next three slides we show this for a specific case, but it

holds in all cases.


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Consider again the 5-year 8.5% coupon bond (annual coupons)

priced to yield 10% pa, with a par value of $100 This bonds has a duration of 4.25 years.

After three months (0.25 years), all times in the duration formulahave decreased by 0.25 years.

We now recalculate the duration (next slide).


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Time

(t)

Cash

flow ($)

PV of cash

flow ($)

PV of cash flow

/Price

(PV of cash

flow/Price) Time

0.75 $8.50 $7.913606 0.081931 0.061449 years

1.75 $8.50 $7.194187 0.074483 0.130346 years

2.75 $8.50 $6.540170 0.067712 0.186208 years

3.75 $8.50 $5.945609 0.061556 0.230836 years

4.75 $108.50 $68.994500 0.714318 3.393006 years

Total $96.588070 1.000000 4.001845 years

0.75 1.75 2.75/ 1 / 1 / 1 / 1

0.75 1.75 2.75 ...

TC i C i C i C Par i

D TP P P P


Duration (D)


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Ignoring a small rounding error, the duration has decreased byexactly 0.25 years.

That is, duration has fallen by one day, every day.

It can be proved that this will always happen.

Now consider what happens when a coupon is paid.


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The numerator of duration is now hardly affected bytomorrows coupon, since the time attached to it is only1/365th of a year.

But the denominator of duration (which is the bond price)includes tomorrows coupon payment in full.

So, when the coupon is paid, the numerator will hardly changeat all, but the bond price will drop instantly by the amount ofthe first coupon.

Hence, with almost the same numerator but a smallerdenominator, the duration instantly increases when thecoupon is paid.


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The Duration of a Bond Portfolio

If the yield curve is flat, the duration of a bond portfolio is theweighted average of the durations of the bonds that comprisethe portfolio, where the weights are the portfolio (value) weights.

Example

Recall the 5-year 8.5% coupon bond (annual coupons) priced toyield 10% pa, with a par value of $100. Its price was $94.313818and its duration was 4.251847 years. Call this Bond A.

Consider buying one Bond A and one Bond B, which is a 2-year6.5% coupon bond priced to yield 10% pa.


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Example (contd.)

We need to calculate the duration of Bond B:

The duration of Bond B is 1.937088 years.

Time(t)

Cashflow ($)

PV of cashflow ($)

PV of cash flow/Price

(PV of cashflow/Price) Time

1 $6.50 $5.909090 0.062912 0.062912 years

2 $106.50 $88.016529 0.937088 1.874176 years

Total $93.925619 1.000000 1.937088 years


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Example (contd.)

Total invested = $94.313818 + $93.925619 = $188.239437.

We first calculate the weighted average of the durations of Bond A andBond B.

Investment in A Investment in BWtd ave +

Total invested Total invested$94.313818 $93.9256194.251847 1.937088

$188.239437 $188.239437

0.50103 4.251847 0.49897 1.937088

3.09685 years

A BD D D


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Example (contd.)

If we calculate the duration of the cash flows in the portfolio, we should

find that the portfolio duration is 3.09685 years.

The portfolio cash flows are:

Time (t)Bond A cash

flows

Bond B cash

flows

Portfolio

cash flows1 $8.50 $6.50 $15.00

2 $8.50 $106.50 $115.00

3 $8.50 $8.50

4 $8.50 $8.50

5 $108.50 $108.50


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Example (contd.)The calculation of portfolio duration is:

Time

(t)

Cash flow

($)

PV of cash

flow ($)

PV of cash flow

/Price(PV of cash

flow/Price) Time

1 $15.00 $13.636364 0.072442 0.072442 years

2 $115.00 $95.041322 0.504896 1.009792 years3 $8.50 $6.386176 0.033926 0.101778 years

4 $8.50 $5.805614 0.030842 0.123368 years

5 $108.50 $67.369963 0.357894 1.789470 years

Total $188.239439 1.000000 3.096850 years

This gives the same result: D= 3.09685.


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This property is often used to design a portfolio that has theduration we require (say, D*).

For example, suppose we wished to invest $1 million in BondA and Bond B and achieve a duration of 2.5 years.

We use wA DA + wB DB = D*, where:

DA = 4.251847

DB = 1.937088

D* = 2.5

wA + wB = 1, which implies wB = 1 wA.


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So we need to solve:

4.251847 wA

+ 1.937088 (1 wA) = 2.5

which solves to give wA = 0.24318

and therefore wB = 0.75682.

So the solution is that we invest:

$243,180 in Bond A and

$756,820 in Bond B.

This portfolio of bonds will have a duration of 2.5 years.


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5. How Duration is Used inPortfolio Management

H D ti i U d i P tf li


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How Duration is Used in Portfolio

Management

Duration is used for three main purposes:

1. As a summary measure of a bonds sensitivity to changes inyield (interest rates).

2. As a guide to act on expectations.

3. As a tool to immunise a bond portfolio.

We will now work through each of these uses.



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Management

1. Duration as a summary measure of bond price sensitivity

Recall our first approximation:

For given iand i, the percentage capital gain or loss is directlyrelated to duration (D).

So high-duration bonds respond more to changes in interestrates (yields).

And low-duration bonds respond less to changes in interestrates (yields).

1

P iD

P i



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Management

2. Duration as a guide to act on expectations

If you expect interest rates to fall, then you:

foresee capital gains

want bonds with high price sensitivity

want high-duration bonds

eglong-term bonds with low coupons.

If you expect interest rates to rise, then you:

foresee capital losses

want bonds with low price sensitivity

want low-duration bonds

egshort-term bonds with high coupons.



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Management

3. Duration as a tool to immunise a bond portfolio

Immunise means to ensure that a bond portfolio achieves a

target rate of return even ifyields change. Suppose you need to invest today to be certain of having $1

million in five years time: how would you do that?

Simple answer: buy a five-year zero coupon bond. (Why?) Problem: In practice, there are very few five-year zeros to buy!

So, in practice, many investors have to invest in couponbonds but then they face the problem of changes in yield.

It turns out that the way to do it is to invest in a bond (or aportfolio of bonds) whose duration is five years.



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Management

Why does this work?

When yields decrease, there is good news and bad news forthe investor:

Good news: capital gain

Bad news: expect a lower reinvestment rate

When yields increase, there is good news and bad news forthe investor:

Good news: expect a higher reinvestment rate

Bad news: capital loss

The duration-matching strategy exactly balances the goodnews and the bad news, so it doesnt matter if yields increaseor decrease.



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Management

Is it as simple as that?

Unfortunately, no.

Unlike buying a zero, duration-matching with coupon bondsisnt a set and forget strategy:

When a coupon is paid, the portfolios duration no longer

matches the time horizon, so the portfolio has to berebalanced.

When yields change, duration changes, so the portfolio hasto be rebalanced.

Technically, it only works if the yield curve is flat (so thatall zero rates equal the yield) and is sure to stay flat.



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Management

Example

(This example is based on Peirson et al, Business Finance,McGraw-Hill, 10th edn., 2009, pp. 103-105).

The current yield curve is flat at 10% pa. An investor needs toinvest today to achieve a target value of $1.275 million in 3

years time.

There is a bond with a term of 3.4 years, paying annualcoupons of 7%.

We will show that by investing in this bond, the investor willachieve the target even if yields decrease to 8% pa, or increaseto 12 % pa.



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Management

Example (contd.)

0.4 1.4 2.4 3.4

0.4 1.4 2.4 3.4

$7 $7 $7 $107

1.1 1.1 1.1 1.1

$95.816022

0.4 $7 1.4 $7 2.4 $7 3.4 $1071.1 1.1 1.1 1.1

$287.740099

$287.740099$95.816022

3.0030 years

denom

num

D P

D

D



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Management

Example (contd.)

If the investor buys one such bond with a par value of $1million, it will cost $958,160.

If its yield of 10% pa can be locked in (immunised) then inthree years time the value of the investment will be:

$958,160 x (1.1)3 = $1,275,311.

If duration matching works, a yield of 10% pa will be achieved,even if yields decrease to 8% pa or increase to 12% pa.



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Management

Example (contd.)

Next we look at what happens if yields decrease (increase) to8% pa (12% pa) immediately after the investment is made andstay at the new level for the next 3 years.

If yields immediately decrease to 8% pa, the bond priceincreases from $958,160 to $1,012,573.

If yields immediately increase to 12% pa, the bond pricedecreases from $958,160 to $907,809.



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Management



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Management1. Because the bondprice is lower .

2. . more bonds

can be bought .

3. . so more coupon

interest is received.



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Management

If yields increase, there is an immediate capital loss, butreinvestment earnings are high.

But if yields decrease, there is an immediate capital gain, butreinvestment earnings are low.

The next slide shows these effects diagrammatically.



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Management


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6. Measuring Portfolio Yield

Measuring Portfolio Yield


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Many institutions (banks, insurance companies etc) calculate anaverage portfolio yield.

For example, if $100mis invested in a 3-year 8% bond yielding10% and $150mis invested in a 10-year 11% bond yielding 13%,the temptation is to say that the portfolio yield is:

This isnt correct. Yields dont just average.

To work out the correct yield we need to calculate every cashflowof the portfolio.

$100 $150

10% 13%$250 $250

0.4 10% 0.6 13%

11.80%

m m

m m



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We first have to calculate the bond prices.

The bond prices are:

and

3 3

$8 1 $1001

0.1 1.1 1.1

$95.02629602

P

10 10

$11 1 $1001

0.13 1.13 1.13$89.14751305

P



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So, with $100minvested in the 3-year bond, the number of 3-year bondsheld is:

and with $150minvested in the 10-year bonds, the number of 10-year bondsheld is:

Therefore, the par values of the bond holdings are:

3-year bonds: $105,234,00010-year bonds: $168,260,400

$100 1,052,340$95.02629602

m

$150

1,682,604$89.14751305

m



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So the annual coupons are:

3-year bonds: 8% $105,234,000 = $8,418,720

10-year bonds: 11% $168,260,400 = $18,508,644

We can now set out the future portfolio cash flows and thenfind the yield-to-maturity (internal rate of return).



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Calculation of portfolio cash flows

Year 3-year bond 10-year bond Portfolio

1 $8,418,720 $18,508,644 $26,927,364

2 $8,418,720 $18,508,644 $26,927,364

3 $113,652,720 $18,508,644 $132,161,364

4 9 $18,508,644 pa $18,508,644 pa

10 $186,769,044 $186,769,044

The yield on the portfolio cash flows is 12.34% pa, which isconsiderably more than the simple average of 11.80% pa.

fnce 30001 week 9 managing fixed income portfolios

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