UBS - Bond Basics

8/2/2019 UBS - Bond Basics

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Bond Basics


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Contents

Bond Examples

Yield to Maturity

Price Risk Versus Reinvestment Risk

Duration

Risk Measures


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Bond Examples

Joe Troccolo, Financial Markets Education


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What is a Bond?

Security that represents a loan

Way for corporations, banks, governments and others to borrowmoney

The borrower (issuer) is obligated to pay interest to the investors

The borrower is also obligated to pay back the amount borrowed

Important aspects of a bond:

cash flows

price

currency credit quality


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Bond Examples

Bonds

....

periodic cash flowsrepayment of principal


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Example

XYV corporation wants to borrow EUR100 million for 5 years

XYV issues (sells) a bond that promises to pay:

EUR 8 million in interest on 15 February

every year for 5 years

EUR 100 million on 15 February 5 years from now

When the bond is first sold the investors pay (collectively) EUR 100

million for the bond

Each investor only buys a part of the bond

Later the bond may trade for more or less than when it was issued


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Bond Terminology

Face Value

Price

Coupon Rate

Coupon Period

Coupon Date

Maturity


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Example

US Treasury 7 1/4s May 16

Coupon is semiannualMay 15, November 15Maturity May 15, 2016

Bond was issued in May 1986

Price: 114 18/32 on October 20


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Invoice Price

Bond buyer pays price plus accrued interest

Price + Accrued interest :

Dirty price (Invoice price)


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Accrued Interest on US Treasury Bonds

Actual/Actual basis

Accrued interest on 7 1/4s May 16

20 October


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Accrued Interest Example I

May 15 Oct 20 Nov 15

Coupon interval: 184 actual daysTime from last coupon: 158 actual days

Accrued interest

7.25 x 158 = 3.11282 184

Invoice Price: 114.56253.1128

117.6753


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Eurobond

Province of Quebec

5.25%

7 July 2004

Currency 500 Million SEK

Price: 99.7500 on 16 October

Coupon is annual

Paid 30/360


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Accrued Interest Example 2

Settlement Date: October 16

30/360 days from 7 July = 99

Accrued Interest: 5.25 x 99/360 = 1.4438

Invoice Price: 99.75001.4438

101.1938


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Summary

Bonds are issued by corporations and governments to borrowmoney

To an investor a bond is a series of promised cash flows

The defining characteristics of a bond are:

face amount

coupon rate

coupon date

maturity date

The buyer of a bond pays the market price plus accrued interest


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Yield To Maturity


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Bonds and Cash Flows

You own a bond that will pay you:

1000 in one year

1000 in two years

1000 in three years 1000 in four years

11000 in five years

You may have paid

10000 for the bond

9000 for the bond

11000 for the bond

You get the same cash flows, whatever you paid!


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Bonds - Yield to Maturity A bond is a series of cash flows

The price of the bond is the price of the cash flows so

The price of the bond is the sum of the present values of the cashflows

We can observe the market price and we know the cash flows so

there must be an interest rate that equates them.


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Yield To Maturity

The single discount rate that equates the net present value of thebonds cash flows to its price.

How do we calculate the price given the yield?

How do we calculate the yield given the price?


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5 Year Bond with 10% Annual Coupon and Yield = 10%

Price = 10 + 10 + 10 + 10 + 1101.10 (1.10)

2(1.10)

3(1.10)

4(1.10)

5

Price = 9.091 + 8.264 + 7.513 + 6.830 + 68.30

Price = 100.00


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Cash Flows for 5 year Bond 10% coupon


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Present Value of Cash Flows with YTM = 10%

9.09 8.26 7.51 6.83 68.30

Yi ld d C


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Yield and Coupon

When the price of the bond is 100 then the YTM = Coupon rate

If the price of the bond is 100, the bond is called a par bond

Terminology: Price > 100: the bond is a premium bond or is trading at a premium

Price < 100: the bond is a discount bond or is trading at a discount

Coupon is determined by the interest rate level when the bond isissued

YTM is determined by the current interest rate level

E l


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Example

A bond with 5 years to maturity has a coupon of 7%

The current level of rates is 10%

What is the bonds price?

We need to discount the cash flows at 10%

PV f C h Fl 5 Y 7% b d ith YTM 10%


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PV of Cash Flows: 5 Year 7% bond with YTM = 10%

6.36 5.79 5.26 4.78 66.44

E l


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Example

A 3 year bond has a coupon of 6%, paid semi-annually

Current interest rates are 5%

We need to discount the cash flows using a 5% semi-annual rate.

65432 1.025103

1.0253

1.0253

025.13

1.0253

1.0253 +++++

3 year semi annual bond 6% coupon YTM 5%


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3 year semi-annual bond 6% coupon YTM = 5%

65432 (1.025)

103

(1.025)

3

(1.025)

3

(1.025)

3

(1.025)

3

1.025

3+++++

Bond Pricing Formula


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Bond Pricing Formula

nt2 )

n

r(1

n

c

100100

)

n

r(1

n

c

100

n

r1

n

c

100Price

+

+++

+

+

+

= L

c = coupon rate

r = yield to maturity

n = coupon frequency

t = years to maturity

Price and Yield


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Price and Yield

price of 10% semiannual bond

0.00

50.00

100.00

150.00

200.00

250.00

300.00

1 4 710

13

16

19

22

25

28

Yield

Price

Premium

Par

Discount

Example: Yield from Price


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Example: Yield from Price

If a 4-year 10% annual coupon bond ispriced at 105, what is its yield?

Yield Price9% 103.24

8% 106.628.5% 104.918.45% 105.08

8.47% 105.02

Yield = 8.4744%

Summary


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Summary

A bond is a series of cash flows

We can observe cash flows and we can observe the price the bondis trading for in the market

The yield to maturity is the interest rate that equates the price ofthe bond with the sum of the present values of the cash flows

The coupon rate is an obligation of the issuer

The market price is what the market will pay

The yield to maturity is a mathematical concept not a promise!


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Price Risk and Reinvestment Risk

Bonds: Price Risk versus Reinvestment Risk


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Bonds: Price Risk versus Reinvestment Risk

9

All yields are 8% 8

7

time to maturity

Buy 20 year annual 8% bond for par.

Scenario 1


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Scenario 1

Yields move to 9% 9

8

Bond Price = 90.87 7

Lose = 9.13% time to maturity

Scenario 2


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Scenario 2

Yields move to 7% 9

8

Bond Price = 110.59 7

Gain = 10.59%

This is price risk

time to maturity

Scenario 1 (again)


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Scenario 1 (again)

time to maturity

Yields move to 9% 9

8

7

We hold the bond to maturity (20 years)and reinvest all the coupons at 9%

Return = 8.48%

Scenario 2 (again)


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Scenario 2 (again)

9

8

Yields move to 7% 7

time to maturity

We hold the bond to maturity and reinvest

all the coupons at 7%

Return = 7.54%

This is reinvestment risk

Holding Period


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g

Holding Period = amount of time the investment is held.During the period, proceeds are reinvested. At the end theinvestment is sold at the current yield or matures.

Holding Period Reinvestment Rate7 8 9

1 year 18.34% 8.00% -0.95%

5 years 9.18% 8.00% 6.93%

10 years 8.08% 8.00% 7.96%

20 years 7.54% 8.00% 8.48%

Duration


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If the bond is held for 10 years, its holding period return will beabout 8%, under all three scenarios

10 years is the bonds duration

Duration is the point where price risk = reinvestment risk

Bond managers attempt to immunise their portfolios byadjusting the duration

View: increase in rates

shorten duration

View: decrease in rates:

lengthen duration

Immunise: protect against changes in interest rates


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Risk Measures for Bonds

Joe Troccolo, Financial Markets Education

Risk Measures


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Modified Duration

Duration and Delta

Convexity and Gamma

Price Change and Yield Change


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Change in P = dP

Price

Change in yield = dr

Yield

is the change in price for a change in yielddr

dP

Delta


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Price Delta = the amount the bonds pricechanges if the yield changes by 1basis point

= price value of a basis point

Example: 20 year 8% semi-annual bond


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Yield Price

8.99% 90.8859% 90.800

9.01% 90.714

Price Delta = = (90.714 - 90.885)/2 = -.0855

Calculating Delta


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Dr1

1-drdP

P1

+= D = Macauleys Duration

Dr1

1Dmod+

= Modified Duration

dP is change in pricedr is change in yield

drDPdP mod=

Price Delta -P Dmod

Delta and Duration


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drDPdPmod

=

In the example

Dmod = 9.41

P = 90.80

dr = .0001

dP = -90.80 x 9.41 x .0001

= -0.085

Price Delta is not constant!


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At higher yields a bond is less sensitiveto a change in yield

Yield Price Dmod pvbp

9% 90.80 9.41 -0.085

11% 75.93 8.48 -0.064

Delta at Different Yield Levels


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Price

dr

dP

dP

Yield

dr

Gamma


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Since = - P Dmod

An increase in yields lowers both Price andDuration.

The change in (due to yield changes) is called

gamma.

= change in delta (due to yield changes)

Duration and Convexity


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As yield changes duration changes.

The change in duration (due to yield changes) is called

convexity.

Gamma and Convexity


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Example: 8% 20 year semiannual bond

Yield Price Duration Delta

8% 100 9.90 -.09907.90 101 9.95 -.1005

For a 10 basis point change in yield:

= .0015 Convexity = .05

Summary


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Fixed Income investments are exposed to risks

A primary risk is the change in value due to changes in the level ofinterest rates

The price value of a basis point measures how much the value willchange for a one basis point change in yield

pvbp is called a bonds delta

The change in delta is called gamma and measures how the pricerisk of the bond changes as yields change

All of these risk measures have assumed that yields move in aparallel fashion

More sophisticated methods use models of the yield curve

UBS - Bond Basics

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