ubs - bond basics
TRANSCRIPT
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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
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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
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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