UBS - Bond Basics

Bond Basics

Contents ♦ Bond Examples ♦ Yield to Maturity ♦ Price Risk Versus Reinvestment Risk ♦ Duration ♦ Risk Measures

Bond Examples  Joe Troccolo, Financial Markets Education

 What is a Bond? ♦ Security that represents a loan

governments and others to borrow ♦ Way for corporations, banks, governments money ♦ 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

Bond Examples

Bonds ....

periodic cash flows repayment of principal

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

Bond Terminology  ♦ Face Value ♦ Price ♦ Coupon Rate ♦ Coupon Period ♦ Coupon Date ♦ Maturity

Example

US Treasury 7 1/4’s May 16 Coupon is semiannual May 15, November 15 Maturity May 15, 2016 Bond was issued in May 1986 Price: 114 18/32 on October 20

Invoice Price

Bond buyer pays price plus accrued interest Price + Accrued interest : “Dirty” price (Invoice price)

 Accrued Interest on US Treasury Bonds

Actual/Actual basis Accrued interest on 7 1/4’s May 16 20 October

 Accrued Interest Example I May 15

Oct 20

Nov 15

Coupon interval: 184 actual days Time from last coupon: 158 actual days Accrued interest 7.25 x 158 = 3.1128 2 184 Invoice Price: 114.5625 3.1128 117.6753

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

 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:

99.7500 1.4438 101.1938

Summary  ♦ Bonds are issued by corporations and governments to borrow

money ♦ 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

 Yield To Maturity 

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!

♦ 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 cash

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

there must be an interest rate that equates them.

 Yield To Maturity  ♦ The single discount rate that equates the net present value of the

bond’s cash flows to its price. ♦ How do we calculate the price given the yield? ♦ How do we calculate the yield given the price?

5 Year Bond with 10% Annual Coupon and Yield = 10% Price = 10 + 10 + 10 + 10 + 110 2 3 4 5 1.10 (1.10) (1.10) (1.10) (1.10) Price = 9.091 + 8.264 Price = 100.00

+ 7.513 + 6.830 + 68.30

Cash Flows for 5 year Bond 10% coupon

Present Value of Cash Flows with YTM = 10%

9.09

8.26

7.51

6.83

68.30

 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 is

issued ♦ YTM is determined by the current interest rate level

Example ♦ A bond with 5 years to maturity has a coupon of 7% ♦ The current level of rates is 10% ♦ What is the bond’s price? ♦ We need to discount the cash flows at 10%

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

6.36

5.79

5.26

4.78

66.44

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.

3 1.025

+

3 1.025

2

+

3 1.025

3

+

3 1.025

4

+

3 1.025

5

+

103 1.0256

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

3 1.025

+

3 (1.025) 2

+

3 (1.025) 3

+

3 (1.025) 4

+

3 (1.025) 5

+

103 (1.025) 6

Bond Pricing Formula

100 Price = 1+

c n r  n

100

+

c

n r  2 (1 + ) n

c = coupon rate r = yield to maturity n = coupon frequency t = years to maturity

100 + 100 +L+

r  nt (1 + ) n

c n

Price and Yield price of 10% semiannual bond 300.00

250.00

200.00

Premium

     e      c        i 150.00      r        P

Par

100.00

Discount 50.00

0.00 1

        0 4         1

        3 7         1

        6         1

 Yield

        9         1

        2         2

        5         2

        8         2

Example: Yield from Price

If a 4-year 10% annual coupon bond is priced at 105, what is its yield? Yield 9% 8% 8.5% 8.45% 8.47%

Yield = 8.4744%

Price 103.24 106.62 104.91 105.08 105.02

Summary  ♦ A bond is a series of cash flows ♦ We can observe cash flows and we can observe the price the bond

is trading for in the market ♦ The yield to maturity is the interest rate that equates the price of

the 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!

Price Risk and Reinvestment Risk 

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

Yields move to 9% 9 8 Bond Price = 90.87 7 Lose = 9.13%

time to maturity

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)

Yields move to 9% 9 8 7 time to maturity We hold the bond to maturity (20 years) and reinvest all the coupons at 9% Return = 8.48%

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

Holding Period

1 year

Reinvestment Rate 7 8 9 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 ♦ If the bond is held for 10 years, its holding period return will be

about 8%, under all three scenarios ♦ 10 years is the bond’s duration ♦ Duration is the point where price risk = reinvestment risk ♦ Bond managers attempt to “immunise” their portfolios by

adjusting the duration ♦ View: increase in rates —

shorten duration

♦ View: decrease in rates: —

lengthen duration

♦ Immunise: protect against changes in interest rates

Risk Measures for Bonds  Joe Troccolo, Financial Markets Education

Risk Measures ♦ Modified Duration ♦ Duration and Delta ♦ Convexity and Gamma

Price Change and Yield Change Price

Change in P = dP

Change in yield = dr Yield

dP dr 

is the change in price for a change in yield

Delta

Price Delta = the amount the bond’s price changes if the yield changes by 1 basis point = price value of a basis point

Example: 20 year 8% semi-annual bond

Yield

Price

8.99% 9% 9.01%

90.885 90.800 90.714

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

Calculating Delta 1 dP P dr 

D mod

=

=-

1 1+ r 

1 1 + r 

D

D

Modified Duration

dP = − P D mod dr 

Price Delta ≡

D = Macauley’s Duration

dP is change in price dr is change in yield

≡ -P Dmod

Delta and Duration dP = −P D mod dr 

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! At higher yields a bond is less sensitive to a change in yield Yield

Price

Dmod

pvbp

9% 11%

90.80 75.93

9.41 8.48

-0.085 -0.064

Delta at Different Yield Levels Price

dP dP dr

dr

Yield

Gamma Since

∆

= - P Dmod

An increase in yields lowers both Price and Duration. The change in ∆ (due to yield changes) is called gamma.



= change in delta (due to yield changes)

Duration and Convexity  ♦As yield changes duration changes. ♦The change in duration (due to yield changes) is called

convexity.

Gamma and Convexity  Example:

8%

Yield 8% 7.90

Price 100 101

20 year semiannual bond Duration 9.90 9.95

For a 10 basis point change in yield: 

= .0015

Convexity = .05

Delta -.0990 -.1005