Lecture 5

V 1.0 – 8-2-2016
 2

Bond basics
• Yield (coupon Y, current Y, YTM)
• Relationship between P and yields
• Interest rate risk and price volatility
 Bond Basics

• A straight bond is a written promise (IOU) that

obligates the issuer of the bond to pay the
holder of the bond:
– A fixed sum of money (called the principal, par value, or
face value) at the bond’s maturity
– Constant, periodic interest payments (called coupons)
during the life of the bond

• Special features may be attached:

– Convertible bonds
– Callable bonds
– Putable bonds
 Bond Basics II

Issuer: government/ corporate

Rating: investment grade / high yield

Type fixed rate/ floating rate / zero coupon/ callable / convertible/…

Seniority: senior / subordinated

Sole 24 ore 4/04/2019
Bloomberg terminal
 Fixed versus floating rate
Fixed rate: constant coupon
Price and market interest rates: inverse relation

yield Price

ZERO COUPON:
coupon = 0

Floating rate
Coupon = Euribor (Libor) + spread
 Yields

• Two basic yield measures for a bond are its

coupon rate and its current yield.

Annual coupon
Coupon rate =
Par value

Annual coupon
Current yield =
Bond price
(Straight) Bond Prices

• The price of a bond is found by adding

together:
– the present value of the bond’s coupon payments,
and
– the present value of the bond’s face value.

What is r ?
 Straight Bond Prices and Yield to Maturity

• The price of a bond is found by adding

together:
– the present value of the bond’s coupon payments,
and
– the present value of the bond’s face value.

• The yield to maturity (YTM) of a bond is the

discount rate that equates today’s bond
price with the present value of all the future
cash flows of the bond. 10-10
 BTP: Price vs YTM

Fonte Bloomberg
 BTP vs CCT

Fonte Bloomberg
 The Bond Pricing Formula

• The price of a bond is found by adding together the present value of

the bond’s coupon payments and the present value of the bond’s face
value.

• The formula is: ⎡ ⎤

C/2 ⎢ 1 ⎥ FV
Bond Price = ⎢1 − 2M
⎥ + 2M
YTM/2 ⎢
⎣ (
1+ YTM
2 ) (
⎥ 1+ YTM
⎦ 2 )
In the formula, C represents the annual coupon payments (in $), FV is the face
value of the bond (in $), M is the maturity of the bond, measured in years, and
YTM is the yield to maturity of the bond, expressed as an annual percent.

10-13
Example: Using the Bond
Pricing Formula
What is the price of a straight bond with: $1,000 face
value, coupon rate of 8%, a YTM of 7%, and a maturity of
20 years?
(Note that we simplified the bond pricing formula a little.)

é ù
C ê 1 ú FV
Bond Price = 1- +
YTM ê
ë
(
1 + YTM
2M

2 û
)
ú 1 + YTM 2M
2
( )
é ù
80 ê 1 ú 1000
Bond Price = 1- +
0.07 ê
ë
(
1+ 0.07
2
)
2´20
ú
û
(
1+ 0.07
2
)
2´20

= (1,142.857 ´ 0.747428) + 252.5725

= $1,106.78.
 Example: Calculating the Price of
this Straight Bond Using Excel

• Excel has a function that allows you to price straight bonds,

and it is called PRICE.

=PRICE(“Today”,“Maturity”,Coupon Rate,YTM,100,2,3)

– Enter “Today” and “Maturity” in quotes, using mm/dd/yyyy format.

– Enter the Coupon Rate and the YTM as a decimal.
– The "100" tells Excel to use $100 as the par value.
– The "2" tells Excel to use semi-annual coupons.
– The "3" tells Excel to use an actual day count with 365 days per
year.

Note: Excel returns a price per $100 face in this example.

Spreadsheet Analysis, I.

10-16
 Premium, Par, and Discount Bonds.
Bonds are given names according to the relationship
between the bond’s selling price and its par value.

Remember: C Y P F (Can You Pass Finance?)

• Premium bonds: If Coupon rate > YTM then Price > Face (par value)

• Discount bonds: If Coupon rate < YTM then Price < Face (par value)

• Par bonds: If Coupon rate = YTM then Price = Face (par value)

Do you see the relationship between C Y and P F?

Domanda

n Siamo nel 2018, Luca decide di investire una parte dei suoi risparmi
nell'acquisto di obbligazioni della Bassetto S.p.A. Ogni obbligazione ha
valore nominale di 100€, scadenza 2021 e cedola con tasso fisso del
2%.
n Siamo nel 2019, Luca incassa la cedola di 2€

n Siamo nel 2020, Luca ha bisogno di liquidità, incassa la cedola e decide quindi di
vendere le obbligazioni prima della scadenza. Il mercato nel frattempo è cambiato
e i tassi sono al 4%

A. 104 €
A quale prezzo Luca
B. 98 €
venderà ogni
C. 96 €
obbligazione?
D. non so
Tasso di rendimento effettivo a scadenza (TRES)
Premium, Par, and Discount Bonds
 Premium, Par, and Discount Bonds

• In general, when the coupon rate and YTM are held

constant:

For premium bonds: the longer the term to

maturity, the greater the premium over par value.

For discount bonds: the longer the term to maturity,

the greater the discount from par value.
Relationships among Yield Measures

For premium bonds:

coupon rate > current yield > YTM

For discount bonds:

coupon rate < current yield < YTM

For par value bonds:

coupon rate = current yield = YTM

Current yield is the annual coupon payment

divided by the bond’s current market price.
10-23
 Calculating Yield to Maturity, I.

• Suppose we know the current price of a bond ($110), its

coupon rate (4%), and its time to maturity (8 years).
How do we calculate the YTM?

• We can use the straight bond formula, trying different yields

until we find the one that produces the current bond price.
é ù
$40 ê 1 ú $100
$110 = 1- +
YTM ê
ë
(
1+ YTM
2 û
)
2´8
ú (
1+ YTM
2
)
2´8

• This process is tedious. So, to speed up the calculation, we

use financial calculators and spreadsheets.

10-24
 Calculating Yield to Maturity, II.

• We can use the YIELD function in Excel:

=YIELD(“Today”,“Maturity”,Coupon Rate,Price,100,2,3)

– Enter “Today” and “Maturity” in quotes, mm/dd/yyyy format.

– Enter the Coupon Rate as a decimal.
– Enter the Price as a percent of face value.
– Note: As before,
• The "100" tells Excel to use $100 as the par value.
• The "2" tells Excel to use semi-annual coupons.
• The "3" tells Excel to use an actual day count with 365 days per year.

• Using dates 8 years apart, a coupon rate of 4%, and a price

(per hundred) of $110 results in a YTM of 2.607%.

10-25
Spreadsheet Analysis, II.

10-26
 A Quick Note on Bond Quotations, I.

• We have seen how to calculate bond prices.

• We have calculated the price as if the next coupon

payment is six months off.

• If you buy a bond between coupon dates, you will

receive the next coupon payment (and might have to
pay taxes on it).
• When you buy the bond between coupon payments,
you must compensate the seller for any accrued interest.
 A Quick Note on Bond Quotations, II.

• The convention in bond price quotes is to ignore

accrued interest.
– This results in what is commonly called a clean price or (i.e., a
quoted price net of accrued interest).
– Sometimes, this price is also known as a flat price or Corso
secco.

• The price the buyer actually pays is called the dirty price
(Prezzo tel quel).
– The buyer must pay accrued interest and the clean price.
– Note: The price the buyer actually pays is sometimes known as
the full price, or invoice price.

10-28
 CALCOLO DEL RATEO
Cedola annuale
6% godimento regolamento stacco cedola

1.1.2013 1.2.2013 1.7.2014

al giorno 1.2.2013 corrisponderanno:

31 6
´ = €0,51381 per €100 di capitale
181 2
Data godimento esclusa e data regolamento inclusa; Valore lordo, si
Approssimazione del rateo di 5 cifre decimali per 100 € di capitale applica ritenuta 12,5%

29
Bond Yields
• Realized Compound Returns versus Yield to
Maturity
– Realized compound return
• Compound rate of return on bond with all coupons
reinvested until maturity
– Horizon analysis
• Analysis of bond returns over multiyear horizon, based
on forecasts of bond’s yield to maturity and investment
options
– Reinvestment rate risk
• Uncertainty surrounding cumulative future value of
reinvested coupon payments
The reinvestment rate risk
Bond Prices Over Time

• Yield to Maturity versus Holding Period

Return (HPR)
– Yield to maturity measures average RoR if
investment held until bond matures
– HPR is RoR over particular investment period;
depends on market price at end of period
 Trade off Risk & Return

Return: coupon + capital gain (loss)

Risks: Interest rate risk Credit risk

Interest Rate Risk and Maturity
 Malkiel’s Theorems, I.

1. Bond prices and bond yields move in opposite

directions.
– As a bond’s yield increases, its price decreases.
– Conversely, as a bond’s yield decreases, its price increases.

2. For a given change in a bond’s YTM, the longer the

term to maturity of the bond, the greater the
magnitude of the change in the bond’s price.
 Malkiel’s Theorems, II.

3. For a given change in a bond’s YTM, the size of the

change in the bond’s price increases at a diminishing
rate as the bond’s term to maturity lengthens.

4. For a given change in a bond’s YTM, the resulting

percentage change change in the bond’s price is
inversely related to the bond’s coupon rate.

5. For a given absolute change in a bond’s YTM, the

magnitude of the price increase caused by a decrease
in yield is greater than the price decrease caused by
an increase in yield.
 Duration

• Bondholders know that the price of their bonds change when

interest rates change. But,
– How big is this change?
– How is this change in price estimated?

• Macaulay Duration, or Duration, is a way for bondholders to

measure the sensitivity of a bond price to changes in bond
yields. That is, given a starting YTM and a change in YTM:

Change in YTM
Pct. Change in Bond Price » -Duration ´
( )
1 + YTM
2
è Two bonds with the same duration, but not necessarily the
same maturity, will have approximately the same price
sensitivity to a (small) change in bond yields.
Duration Properties

1. All else the same, the longer a bond’s

maturity, the longer its duration.
2. All else the same, a bond’s duration increases
at a decreasing rate as maturity lengthens.
3. All else the same, the higher a bond’s coupon,
the shorter is its duration.
4. All else the same, a higher yield to maturity
implies a shorter duration.
 Modified Duration

• Some analysts prefer to use a variation of Macaulay’s

Duration, known as Modified Duration.

Macaulay Duration
Modified Duration =
æ YTM ö
ç 1 + ÷
è 2 ø

• The relationship between percentage changes in bond

prices and changes in bond yields is approximately:

Pct. Change in Bond Price » - Modified Duration ´ Change in YTM

 Calculating Duration
Using Excel, Explanation

• We can use the DURATION and MDURATION functions in

Excel to calculate Macaulay Duration and Modified
Duration.

• The Excel functions use arguments like we saw before:

=DURATION(“Today”,“Maturity”,Coupon Rate,YTM,2,3)

• Example: Verify that a 5-year bond, with a 9% coupon and

a 7% YTM, has a Duration of 4.17 and a Modified Duration
of 4.03.
Calculating Macaulay Duration
and Modified Duration, using Excel
 Example: Using Duration
• Example: bond with a Macaulay Duration of 11 years and a
current yield to maturity of 8%.

• If the yield to maturity increases to 8.50%, what is the

resulting percentage change in the price of the bond?
Pct. Change in Bond Price » - 11´
[(0.085 - 0.08 )]
(1+ 0.08 2)
» -5.29%.

Note that the starting YTM appears in the denominator.

Properties of Duration
 44

RESULTS

CLASSROOM TEST
 55

BOND CHARACTERISTICS
Bond Characteristics
• International Bonds
– Foreign bonds
• Issued by borrower in different country
than where bond sold, denominated in
currency of market country
– Eurobonds
• Denominated in currency (usually that of
issuing country) different than that of
market
Innovation in the Bond Market
– Inverse floaters
• Coupon rate falls when interest rates rise
– Asset-backed bonds
• Income from specified assets used to service
debt
– Pay-in-kind bonds
• Issuers can pay interest in cash or additional
bonds
– Catastrophe bonds
• Higher coupon rates to investors for taking on
risk
Innovation in the Bond Market
Bond Characteristics (corporate bonds)
– Call provisions
• Callable bonds: May be repurchased by
issuer at specified call price during call
period
– Puttable bonds
• Holder may choose to exchange for par
value or to extend for given number of
years
– Convertible bonds
• Allow bondholder to exchange bond for
specified number of common stock shares
Callable Bonds

• Thus far, we have calculated bond prices

assuming that the actual bond maturity is the
original stated maturity.

• Most bonds, however, are callable bonds.

• A callable bond gives the issuer the option to

buy back the bond at a specified call price
anytime after an initial call protection period.

• For callable bonds, YTM might not be useful.

 Yield to Call
• Yield to call (YTC) is a yield measure that assumes a bond will be called
at its earliest possible call date.

• The formula to price a callable bond is:

é ù
C ê 1 ú+ CP
Callable Bond Price = 1-
YTC ê
êë ( 1+ YTC )2T ú

2 úû
(1+ YTC
2
)
2T

• In the formula, C is the annual coupon (in $), CP is the call price of the
bond, T is the time (in years) to the earliest possible call date, and YTC
is the yield to call, with semi-annual coupons.

• As with straight bonds, we can solve for the YTC, if we know the price
of a callable bond.
Spreadsheet Analysis.
Callable and Straight Debt
Callable Price = $ 1100
 Convertible Bonds
Are basically Corporate Bond with a call option:

If the value of the shares rises, then the investor find it profitable to convert
the bond, while if it falls, then the investor is left with the bond.

We can say that there is convexity (bond protection and equity appreciation)

May be less liquid that usual bonds

Amt dur_adj_ YLD_cnv_ Z_SPRD_ rtg_ rtg_ rtg_ MIN

ISIN Description px_mid ISSUER cpn typ Payment rank MTY TYP
Outstanding mid MID MID SP MOODY FITCH PIECE
XS1238034695 AMXLMM 0 05/28/20 3.000.000 1,05 0,54 99,43 104 AMERICA MOVIL SABA-DE CV A3 A- ZERO COUPON Sr Unsecured CONVERTIBLE 100.000
XS0802174044 AMXLMM 3 07/12/21 1.000.000 2,10 0,00 106,56 18 AMERICA MOVIL SABA-DE CV A3 A- FIXED Sr Unsecured AT MATURITY 100.000
XS0519902851 AMXLMM 4 ¾ 06/28/22 750.000 2,90 0,22 114,18 34 AMERICA MOVIL SABA-DE CV A3 A- FIXED Sr Unsecured AT MATURITY 50.000
XS0954302104 AMXLMM 3.259 07/22/23 750.000 3,92 0,31 112,30 35 AMERICA MOVIL SABA-DE CV A3 A- FIXED Sr Unsecured AT MATURITY 100.000
XS1379122101 AMXLMM 1 ½ 03/10/24 850.000 4,68 0,52 104,68 50 AMERICA MOVIL SABA-DE CV A3 A- FIXED Sr Unsecured AT MATURITY 100.000

Here above the convertible bond offers higher yields than its
counterparts.
 Convertible Bonds
Let’s say ABC Company issues a five-year convertible bond with a $1,000 par
value and a coupon of 5%. The “conversion ratio”—or the number of
shares that the investor receives if they exercises the conversion—option
is 25. The effective conversion price is, therefore, $40 per share ($1000
divided by 25).

The investor holds on to the convertible bond for three years and receives $50
dividends each year. At that point, the stock has risen well above the
conversion price and is trading at $60. The investor converts the bond and
receives 25 shares of stock at $60 per share, for a total value of $1,500. In this
way, the convertible bond offered both income and a chance to gain from the
upside risk of the underlying stock.
 67

Seniority
– Subordination clause
• Restrictions on additional borrowing stipulating senior
bondholders paid first in event of bankruptcy
 Trade off Risk & Return

Return: coupon + capital gain (loss)

Risks: Interest rate risk Credit risk

Default Risk and Bond Pricing
• Determinants of Bond Safety
– Coverage ratios: Company earnings to fixed costs
– Leverage ratio: Debt to equity
– Liquidity ratios
• Current: Current assets to current liabilities
• Quick: Assets excluding inventories to liabilities
– Profitability ratios: ROE, ROA
– Cash flow-to-debt ratio: Total cash flow to
outstanding debt
Financial Ratios and Default Risk
 Yield Spreads between Corporate and 10-Year
Treasury Bonds
20
Aaa -rated
18
Baa-rated
16 B-rated

14
Yield spread (%)

12

10

0
1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

2012
Default Risk and Bond Pricing

• Credit Default Swaps (CDS)

– Insurance policy on default risk of corporate
bond or loan
– Designed to allow lenders to buy protection
against losses on large loans
• Later used to speculate on financial health
of companies
Prices of CDSs, U.S. Banks
Prices of CDSs, German Sovereign Debt
The Yield Curve
• Yield Curve
– Graph of yield to maturity as function of
term to maturity
• Term Structure of Interest Rates
– Relationship between yields to maturity and
terms to maturity across bonds
Treasury Yield Curves
 Theories of Term Structure (1)

• The Expectations Hypothesis Theory

– Observed long-term rate is a function of today’s
short-term rate and expected future short-term
rates
– fn = E(rn) and liquidity premiums are zero
Returns to Two 2-Year Investment Strategies
 The Yield Curve and
Future Interest Rates
• Forward rates
(1 + yn ) n
(1 + f n ) =
(1 + yn -1 ) n -1
– fn = One-year forward rate for period n
– yn = Yield for a security with a maturity of n

n -1
(1 + yn ) = (1 + yn-1 ) (1 + f n )
n
 Forward Rates
• The forward interest rate is a forecast of a
future short rate.
• Rate for 4-year maturity = 8%, rate for 3-year
maturity = 7%.

1+ f4 =
(1 + y4 )
4
=
1.08 4
= 1.1106
(1 + y3 )3
1.07 3

f 4 = 11.06%
 Theories of Term Structure (2)

• Liquidity Preference Theory

– Long-term bonds are more risky; therefore, fn
generally exceeds E(rn)
– The excess of fn over E(rn) is the liquidity
premium
– The yield curve has an upward bias built into the
long-term rates because of the liquidity premium
Illustrative Yield Curves
 Interpreting the Term Structure

• The yield curve reflects expectations of

future interest rates
• The forecasts of future rates are clouded by
other factors, such as liquidity premiums
• An upward sloping curve could indicate:
– Rates are expected to rise
and/or
– Investors require large liquidity premiums to hold
long term bonds
 Interpreting the Term Structure

• The yield curve is a good predictor of the

business cycle
– Long term rates tend to rise in anticipation of
economic expansion
– Inverted yield curve may indicate that interest
rates are expected to fall and signal a recession
 Term Spread: Yields on 10-Year versus 90-day
Treasury Securities

16
10-year Treasury
90-day T-bills
12 Difference
Interest rate (%)

-4
1970

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2012