SEMINAR ON BONDS

1
SEMINAR ON BONDS

• Day 1
• Some Preliminaries
• The Time Value of Money
• Compounding
• Types of Bonds
• Yields Pricing
• Yield Curves
• Duration (Macauleys duration , Modified
  Duration, PVBP)
• Day2
• Effective Duration. Callable/ Putable Bonds
• Convexity
• Bond Price Volatility
• Inflation Linked Bonds

2
Present Value and Future Value(One Interest
period)

• For securities with remaining maturity lt 1 year
  (one cash flow)
• PV present value ( i.e. the cash amount we pay
  to buy the security)
• FV future value ( i.e. the redemption amount
  of the security including coupon payment, if any)
• Days no of days from purchase to maturity
• Base 360, or 365, or actual, depending on the
  market convention
• Y yield annualized.

3
Present Value and Future Value(continued)

• Example Investing 100 now (PV) for one year at
  a yield of 10 we get on maturity 110 (FV)
  assuming 365 days and 365 base
• Investing though, today 90 (PV) for one year
  and getting on maturity 100 ,(FV) we realize a
  yield of ?
• 11.11

4
Present Value and Future ValueMultiple interest
periods

• The effect of compounding
• Example We invest 100 on a 3 years security
  paying an annual coupon 3 once a year each year.
  What will be the future value of our 100
  investment?
• This is because each coupon payment we assume we
  reinvest at the initial yield.
• The same security paying a semiannual coupon
  yields us a future value of

5
Interest Calculations

• Periodic vs. Continuous compounding
• Periodic Compounding
• FVFuture Value (Principleinterest)
• PVPresent Value (Principle)
• minterest frequency per year
• nyears
• Continuous compounding the limit of periodic
  compounding with m
• i.e.
• e2.71828

6
Interest Calculations (continued)

• Converting periodic to continuously compounding
  interest rate
• Let R1 continuously compounded interest rate
• R2 periodic compounded interest rate

7
Interest Calculations (examples)
8
Definitions and Concepts

• BOND a financial obligation for which the issuer
  promises to pay the bondholder a specified stream
  of future cash flows, including periodic interest
  payments (coupons) and a principal repayment.

Default or Credit Risk
BONDHOLDER RISK
Market (interest rates risk)
Liquidity Risk
9
Why buy bonds?

• Investors have traditionally held bonds in their
  portfolios for three reasons
• Income. Most bonds provide their holders with
  fixed income. On a set schedule, annually or
  semiannually or quarterly the issuer sends the
  bondholder a fixed payment
• Diversification. Although diversification does
  not ensure against loss, an investor can
  diversify a portfolio across different asset
  classes that perform independently in market
  cycles to reduce the risk of low, or even
  negative, returns.
• Protection against Economic Slowdown or
  Deflation.

10
Types of Bonds

• Bonds are differentiated according to the issuer,
  and according to the type of their cash flows.
• The major bond markets are
• 1) sovereign bonds (issued by governments)
• 2) corporate bonds (issued by corporates)
• There are many types of bonds, depending on their
  cash flows among which are
• Fixed rate bonds (bonds paying periodically fixed
  coupon)
• Floating rate notes (bonds that their coupons are
  linked to an index, e.g Libor)
• Callables I.e. bonds giving the right to the
  issuer to buy them back before their maturity
  (e.g. mortgages)
• Bonds with Sinking Funds for example a 7 10
  years corporate bond, paying down 10 of the
  principal amount annually beginning in year 3
• Zero Coupon bonds bonds that are just redeemed
  on maturity without any other periodic payment of
  interest

11
Market conventions

• Bonds are quoted in the secondary market as a
  percentage of their face value (Clean or Flat
  Price) without including the accruals (if any).
• Example
• Market quote for Hell Rep 3.10 due 20 April 10
  on 7th Oct 2005 with settlement 12th Oct 2005
  was
• 101.12 - 101.14 This means that to buy Face
  value 1 million Euro of this bond we have to pay
  1,011,400 Euro 175 days (days from 20 April
  2005 to 12 Oct 2005) accrued interest
  (3.101751,000,000/365) 14863.01 Euro total
  1,026,263.01 Euro
• Dirty or Full Price of the Bond Clean Price
  Accruals.
• Coupon frequency (annual or semiannual), and the
  days count convention (e.g. 30/360, Act/360,
  Act/Act, etc..) is predetermined by the issuer.
• Most Euro zone government bonds have annual
  coupons and Actual/Actual year fraction
• US Treasuries, Italian BTPs, U.K. Gilts have
  semiannual coupons and Actual/Actual year
  fractions.

12
Yields

• The main types of yields are the following
• Current Yield
• Yc current yield
• c annual coupon payment
• P Clean price
• Simple Yield to maturity (or Japanese Yield)
• Yssimple yield to maturity
• Rredemption value (most cases 100)
• PClean Price
• Tsmdays from settlement to maturity
• cannual coupon
• Yield to maturity is the value of the discount
  rate in the bond equation that equates the
  present value of all future cash payments of the
  bond to the current market price

13
Yield to Maturity. The Bond Equation

• B dirty price of the bond Present Value of the
  bonds cash flows discounted by the yield to
  maturity
• Ci cash flow of the bond at time ti with 1ltiltn
• Yyield to maturity
• Analytically. Having a bond paying an annual
  coupon c for n years and redeems at par (100),
  yielding to maturity y its present value is
  given by

14
Yield to Maturity. The Bond Equation (continued)

• For any coupon frequency the present value of the
  bond ( i.e. its dirty price is given by the
  formula
• B dirty pricePV
• R redemption value (usually 100)
• Y Yield to maturity
• m coupon frequency
• DSC days from settlement to next coupon payment
• E no of days between coupon dates

15
Yields (continued)

• Example
• Bond Hell Rep. 3.10 due on 20th April 10. Market
  price 101.14 as at 7th Oct 2005, settlement 12th
  Oct 2005. Calculate its current yield, simple
  yield and yield to maturity.

16
Current Yield, Simple Yield, Yield to Maturity
Examples
17
Bloomberg Yield Analysis screen for GGB 3.60
JUL 16
18
Current vs. Japanese and Yield to Maturity

• Current yield is the quickest check of the yield
  of a bond but not reliable as it ignores any
  change in the value of the capital invested.
  Investors today do not rely on current yield.
• Simple Yield to Maturity although more accurate
  than current yield is still not the ideal measure
  of yield because 1)it assumes constant capital
  gain (for bonds trading in discount) or capital
  loss (for bonds trading at premium), and 2) it
  ignores the time value of money. It is simple.
• Yield to Maturity is the most accurate measure of
  a bonds yield as it explicitly recognizes the
  importance of points in time at which different
  cash payments from a bond are to be received.
  Implicit in the definition of the yield to
  maturity is the assumption that the investor will
  be able to reinvest all coupon payments at at a
  rate equal to yield to maturity at which he
  bought his bond. This risk is known as the
  reinvestment risk.

19
ZERO COUPON BONDS

• Zero coupon bonds are bonds that pay no coupons,
  or bonds of which their coupons have been striped
• Cash flows wise are the simplest fixed rate bonds
  as they have a single cash flow on maturity. The
  bond equation is simplified as there are no
  coupon payments to
• BBond price (dirtyclean)
• Rredemption value (usually 100)
• Y yield to maturity
• DSM days from settlement to maturity
• E base of the year (360,365)

20
ZERO COUPON BONDS (examples)
21
The PriceYield Function
22
Yield Curves

• A yield curve plots the yields to maturity of a
  series of bonds of the same quality ( i.e. same
  credit) against their respective terms to
  maturity.
• A yield curve can exhibit the following 4 shapes
• Normal or positively sloped yield curve A curve
  in which short-term interest rates are lower than
  longer term interest rates
• Inverted yield curve or negatively sloped A
  curve in which long -term interest rates are
  lower than that of shorter term interest rates.
• Flat yield curve Short-term and long term
  interest rates are roughly equal.
• Humped yield curve A humped yield curve is
  positively sloped from the short maturity sector
  to the intermediate sector, but negatively sloped
  from the intermediate to the long sector.

23
Normal Yield Curve
24
Inverted Yield Curve
25
Flat Yield Curve
26
Humped Yield Curve
27
Theories explaining the yield curve shape

• Liquidity preference theory. This theory states
  that that the yield curve will be upward sloping
  because of the preference of investors for
  liquidity. Liquidity here is defined as the
  ability to recover the principal of the bond in a
  reasonably short period of time.
• Market segmentation theory. This theory views
  the fixed income market as a series of distinct
  markets, segregated by maturity. Individual
  investors and issuers are restricted to specific
  maturity sectors. Thus investors and issuers do
  not have complete maturity flexibility.
• Expectations theory. According to the
  expectations theory the shape of the yield curve
  reflects the market consensus forecast of future
  interest rate levels.

28
Summing up theories of the yield curve shape

• Yield curves tend to exhibit a modestly positive
  slope over long periods of time, reflecting the
  market participants desire for liquidity.
  Liquidity preference
• Market segmentation shows up as an influence on
  yield curve shape, particularly over short term
  horizons (e.g. auction periods) and within
  specific issuer sectors. Imbalances of supply
  and demand create bumps on the yield curve at
  various maturity points for specific periods of
  time.
• Finally there are an adequate number of investors
  with maturity flexibility (e.g. mutual funds) to
  validate a degree of expectations reflected in
  the yield curve shape. Inflation fears tend to
  steepen the slope of the yield curve while
  disinflation expectations act to flatten or
  invert the yield curve.

29
The higher credit quality, the flatter the yield
curve
30
Germany (AAA)- Belgium (AA) Greece (A)
31
Determinants of the absolute level of the yield
curve

• A nominal interest rate can be dissected into
  three basic components.
• Nominal interest rate real interest rate
  inflation premium risk premium
• The real interest rate is the compensation to
  the investor for deferring consumption to a
  future period. Even if inflation is stable at 0
  a risk less investment (e.g US Treasury) must
  offer a positive rate of return.
• The inflation premium is intended to preserve the
  purchasing power of the investor over time. This
  premium reflects an expectation of the future
  inflation level over the lifespan of the
  investment.
• The risk premium protects the investor against
  all other potential negatives, including a)
  credit or default risk, b) call or early
  redemption risk, c) liquidity or marketability
  risk, d) risk of unexpected changes in inflation
  (i.e. the degree of unpredictability in assessing
  future inflation.

32
DURATION

• The weighted average maturity of a bonds
  cashflows, where the present values of the cash
  flows serve as weights
• the term to maturity of the equivalent zero
  coupon bond
• the balancing point of a bonds cash flow stream,
  where the cash flows are expressed in terms of
  present value

Cash flows
Time in years
0
Duration
33
Calculation of duration of a 10 year bond with
coupon 4 priced at par to yield 4 on maturity
34
Factors influencing Duration

• Term to maturity
• Coupon rate
• Accrued interest
• Market yield level
• Sinking fund features
• Call provisions
• Passage of time

35
The influence of each factor on duration
36
Factors influencing duration- A rule of thumb

• As a rule of thumb
• Long Maturity, Low Coupon, Low Yield High
  Duration

37
Duration of 7 coupon bonds of various
maturitiesEach bond is priced at par YTM 7
38
The durationterm to maturity relationship for
coupon bearing bonds
39
The durationterm to maturity relationship for
zero coupon bonds
40
The durations of 15 year Govt. Bonds for several
coupon rates. Yield environment 7
41
The durationcoupon rate relationship
42
The Durationaccrued interest relationship
A bonds duration is inversely related to the
amount of accrued interest attached to the bond.
Hellenic Republic due 20/5/13, 7.50 price 111,
settlement 19/5/99, YTM6.29, Duration
8.80 price 111, settlement 20/5/99, YTM6.29,
Duration9.39
43
The Durationmarket yield relationship
44
The duration-market yield relationship
45
Calculation of the duration of a 7 coupon,
10-year corporate bond with a sinking fund paying
down10of the principal amount annually,
beginning in year 3. The bond is priced at par to
yield 7 to maturity
46
The relative contributions to the price of a 7
coupon, 10year corporate bond with (1) no sinking
fund provisions and (2) a 70 sinker. Each bond
is priced at par to yield 7 to maturity.
Sinking fund provisions lower the duration of a
bond by reducing the average maturity of the
principal repayment.
47
The durationpassage of time relationship

• As time passes a bonds duration falls at an
  increasing rate. The duration decline is a
  natural consequence of the progressively smaller
  set of remaining coupon cash flows and the
  approaching principal repayment.

48
Modified Duration

• Macaulays duration can be used as a measure of a
  bonds risk. A longer duration implies a higher
  degree of price sensitivity and therefore,
  greater market risk.
• Macaulays duration in order to be more accurate
  as a measure of bond risk requires a
  modification. This revised version of duration is
  called modified duration and is calculated as
  follows
• MDmodified duration
• Dduration
• Yyield to maturity

49
Modified Duration (..continued)

• Modified duration calculates the percentage
  change in a bond price for one basis point change
  in yield.
• Bdirty price of the bond
• Yyield to maturity

50
Price Value of a Basis Point (PVBP,or PV01)

• Price Value of a Basis point is a measure that
  shows how much the price of the bond (or a
  portfolio) will change for a shift of 1 b.p. in
  yield . It is simply the Modified Duration times
  the dirty price of the bond
• PV01 is widely used as it shows with good
  approximation how much money a bond or a
  portfolio of bonds will profit (loose) from a
  favorable (adverse) shifts in yield(s).

51
Mathematics of Duration

• Bdirty price of the bond
• ci cash flow of the bond at time ti, 1lt i lt n
• yyield to maturity
• D Duration, MD Modified Duration

52
continued Mathematics of Duration

• Bdirty price of the bond,
• MDModified Duration,
• PV01Present Value of a basis point.

53
Duration, Modified Duration, PV01 (example)
54
Effective Duration

• Effective Duration is
• For an option-free bond the bonds modified
  duration
• A measure of the average maturity of a bonds
  cash flows. For a callable bond the average
  maturity of the cash flows is shortened by the
  possibility of early redemption of the issue.
• A sophisticated weighted average of the modified
  durations that a callable (putable) bond can
  have.
• A simple weighted average of the modified
  duration of the option-free component and the
  modified duration of the option component
• For a callable (putable) bond effective duration
  lies between the modified duration to call (put)
  and the modified duration to maturity. It
  approaches the MD to call(put) as yields
  fall(rise), and the MD to maturity as yields
  rise(fall).

55
Callable Bonds

• A callable bond is a bond that gives the right to
  the issuer to buy it back at a pre-specified
  price over a predetermined period
• The price of a callable bond is calculated as the
  price of a an equivalent non callable bond of
  similar structure less the value of the call
  option(s) attached
• The call option value is subtracted because the
  bondholder implicitly sells the call to the
  issuer of the bond.

Noncallable Bond Price
Call Option Value
Callable Bond Price

-

• The crossover price is the price at which the
  Yield to Call and the yield to Maturity are
  equal. The yield level is termed crossover yield

56
The price yield curves for a callable bond and
for two non callable counterparts a non callable
maturing on the first call date and the a non
callable maturing on the final maturity date
MMMaturity Date Bond C1C1Call Date
Bond C2C2Callable Bond
M
Crossover Yield, Crossover Price
C1
Bond price
C2
C1
M
C2
Market Yield ()
57
Assessing the Duration of a Callable Bond

• There are 3 ways of assessing a callable bond
  duration
• Calculate the duration to call (DTC) and duration
  to maturity (DTM). Use DTC if the bonds market
  price exceeds the crossover price, and use DTM if
  the bonds market price exceeds the crossover
  price
• Calculate a weighted average duration based on a
  subjective assessment of the probability of call.
  
• Calculate an effective, or option- adjusted
  duration by using an option valuation model.

58
Factors influencing a callable bonds duration

• Selecting either DTC or DTM
• Market price gt Crossover price Modified DTC
• Market priceltCrossover price
  Modified DTM

Under this (more naive) approach the factors
influencing duration are the market price and the
bonds crossover price
59
Factors influencing duration of a callable bond

• B. The weighted average duration approach
• The modified DTC and the modified DTM that form
  the lower and upper boundaries respectively of
  the bonds modified duration.
• The current yield environment. A low yield
  environment increases the probability of future
  calls.
• The expected trend in interest rates. A trend to
  lower rates increases the probability of early
  redemption
• The expected variability in interest rates. The
  greater the variability the more probable it is
  that the bond will be redeemed early.

DTC
DTM
AVGDUR
60
Factors influencing the duration of a callable
bond

• C. Effective duration approach
• Call date(s). An early call date reduces a
  bonds effective duration.
• Maturity date. A short remaining term to
  maturity lowers a bonds effective duration.
• Call (i.e. strike) price. A low call price
  shortens effective duration.
• Market price. A high market price (i.e. low
  yield) decreases duration.
• Market yield volatility. A high degree of yield
  volatility reduces duration.

61
Example of Calculating either DTC or DTM
A 10 year 6 corporate bond NC5 at 103.00 with a
market price at 106.00. At market price 106.00
the modified duration of a non callable bond
maturing in 10 years is 7.48 At the same price
(106.00) the modified duration of a non callable
bond with maturity 5 years and redeeming at
103.00 is 4.26
Yield Environment Price to Maturity Price to Call
5.21 106.00 105.71
5.30 105.32 105.32
6 100.00 102.24
Crossover price and Yield
Market price (106.00) gt than crossover price
(105.32) therefore the duration of the callable
bond is 4.26
62
Example of calculating the weighted average
duration
A 10 year 6 corporate bond NC5 at 103.00 with a
market price at 106.00
Pc probability of call DTC 4.26 DTM 7.48
Yield Environment Bond Price Probability of Call
2 135.93 1
3 125.59 0.9
4 116.22 0.8
5 107.72 0.7
5.21 106.00 0.600
6 100.00 0.4
7 92.98 0.3
8 86.58 0.2
9 80.75 0.1
10 75.42 0
AVGDUR4.26X0.607.48X0.405.55
63
Example of calculating the effective duration
A 10 year 6 corporate bond NC5 at 103.00 with a
market price at 106.00
Effective Duration(modified duration of a
noncallable X Wb)( modified duration of the call
option X Wc) Wb the market value weight of the
bond component (expressed in decimal form) Wc
1-Wb the market value weight of the call option
(expressed in decimal form) Wb Wc 1 A non-
callable bond with same maturity of the callable
trades at 110.00 yielding 4.72 and having a
modified duration of 7.56 The value of the call
option is Callable bond price noncallable bond
price call option value 106 110 call option
value, therefore call option value - 4 An
option is extremely price sensitive and therefore
has a large duration. If the call option has a
modified duration of 50 then Effective duration
of the callable bond 7.56X110/10650X(-4/106)7.5
6X1.037736-50X1.88679 5.96
64
Putable Bonds

• A putable bond is a bond that gives the right to
  the holder to sell it back at a pre-specified
  price on a predetermined put date.
• The price of a putable bond is calculated as the
  price of a an equivalent non putable bond of
  similar structure plus the value of the put
  option attached
• The put option value is added because the
  bondholder implicitly buys the put option from
  the issuer of the bond.

Putable Bond Price Option-free bond price put
option value

• A putable bond is attractive because the holder
  (not the issuer) has the discretion to exercise
  the put option.
• In a bull market it behaves like an option free
  bond, (with significant price gains) whilst in a
  bear market downside price losses are limited.

65
Comparison of callable and putable bonds
Feature Callable Putable
Option definition A call option gives its holder the right to buy the bond at a pre-specified price on a pre-specified date. A put option gives its holder the right to sell a bond at a pre-specified price at a pre-specified date.
Option holder Issuer Investor
Market availability Widespread Limited Supply
Option Strike Price Typically at a premium Typically at par
Option Exercise A series of call dates and call prices A single put date
In the money when Market price gt call price Market price lt put price
Out the money when Market price lt call price Market gt put price
Yield vs. an option-free bond Higher yield Lower yield
66
The Limitations of Modified Duration

1. Instantaneous yield change. Although it is
   possible to experience sizable intraday, yield
   shifts in turbulent financial markets, yield
   shifts typically occur over time.
2. Small change in yield. Modified duration is a
   better approximation of the price behavior of the
   bond for small yield changes (10 bps or less)
3. Parallel shifts in yield. It would be a rarity
   to find all bond yields moving in tandem.
   Short-term bond yields fluctuate more than
   long-term bond yields. Nonparallel yield shifts
   often occur.

67
The Limitations of Modified Duration (example)
68
CONVEXITY
Positive Convexity Region
69
Definitions of Convexity

• Convexity is the second derivative of the
  priceyield function and it shows the rate of
  change of modified duration as yields shift
• Convexity can be defined as the difference
  between the actual bond price and the bond price
  predicted by the modified duration line.
• The term convexity arises from the fact that
  priceyield curve is convex to the origin of the
  graph. This curvature creates the convexity
  effect
• Convexity enhances a bonds price performance in
  both bull and bear markets but not in a uniform
  manner.
• The larger the changes in yield the greater the
  convexity effect.
• A decline in yields creates stronger convexity
  impacts than does an equivalent rise in yields.

70
Convexity and the PriceYield function

• The nonlinear priceyield function can be
  analyzed as a Taylor series of derivatives

71
Factors influencing Convexity

1. Duration. Convexity is positively related to the
   duration of the underlying bond. It is also an
   increasing function of duration
2. Cash flow distribution. Convexity is positively
   related to the degree of dispersion in a bonds
   cash flows.
3. Market yield volatility. High Volatility in
   interest rates creates large convexity effects.
4. Direction of yield change. Convexity is more
   positively influenced by a downward movement in
   yields than by an upward surge in yields

72
The ConvexityDuration relationship

• Price,yield curves for 3,10, 30 years bonds with
  a coupon 7 and price at par to yield 7

73
Convexitycash flow distribution relationship
The priceyield curves of a 30year Bond with a
coupon 7 priced at par to yield 7 and a
duration of 13.25 and a zero coupon bond of same
duration(I.e. 13.25 years to maturity) at 7 yield
P30,30years coupon bond
P0,zero coupon bond
74
The convexityyield volatility relationship
High Volatility in interest rates creates large
convexity effects
75
The Convexitydirection of yield change
relationship

• Convexity effects are greater in declining yield
  environment than in rising yield environment.