Bonds

1
Bonds

• A bond is a long-term debt instrument in which a
• borrower agrees to make payments of principal and
  
• interest, on specific dates, to the holders of
  the bond.

2
Bond Characteristics

• Par (face) value principal amount to be repaid
  at maturity
• Coupon rate, coupon payment Each period (may be
  every 6
• months, every year etc.) there is coupon payment.
  It is a fixed
• amount (unless it is a floating rate bond)
• Floating rate may be tied to t-bill rate. It has
  caps or floors, it may
• be convertible to fixed rate.
• Another type zero coupon bonds
• Maturity maturity date on which the contract
  expires and
• principal (face value) is repaid

3
Call provision

• issuer may call the bond for redemption before
  maturity. It is an
• option given to the issuer. Callable bonds
  therefore sell at a
• Discount
• Call premium cost to the issuer of calling a
  bond issue. It
• typically declines over time
• Call protection exists if a bond can not be
  called for a period of
• time after its issuance

4
Sinking fund arrangement

• requires the issuer to retire a portion of the
  bond each year
• How
• Redeem a portion, determine by lottery (no call
  premium)
• Buy on the open market
• issuer may deposit cash with a trustee rather
  than repurchasing bonds
• Sinking funds are designed to protect bondholders
  
• Large amounts of cash outflow at maturity may
  create a problem
• for the issuer

5
Other features

• Bond may be convertible to common stock
• Bonds may be packaged together with warrants
  (warrant a long-term option to buy a stated
  number of common stock at a specified price)
• Income bond A company agrees to pay interest
  only if it meets a threshold income requirement
  (e.g. interest will be paid at 12 if income is
  greater than 15 million)
• Indexed bond interest paid is based on Consumer
  Price Index (plus real rate)

6
Bond Value

• Bond Value PV of cash flows it will generate
• Given kd, N, INT, M
• VB INT PVIFAkd,N M PVIFkd,N
• Coupon rate vs kd (current market rate)
• The discount rate (kd ) is the opportunity cost
  of
• capital, and is the rate that could be earned on
• alternative investments of equal risk.
• ki k IP MRP DRP LP

7
Semiannual Coupons

• Coupon rate is given as APR (annualized rate)
• plus frequency of coupon payments

8
Relationship between coupon rate, required yield,
and price

• As yields in the marketplace change, the only
• variable that can change to compensate an
• investor for the new required yield in the
• market is the price of the bond.
• When the coupon rate is equal to the required
• yield, the price of the bond will be equal to its
  
• par value.

9
Relationship between coupon rate, required yield,
and price

• When yields in the marketplace rise above the
  coupon rate at a
• given point in time, the price of the bond
  adjusts so that an
• investor thinking of the purchase of the bond can
  realize some
• additional interest. If it did not, investors
  would not buy the issue
• because it offers a below market rate. The
  opposite holds if yields
• in the marketplace fall below the coupon rate.
• when kdltkc VB gt M Selling at a premium
• kdkc VB M Selling at par
• kdgtkc VB lt M Selling at a discount

10
Evolution of bond value over timeSpecial case
kdconstant over time
Example M1,000 N30
years kc 10
annual coupon
Special case kdconstant over time
11
The graph shows
advantage or disadvantage will last for a shorter
period time as time passes
12
Bond Yields

• YTM yield to maturity
• YTC yield to call
• CY current yield
• YTM current required rate of return on a bond.
  It is
• same as kd, or promised return
• Given P(current price), N, INT, M
• P INT PVIFAkd,N M PVIFkd,N

13
yield to call

• YTC rate of return earned on a bond if it is
  called
• before its maturity date
• If current YTM lt kc (i.e. premium bond) and bond
  is
• callable. then it is likely to be called
• P INT PVIFAk,TTC Pc PVIFk,TTC
• P is current market price of the bond
• Pc is call price (usually MINT)
• Solve for k, it is YTC

14
Current yield

• Current yield CYt
• Current yield relates the annual coupon interest
  to the
• market price.
• The current yield calculation takes into account
  only the
• coupon interest and no other source of return
  that will
• affect an investors yield. No consideration is
  given to
• capital gain/loss. The time value of money is
  also ignored.

15
What is YTM

• The yield-to-maturity on a bond is the single
  interest
• rate that, if paid by a bank on the amount
  invested,
• would enable the investor to obtain all the
  payments
• promised by the security in question.
• Equivalently, YTM is the discount rate that makes
  the
• present value of the promised future cash flows
  equal
• in sum to the current market price of the bond.

16
YTM

• so YTM is a promised yield
• but it is not the expected yield unless
  P(default)0
• it is an ex ante return

17
YTM

• YTM calculations do not take into account any
  changes in the market value of a security before
  maturity.
• This fact might be interpreted as implying that
  the owner has no interest in selling the
  instrument before maturity, no matter what
  happens to his or her situation.
• The calculation also fails to treat intermediate
  payments in a fully satisfactory way.
• An owner who does not wish to spend interest
  payments might choose to buy more of these
  securities. But the number that can be bought at
  any time depends on the price at that time, and
  YTM calculations fail to take this consideration
  into account.

18
Holding-Period Return

• A measure that can be used for any investment is
  its holding-
• period return. The idea is to specify a holding
  period and then
• assume that any payments received during that
  period will be
• reinvested. Holding period is defined as the
  length of time over
• which an investor is assumed to invest a given
  sum of money.
• Although assumptions may differ from case to
  case, the usual procedure assumes that any
  payment received from a security will be used to
  purchase more units of that security at the then
  current market price.
• When this procedure is applied, the performance
  of a security can be measured by comparing the
  value obtained in this manner at the end of the
  holding period with the value at the beginning.

19
Holding-Period Return

• if there are N years in the holding period, rhp
  can be
• converted into an equivalent annual rate
• (1rhp,annual )N rhp

20
Example

• Consider a 5 annual coupon bond
• with 1,000 face value and 3 years
• to maturity. Its current market price
• is 922.69.

YTM is 8. It is the interest rate that solves
the following equation

• YTM calculation assumes
• that the owner will receive those 3 cash flows,
  i.e. bond will be held until maturity.
• coupons will be reinvested
• the reinvested coupons will earn k percent
  return per year (which is the current YTM we will
  find).

21
Example

• It is easier to see this as follows
• Original investment 922.69
  at t0
• Final Value 50(1k1)250(1k2)1,050 at
  t3
• Our choice of k1 and k2 reflect our reinvestment
  rate assumption

22
Example

• Question What is the average annual rate earned
  from
• this investment?
• 922.69(1kavg)3 50(1k1)250(1k2)1,050
• YTM concept solves this equation by assuming
  kavg k1 k2
• Using bank analogy above

Generates same cash flows
23
Holding period return

• does not make the above assumptions
• It does not assume bond will be held till
  maturity
• It requires you to know in advance the
  reinvestment rate(s) for
• your coupons
• But it is flexible about those rates i.e. it does
  not assume
• kavg k1 k2
• So compared to YTM calculation we need to know
  more
• Our horizon when we will sell the bond
• Reinvestment rates
• Selling price of the bond at the end of our
  horizon

24
Holding period return

• For example, assume we choose 2 years as our
  horizon.
• i.e. we will sell the bond after 2 years
• If we further assume that we will deposit coupons
  into a
• bank account (one year time deposit)

25
Holding period return

• Original investment 922.69 at t0
• Final Value 50(13) 50990.65 1,092.15 at
  t2
• Question What is the average annual rate earned
  from this investment?
• 922.69(1kavg)2 1,092.15
• kavg 8.80 is the holding period return

26
different reinvestment rate

• If we use a different reinvestment rate
• assumption, for example reinvesting
• coupons on the same bond, then we
• need the following information

At time 1 we buy 0.052 shares of
the same bond At time 2 we will get coupon
1.0525052.60 Sell our 1.052 shares of the
bond for 1.052990.651,042.16 Original
investment 922.69 Final Value
52.601,042.161,094.76 922.69(1kavg)2
1,094.76 kavg 8.93
27
Example
Assume that all coupons are reinvested at the new
YTM Investors horizon i.e. when liquidation
occurs is important
28
Example

• Note that if your holding period is 5 years, your
  equivalent annual
• holding period return is 9.00 which is equal to
  the YTM at the
• time of purchase

29
Example
30
Example

• Note that if your holding period is 5 years, your
  equivalent annual
• holding period return is 9.00 which is equal to
  the YTM at the
• time of purchase

31
What does the example show

• When you buy the bond you do not know if ytm is
  going to change and if it is in what direction.
• Note that 5 year horizon gives you 9 average
  annual holding period return no matter if ytm
  falls or rises by 1 percent.
• 5 year is the value of another measure (which we
  will not discuss in this course) called the
  duration. If your horizon is 5 years, then
  reinvestment rate and price effects cancel each
  other and your average annual holding period
  return equals ytm at the time of the purchase.

32
Yield to Call

• Example 10 year bond 10 semiannual coupon
  payment selling for 1,133.9
• Can be called after 4 years. If YTM stays at this
  rate (or falls) the
• issuer may call it. (To buy it back at a price
  lower than the market price)
• Find YTC
• P1,133.9 50 PVIFAk,8 1,050PVIFk,8

what is YTM at this time P1133.9 50 PVIFAk,20
1000PVIFk,20
As YTM decreases bond is more likely to be
called The company may use refunding strategy
33
Yield to Call

• note that in the example above kcgtYTMgtYTC
• What happens to YTC as YTM decreases?
• P increases so YTC has also to fall
• Example if we assume current price885.30
• ? YTC 14
• ? YTM 12
• note that in this example kcltYTMltYTC

34
Riskiness of a bond

• Interest rate risk
• Interest rate risk is the concern that rising kd
  will cause the value of
• a bond to fall.
• Means as kd ? VB ?
• Exposure is higher on bonds with longer
  maturities ceteris paribus

• Example 10 annual coupon bonds with 1 year and
  10 years to maturity. When yield to maturity
  changes

35
Reinvestment rate risk

• Reinvestment rate risk is the concern that kd
  will fall,
• and future CFs will have to be reinvested at
  lower rates,
• hence reducing income.
• As kd ? funds will be reinvested at a lower rate
  many
• bonds will be called
• Even if they are not called, funds will be
  reinvested at a
• lower rate at maturity
• Exposure is higher on bonds with shorter
  maturities
• ceteris paribus

36
Default Risk

• Recall interest rate k IP DRP LP MRP
• Types of corporate bonds
• Bond ratings
• Junk bonds
• Bankruptcy and reorganization
• Default risk depends on financial strength of
  issuer
• Terms of bond contract
• Same corporation may have several types of bonds
  outstanding.
• They may have different default risks due to
  differences in
• seniority and collateralization.

37
types of bonds

• Mortgage bond Represents debt that is secured by
  the pledge of
• specific property. In the event of default, the
  bondholders are
• entitled to obtain the property in question and
  sell it to satisfy their
• claims on the firm.
• Debenture An unsecured debt backed only by the
  credit
• worthiness of the borrower. There is no
  collateral, and the
• agreement is documented by an indenture. To
  protect the holders
• of such bonds, the indenture will usually limit
  the future issuance
• of secured as well as any additional unsecured
  debt.
• Subordinated debenture

38
Why are ratings important?

• They affect cost of borrowing for firms
• Institutional investors can only buy investment
  grade bonds
• Individual issue ratings may change over time (up
  or downgrading
• possible)
• Junk bonds (speculative grade bonds)
• Have high default risk and therefore have
  required return
• Junk bond market was developed in early 1980s and
  it collapsed in early
• 1990s
• These bonds are used by companies to finance
• a leveraged buyout
• a merger
• a troubled company

39
Bankruptcy and reorganization

• bankruptcy may lead to either liquidation or
  reorganization
• reorganization calls for the restructuring of
  existing debt
• interest rate ?
• maturity ?
• bond holders may get equity stake
• decision depends on
• value of reorganized firm gt or lt liquidation
  value of firms assets