bonds

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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.

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Bond Characteristics

Par (face) value: principal amount to be repaid at maturityCoupon rate, coupon payment: Each period (may be every 6 months, every year etc.) there is coupon payment. It is a fixedamount (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 bondsMaturity: maturity date on which the contract expires and principal (face value) is repaid

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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

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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

problemfor the issuer

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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)

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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

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Semiannual Coupons Coupon rate is given as APR (annualized

rate) plus frequency of coupon payments

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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 themarket 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.

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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 kd<kc VB > M Selling at a premiumkd=kc VB = M Selling at parkd>kc VB < M Selling at a discount

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Evolution of bond value over timeSpecial case: kd=constant over time

Coupon Rate 10%Face Value 1,000$ TTM 30# Coupons per year 1YTM 13%

Bond Value $775.13

Example: M=1,000 N=30 years kc = 10%

annual couponSpecial case: kd=constant over time

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The graph shows

advantage or disadvantage will last for a shorter period time as time passes

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Bond Yields YTM yield to maturityYTC yield to callCY 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

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yield to call

YTC : rate of return earned on a bond if it is called before its maturity date

If current YTM < 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 bondPc is call price (usually M+INT)Solve for k, it is YTC

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Current yield

Current yield: CYt: Current yield relates the annual coupon interest to themarket price.

The current yield calculation takes into account only the

coupon interest and no other source of return that will affect an investor’s yield. No consideration is given to capital gain/loss. The time value of money is also

ignored.

price

coupon dollar annual yieldcurrent

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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 thepresent value of the promised future cash flows equal in sum to the current market price of the bond.

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YTM

so YTM is a promised yield but it is not the expected yield unless

P(default)=0 it is an ex ante return

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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.

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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.

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Holding-Period Return

if there are N years in the holding period, rhp can be

converted into an equivalent annual rate:

(1+rhp,annual )N = rhp

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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).

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Example

It is easier to see this as follows: Original investment: $922.69 at

t=0 Final Value: $50(1+k1)2+$50(1+k2)+$1,050 at

t=3 Our choice of k1 and k2 reflect our reinvestment

rate assumption

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Example

Question: What is the average annual rate earned from

this investment?

$922.69(1+kavg)3 = $50(1+k1)2+$50(1+k2)+$1,050

YTM concept solves this equation by assuming kavg = k1 = k2

Using bank analogy above:

Generates same cash flows

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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

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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)

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Original investment: $922.69 at t=0Final Value: $50(1+3%) +$50+$990.65 =$1,092.15 at t=2

Question: What is the average annual rate earned from this investment?

$922.69(1+kavg)2 = $1,092.15

kavg =8.80% is the holding period return

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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 =1.052*50=$52.60Sell our 1.052 shares of the bond for 1.052*990.65=$1,042.16Original investment: $922.69Final Value: $52.60+$1,042.16=$1,094.76$922.69(1+kavg)2 =$1,094.76kavg = 8.93%

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Example

Assume that all coupons are reinvested at the new YTMInvestor’s horizon i.e. when liquidation occurs is important

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Example

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

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Example

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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

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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.

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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 YTCP=1,133.9= 50 PVIFAk,8 + 1,050PVIFk,8

what is YTM at this timeP=1133.9= 50 PVIFAk,20 + 1000PVIFk,20

As YTM decreases bond is more likely to be calledThe company may use refunding strategy

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Yield to Call

note that in the example above kc>YTM>YTC What happens to YTC as YTM decreases?

P increases so YTC has also to fall

Example: if we assume current price=$885.30 YTC= 14% YTM= 12%note that in this example kc<YTM<YTC

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Riskiness of a bond

Interest rate riskInterest rate risk is the concern that rising kd will cause the value

ofa 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

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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 alower rate at maturity

Exposure is higher on bonds with shorter maturities ceteris paribus

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Default Risk

Recall interest rate= k* + IP + DRP + LP + MRP

Types of corporate bondsBond ratingsJunk bondsBankruptcy 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.

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types of bonds

Mortgage bond: Represents debt that is secured by the pledge ofspecific 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:

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Why are ratings important?

They affect cost of borrowing for firmsInstitutional investors can only buy investment grade bondsIndividual issue ratings may change over time (up or downgrading possible)

Junk bonds (speculative grade bonds)Have high default risk and therefore have required returnJunk 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

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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 > or < liquidation value of firm’s

assets

bonds

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