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Chapter Twelve: Bond Markets and Fixed-Interest Securities (Econ 512): Economics of Financial Markets Dr. Reyadh Far

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Chapter Twelve: Bond Markets and Fixed-Interest Securities(Econ 512): Economics of Financial Markets Dr. ReyadhFaras

12.1 What Defines a Bond?

- The prototypical bond is a contract that commits the issuer to make a definite sequence of payments until a specified date.
- Bonds are commonly traded in secondary markets.
- Bonds are issued by governments and incorporated companies.
- Bonds provide companies with a way of acquiring capital at a known cost, without sacrificing rights of control over the company if it fulfilled the terms of the contract.

Econ 512 Dr. Reyadh Faras

- Maturity Date: the date on which the principal is paid back to the lender.
- Face Value (par value): the price at which the bond is sold and the amount paid back to the lender.
- Coupons: the amount of interest paid to the lender at periodical intervals.
- Indenture: the formal contract between the lender and the borrower that specifies all provisions.
- Default: when the issuerer fails to pay the coupons or the principal to the borrower.

Econ 512 Dr. Reyadh Faras

12.1.1 Maturity (redemption) Date

- Let T denote the maturity date, t the present day (today), n the life or time to maturity (n = T- t).
- Maturity may be fixed, though it need not be. Other possibilities include the following:
- Callable bonds, which include provisions specifying conditions in which the issuer can terminate the contract before T, typically by paying the face value of each bond to its current owner.
- Convertible bonds, which allow holders to exchange the bond for another asset.

Econ 512 Dr. Reyadh Faras

A bond may be redeemed to either: (a) in cash at face value, (b) with a unit of the issuer’s ordinary shares, at the discretion of the holder. Alternatively, the holder can convert the bond into shares over a specified period during the bond’s life.

- In addition, it can be interpreted as a bundle comprising an inconvertible bond and an option.
- Perpetuities, for which T → ∞. A perpetuity is a promise to make a coupon payment every time period, indefinitely into the future.
- Sinking bonds, it oblige the issuer to redeem existing bonds over an extended period of time, typically by the purchase of outstanding bonds at current market prices.

Econ 512 Dr. Reyadh Faras

- Denote the sequence of coupons by ct+1, ct+2 , .., cT per unit of the bond.
- The simplest, and most common, bond is one for which the coupons are constant: c, c, .., c.

Timing of Coupon Payments

- Although coupons are expressed at annual rates, their payment is split into installments.
- Bonds traded between dates of coupon payments at a dirty price: the price that reflects an element of accrued interest.
- A clean price is the price after subtracting the estimated value of accrued interest.

Econ 512 Dr. Reyadh Faras

- Also called ‘pure discount bonds’ or ‘bullet bonds’
- These are bonds for which c = 0, and pay a lump sum, the face value, at maturity.
- While they exist, they are less commonly issued.
- They are created synthetically as stripped bonds, or strips: a financial intermediary purchases a coupon-paying bond and repackages it in the form of a sequence of zero-coupon bonds, one for each coupon, and one for the face value paid at maturity.
- A coupon-paying bond is similar to a portfolio of zero-coupon bonds.

Econ 512 Dr. Reyadh Faras

- Rather than promising a constant coupon, the bond indenture might include a rule to calculate regular payments over the life of the bond.
- Examples include:
- Floating rate bonds, for which the coupon is linked to an observed interest rate that varies across time; and
- Index-linked bonds, for which the coupon is linked to a specified index of prices (e.g. CPI).

Econ 512 Dr. Reyadh Faras

- When the issuer defaults on any clause of the contract, it is at the discretion of the bondholders to make a legal claim on the issuer’s assets.
- Bond indentures include clauses that place restrictions on, or provide privileges for, parties to the contract. Below are two examples:
- The contract could give priority to some bonds over others with respect to their claims on the issuer’s assts.

Econ 512 Dr. Reyadh Faras

A specific asset, or group of assets, may be identified as collateral for the bond.

- In the case of collateralized bonds, the specified asset alone constitute security for the bond.
- By specifying particular assets as collateral, the issuer may make loans more marketable.
- This process can lead to the securitization of loans.
- For instance, loans on real estate can be packaged together and traded as bonds backed by the property that was mortgaged to obtain the loan.

Econ 512 Dr. Reyadh Faras

Default: Who gets what?

Econ 512 Dr. Reyadh Faras

12.2.1 Nominal Zero-Coupon Bonds

- Any Zero-Coupon (ZC) bond can be specified with just two parameters: its face value, m, and the date, T, at which the issuer pays m to the bond’s holder.
- ZC bonds are assumed to be nominal in the sense that the redemption value is fixed in units of account (e.g. m= $100).
- The price (market value) of a ZC bond can be expressed as a proportion of m.

Econ 512 Dr. Reyadh Faras

Let pn denote the market price today of an n-period ZC bond.

- The yield to maturity, or spot yield, on this n-period ZC bond is defined as the constant annual rate of return, yn, that would be received if the bond is held until maturity: yn = (m/ pn)1/n - 1
- Thus, a bond with m= 100, n= 4, pn= 83 has a spot yield = (100/83)(1/4) - 1= 4.77% per annum.
- Equivalently, yn can be defined to satisfy:

pn = m/ (1+yn)n

- In words: the spot yield on a ZC bond is the rate of return that equates its market price to the net present value of its value.

Econ 512 Dr. Reyadh Faras

At each date there exists a sequence of spot yields (yi) one for each maturity.

- The spot yield, yn, is the rate of return on the bond only if it is held to maturity.
- If the investor’s holding period differs from n, the bond is risky because either:
- For a holding period less than n, the bond will be sold before maturity; or
- For a holding period greater than n, m will be reinvested at date T for a subsequent return that is not known until T, or later.

Econ 512 Dr. Reyadh Faras

Consider the holding period yield on an n-year bond over the coming year.

- Let pn,t denote the price of an n-year bond today (date (t) and pn-1, t+1 denote its price at t+1.
- Then the one-year holding period yield is defined as: (pn-1, t+1 ) /(pn,t) -1
- Except for bonds about to mature, the future value of the bond is uncertain: pn-1, t+1, for n>1, depends on market conditions at date t+1 in the future.
- Hence, the holding period yields are uncertain and bonds are like any other asset the FV for which is uncertain.

Econ 512 Dr. Reyadh Faras

The curve depicting the relationship between a ZC bond’s price and its spot yield, has 2 properties:

a) negatively sloped: that is the higher the yield, the lower the bond price, and

b) convex: for successive increases in the yield, the smaller are the reductions in price.

- It is recognized that bond yields are intimately related to one another and to the risk-free interest rate (which is controlled by the Central Bank).
- The policy-determined-risk-free rate then impacts upon bond yields and, hence, their prices.

Econ 512 Dr. Reyadh Faras

In the simplest case, if the yields on all bonds equal the risk-free interest rate, then monetary policy directly determines all bond prices.

- Restrictive monetary policy (higher interest rates) is associated with a fall in bond prices.

Econ 512 Dr. Reyadh Faras

- Consider a bond that promises to pay to its holder a coupon of c per year for n years plus the face value, m, when the bond terminates at maturity.
- If the current market price of the bond is p, then its yield to maturity, y, is defined as the solution to

p = (12.6)

- The yield to maturity can be understood as the internal rate of return on the bond.
- Notice that both p and y depend upon: i) the time to maturity, n; ii) the coupon, c; and iii) the face value, m.

p =

Econ 512 Dr. Reyadh Faras

The yield to maturity on a coupon-paying bond does not have the same interpretation as the spot yield on a ZC bond.

- For coupon-paying bonds, the value of y is, at best, an approximation to the rate of return from holding the bond from the present until it matures
- Unless forward contracts are available to guarantee the rates at which future coupons can be reinvested, then the rate of return on a coupon paying bond is inherently uncertain.
- The risk associated with the rates at which coupons can be reinvested is referred to as reinvestment risk.

Econ 512 Dr. Reyadh Faras

Two other concepts of yield are occasionally useful for coupon-paying bond:

- First, if the bond’s price equals its face value (p=m in 12.7), then the resulting y is termed its par yield, which equals c/m.
- Second, a bond’s flat (or current) yield is defined as c/p, it is a misleading measure of the return on a bond except for perpetuities.
- Conclusions:

1) the holding period yields are uncertain, partly a result of reinvestment risk.

2) the price a negative and convex function of its yield to maturity.

Econ 512 Dr. Reyadh Faras

- The analysis focuses on the responsiveness of p to y
- One possible measure is ∂p/∂y, the rate of change of price with respect to yield.
- The Macaulay duration provides a more robust measure for the responsiveness of p to y.
- The Macaulay duration, D, is defined as:

D = (12.8)

- The value of D depends upon: n, c and m.
- The main purpose of constructing D is to obtain a single number to measure the responsiveness of p to y, allowing for differences in c and n among bonds.

Econ 512 Dr. Reyadh Faras

Example

Econ 512 Dr. Reyadh Faras

An important aspect of D is its time dimension.

- For a coupon-paying bond, D < n, always.
- Intuitively, this is because portion of the payoff on the bond is received before the bond matures at n.
- The payoff on ZC bond occurs entirely at maturity. Consequently, D = n.
- For coupon-paying bonds with the same time to maturity and with the same yield, the one with the higher coupon has the smaller D (∂D/∂c <0).
- In words: higher coupons mean that a higher proportion of the bond’s payoff occurs before maturity and hence its average period is smaller.

Econ 512 Dr. Reyadh Faras

- The previous analysis is applicable to bonds with observed prices as the outcome of open market trading.
- Fortunately, it can be adapted to prescribe valuing non-tradable bonds or traded infrequently.
- Bond values are computed as a function of observed prices of actively traded bonds.
- The arbitrage principle provides the necessary link between bond valuation and observed prices.
- Example: a ZC bond pays $100 one year from now, the risk-free interest rate is 25%. In a frictionless market, the bond is traded at $80, why?

Econ 512 Dr. Reyadh Faras

To generalize the analysis, note that the spot yield is y= (m/p) – 1 (for n=1)

- In words, the arbitrage principle implies that the spot yield on a one-period ZC bond equals the risk-free interest rate.
- The rule: the bond’s value equals the net present value, discounting at the risk-free interest rate.
- In the example,
- the discount factor equals 1/(1+0.25) and

the bond’s valuation is $80 = $100 x 1/(1+0.25)

- The trivial reasoning can be applied to more realistic bond contracts.

Econ 512 Dr. Reyadh Faras

Consider a bond contract B that pays a coupon c for the next n years, plus m at maturity.

- Bond B is regarded as n ZC bonds, the 1st paying c after one year, the 2nd paying c after two years, and so on to the one paying c+m after n years.
- In the absence of arbitrage opportunities, value of B equals the sum of its stream c, c, …, c+m weighted by the ZC bond prices:

value of B = p1c + p2c + p3c + …+ pn(c+m) (12.12)

- Where pj denotes the price of a ZC bond paying one unit of account (say, $1) after j years.

Econ 512 Dr. Reyadh Faras

If bond B could be purchased or sold for a value different from that given by (12.12), then an investment strategy (for B and ZCs) could be devised to guarantee arbitrage profits. (12.12) can be rewritten as:

value of B = (12.13)

- In summary, it is possible to derive a rule such as(12.13), to value a bond as a function of the realized prices (or spot yields) of other bonds. The result is called the “fair” value of the bond.

Econ 512 Dr. Reyadh Faras

- Not totally true that bond portfolios are risk free.
- Holding bonds is associated with two broad risk categories: (a) interest rate risk and (b) basis risk.
- Interest rate risk reflects the impact of market-wide credit conditions on bond prices.
- If bond yields tend to move broadly together, then a general rise in the cost of borrowing raises bond yields, thus reducing bond prices and the market value of bond portfolios.
- The Macaulay duration serves to measure the responsiveness of bond prices to their yields, and hence provides an index of the magnitude of interest rate risk.

Econ 512 Dr. Reyadh Faras

Basis risk encompasses all sources of risk other than interest rate risk, including the following:

- Credit risk: reflects the possibility of default. Event risk forms a subset of credit risk associated with specific incidents that could result in default
- Reinvestment risk: reflects unforeseen changes in future interest rates at which coupon receipts from a bond can be reinvested.
- Timing risk: reflects the contingency that a cash flow of a bond is altered during its lifetime.
- Exchange rate risk: reflects unforeseen fluctuations in the exchange rtes among currencies.
- Purchasing power risk: reflects unanticipated changes in the future value of money (i.e. real value of returns).

Econ 512 Dr. Reyadh Faras

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