Chapter 9 Interest Rates and Bond Valuation

9.1 Bonds and Interest Rates

In this chapter, we will focus on bond instruments, which are one way to securitizeTo make a series of cash flows into a tradable instrument. (to make a series of cash flows into a tradable instrument) a loan. A bondA loan packaged as a tradable security, typically with a periodic interest payment and a repayment of principal at maturity. is a loan packaged as a tradable security, typically with a periodic interest payment and a repayment of principal at maturity.

Bonds originally were pieces of paper, stating all of the necessary information in the center with coupons (yes, just like the Sunday paper!) around the edge. The owner of the bond would cut off the coupon and turn it in to receive each periodic interest payment. The terminology stuck, so even with today’s electronic bonds, the periodic payments are called coupon paymentsThe periodic payments of a bond, typically paid semiannually.. Most bonds pay these coupon payments twice a year (semiannually), though once a year (annual) payments are not uncommon.

These coupon payments typically occur until the maturity dateThe date on which the final coupon payment and the principal of the bond is due., at which time the final coupon payment and the principal of the bond would be due. This principal amount is called the par valueThe pricipal amount of the bond.. Currently, the most common par value for corporate bonds in the USA is $1,000. When describing a bond, we typically list the coupon rateThe annual total of the coupon payments expressed as a percentage of the par value., which is the annual total of the coupon payments expressed as a percentage of the par value.

For example, a bond which pays $50 semiannually would be paying a total of $100 ($50 × 2) every year. If the par value of the bond is $1,000, then we say that the bond has a coupon rate of $100 / $1,000 = 10%.

All bonds are governed by a bond indentureThe contract which stipulates all of the sepcific details of a bond, including conditions for repayment and default., which is the contract which stipulates all of the specific details of the bond, including conditions for repayment and default, when bondholders can usually seek legal recourse for greater control of the company. Specific items in the bond indenture are referred to as covenantsSpecific items in a bond indenture..

9.2 Credit Risk

Not all bonds are created equal. For example, which seems like the safer investment: loan $1,000 to the US government, or loan $1,000 to a small company that hasn’t paid interest to its current investors for over a year? Investing in the world’s largest economy probably seems like the better bet, all other things being equal. But what if the small company offered you three or four times the return on your investment? Now the decision is not so straightforward….

The major difference between our government bond and the small company’s bond is our expectation of the borrower’s creditworthiness. We label the uncertainty in future cash flows due to possibility that the borrower will not pay credit riskThe uncertainty in future cash flows due to possibility that the borrower will not pay.. The most common credit risk is default riskThe most common credit risk, which in the extreme occurs when the company doesn’t have the cash to pay the necessary scheduled payment (or chooses not to)., which in the extreme occurs when the company doesn’t have the cash to pay the necessary scheduled payment (or chooses not to). This is not the only reason; bonds can be determined to be in default for breeching specific covenants (for example, a covenant specifying that certain liquidity ratios be maintained), even if all payments have been made.

Since, as of the time of this writing, the US government is the largest government on Earth with the largest potential revenue stream, it has been considered the borrower with the least amount of credit risk. While not truly without credit risk (in 1979, for instance, the US government was late in a small portion of its debt payments, due to a supposed paper error!), as a proxy, we consider US government bonds to be the “risk-free” asset, in terms of credit risk.

Sometimes, credit risk can be reduced through the use of collateralAssets pledged to secure repayment., or assets pledged to secure repayment. The most familiar collateral arrangement is the standard home mortgage, where the property itself is collateral against the amount borrowed. Corporations can also pledge certain assets when they choose to issue bonds, with the anticipation that the lower credit risk will correspond to a lower cost of borrowing.

To aid investors in their assessment of the credit risk of bond issuers, there exist ratings agencies which attempt to qualify the creditworthiness of each bond issue. Some of the larger agencies include: Standard and Poor’s (S&P), Moody’s, and Fitch. Ratings range from AAA (or Aaa, as different agencies have slightly different scales) for the most creditworthy debt, to D for debt that is already in default. Bonds rated at least BBB− (or Baa3) are considered “investment grade”, which can be an important distinction for many mutual funds that invest in bonds to allow the bonds to be held in their portfolios. Any bonds rated lower (BB+/Ba1 or below) are considered not investment grade, or “speculative”. These bonds are sometimes called “junk bonds” as well (though it would be important to remember that even “junk bonds” have a higher claim on a company’s assets than any shares of stock).

There have been some concerns over the accuracy of the ratings provided by these providers, especially after the credit crisis of 2008. Some highly rated securities ended up having default rates much higher than would have been expected, given their assigned ratings. One argument is that these securities were so complex that the ratings agencies couldn’t accurately evaluate them. Another claim was that, since ratings agencies earn their revenues by charging for assigning a rating to a security’s issue, there is a potential conflict of interest that encourages the agencies to give higher ratings than they otherwise should. Regardless of the cause, confidence in ratings agencies has been shaken, and, while the information provided by such companies is undoubtedly useful, an investor is wise to not blindly take ratings as absolute truth, especially when more complex instruments are involved.

9.3 Bond Yield

All investments have a certain amount of risk (even stuffing dollars under your mattress is vulnerable to house fire!), and, in a perfect world, we would be able to completely assess the risk of each investment. There are many employees, companies, and consultants whose entire focus is on evaluating risk. Arguably, all of finance boils down to the study of risk and return.

The value of a bond, like any series of future cash flows, is intrinsically related to the perceived opportunity cost of the investment, as we discussed in Chapter 6 "Time Value of Money: One Cash Flow". This cost, called the real interest rateThe interest rate that only compensates for the perceived opportunity cost of the investment., is determined by the market by causing the price of the bond to rise or fall. When we hear newscasters speaking of “interest rates falling”, what they are really saying is that the relevant bonds have changed in price, exposing a belief by investors that the opportunity costs of holding the bond are going down. So, as a shortcut we can talk about interest rates changing, but we should remember that, ultimately, these rates are derived from the price of the bonds! An investor buying a bond at a certain price point will yield a certain return on her investment; in bond parlance, this expected return is called the bond’s yieldA bond’s expected return.. Specifically, yield to maturity (YTM)The annualized rate of retun on a bond, assuming the bond is held to maturity and all the expected cash flows occur. is the annualized rate of return on a bond, assuming the bond is held to maturity and all the expected cash flows occur.

Each bond faces a host of additional factors, all of which must be accounted for to properly assess the value of a bond. As time goes by, the purchasing power of a dollar (or whichever unit of currency we are using) could deteriorate; this we call inflationThe deterioration of the purchasing power of a currency.. Opportunity cost and inflation factored together are typically labeled the nominal interest rateThe interest rate that compensates for opportunity cost and inflation factored together..

If both the inflation rate and the real rate of interest are low (that is, < 5% each), then we can safely approximate the nominal interest rate as:

We use r* (pronounced “r-star”) to represent the real rate of interest and π (the Greek letter “pi”), which here is not the circle constant of 3.14, to represent inflation. Why have finance scholars chosen to use π? Because we thought using i would be too confusing, since both “interest” and “inflation” start with the same letter….

This equation has the benefit of being easy to remember and can be used for fairly accurate results. While historically it has been safe to assume a real interest rate at or below 5%, there have been many occurrences of large rates of inflation much larger than 5% throughout the world. When this is the case, it is more prudent to utilize the more accurate Fisher equation for interest rates:

While a little more cumbersome to remember and use, it is the more accurate, especially if inflation is high.

This combination of the real rate of interest and the inflation rate should be the return required on an asset that is free of any other risks; we call this the risk-free rate of return and designate it: r_RF. As of the time of this writing, standard practice is to use US government bonds as a proxy for such a risk-free asset.

We already know of one risk, credit risk, that will cause investors to demand higher yields, but there are many other potential sources of risk. For example, if a bond isn’t likely to be easily tradable, then investors could want higher yields to compensate. All of these will contribute to the investment’s risk premiumThe amount of return required by investors over the risk-free asset., or amount of return required by investors over the risk-free asset.

9.4 Bond Valuation

The financial value of any asset is the present value of its future cash flows, so we already have the tools necessary to start valuing bonds. If we know the periodic coupon payments, the par value, and the maturity of the bond, then we can use our time value of money skills from Chapter 7 "Time Value of Money: Multiple Flows" to solve for either price or YTM, given the other.

For example, if I know that a bond with a 5% annual coupon has 7 years to maturity, a $1,000 par value, and has a YTM of 6.5%, I can figure out its price. Since payments are yearly: n = 7 years, r = 6.5% per year, PMT = ($1,000 × 5%) = $50 per year, and FV = $1,000.

Figure 9.1 Bond Timeline—Annual Coupon

Using a financial calculator or excel, or solving by hand, we should get a PV = $917.73.

If our bond paid its coupon semiannually, we need to calculate the value in terms of semiannual (six month) periods. By convention, bond yields are quoted like an APR, in that they are always equal to the rate over the period times the periods in a year. Thus, our inputs should be: n = 7 × 2 = 14 semi-years, r = 6.5% / 2 = 3.25 % per semi-year, PMT = ($1,000 × 5% / 2) = $25 every semi-year, and FV = $1,000.

Figure 9.2 Bond Timeline—Semiannual Coupon

Our result is a PV of $916.71, which is only slightly different than our annual coupon bond.

Solving for YTM is similar. If the price of our above semiannual coupon bond rose to a quoted 94.35, what is the YTM?

Figure 9.3 Bond Timeline—Semiannual Coupon Solving for YTM

n = 14 semi-years, PMT = $25, FV = $1,000, and PV = − ($1,000 × .9435) = −$943.50. We represent the price as a negative number since it is a cash outflow. Solving for r gives a result of 3.00% for the semiannual period, or 3.00% × 2 = 6.00% for the quoted YTM.

Excel has two functions that can be used to directly solve for bond prices and yields. Both require actual calendar dates for settlement and maturity of the bond; since all we know of our bond is that it has 7 years to maturity, we’ll pick dates that are 7 years apart. Both also require an input of how much of the par value is received at redemption, quoted as a % of par. Since we are receiving all of the par value at redemption, this number will be 100.

=PRICE(settlement date, maturity date, coupon rate, yield, redemption, pmts per year)

=PRICE(“1/1/2010”, “1/1/2017”, 5%, 6.5%, 100, 2)

91.67

=YIELD(settlement date, maturity date, coupon rate, price, redemption, pmts per year)

=YIELD(“1/1/2010”, “1/1/2017”, 5%, 94.35, 100, 2)

.0600 or 6.00%

9.5 The Bigger Picture

Bonds allow entities to borrow money easily from investors. When banks demand too much return or are unwilling to accept more risk, bonds can be the avenue to raising more capital. Corporations (or govenments, municipalities, etc.) can then use the capital raised to invest in new projects or pay for ongoing projects.

As managers, we need to consider how our actions will influence our ability to issue bonds and raise capital. As investors, we need to consider bonds as an investment class.

One side benefit of bond markets is that we can witness the perceived current value of the bond as they trade. From these prices, we can deduce the return demanded by the market, and use this information to learn more about the market (for example, risk tolerance or inflation expectation) or the issuer (for example, perceived risk of default) of the bond.

9.6 End-of-Chapter Problems