Lecture 4. Time Value of Money and Bond Valuation

Lecture 4. Time Value of Money and Bond Valuation

Readings: Chapters 8 and 9

Most students have learned about the time value of money in another class, accounting, finance, or economics. This section, therefore, will be brief and should be considered review. Those of you who have not the time value of money will need to spend more time on Chapter 8. All students will need to be able to use EXCEL to find the present and future value during the remainder of this term.

Time value of money tools are among the most important in the finance field. Finance has been described as the ‘art of finding the current value of future sums of money’. This is a reasonable definition because, in finance, we spend lots of time determining values of stocks, bonds, companies, or other projects. We define value as the present value of expected net cash flows in future years. An investment with no expected cash flows in the future is not worth considering.

Other questions that the time value of money helps to answer are: How much do I need to save every year to retire at the age of 65 or to put my new child through college 18 years from now or even to buy a car next year? What will be the payment on my mortgage or my car loan? How much should I put into a bank account if I want to have $1000 at the end of the year? In order to answer any of these questions, you need to understand the time value of money and how to use the concept to solve problems.

By the end of this unit, you will be able to:

• Find the future value and present value of a lump sum of dollars,
• Find the present and future values of an annuity,
• Use EXCEL to find the solutions to TVM problems, and
• Apply TVM concepts to bonds and find their values and yields.

The concepts that you learn in this chapter will be used throughout this course to value not only bonds, but also stocks and corporate investments in real assets. Hopefully you will apply them in making personal decisions as well.

TIME LINES

Chapter 8 of the textbook is very thorough in explaining the concepts of money and time. A few additional thoughts at the beginning may also be helpful.

Understanding time is crucial to the process. Money held today is worth more than money held tomorrow, which is worth more than money held a year from now. The reason is not that “a bird in the hand is worth more than one in the bush,” or that you might die tonight and never see the money tomorrow. The reason is that if you hold money today, you can invest it and earn more money in the future. Unless interest rates are 0%, having money to invest today means that you will have more tomorrow and even more the next year.

The previous explains the importance of opportunity cost in understanding the time value of money. If I borrow $100 from you today and promise to return it one year from now, you give up the opportunity to invest that money over the course of that year. Therefore, if I ask to borrow money and promise to pay you $100 a year from now, you would give me less than $100 today as a way to compensate for that lost opportunity. The further into the future I offer to pay you back, the less you would give me today. Therefore to value the promise you must be clear about the time sequence. The easiest way to keep time sequences straight is to draw a time line such as the one below.

Today time (T) = 0. A year from now T=1, two years from now T=2. At the maximum length of time for the problem T=N. Whenever you work a TVM problem, draw a time line first and then fill in the money values that you know and an X for the one for which you are looking. That way the sequencing will be clear in your mind. It is important to note here that we are defining a unit of time as one year. In fact, a unit of time can be any length. We could define the time period as one year, one quarter, one month or virtually any other period depending upon the problem we are addressing. The period of one year is used here for the sake of simplicity.

For example, I might ask you how much would be in a bank account 5 years from now, if I deposit $100 today, earn 5% per year and make no additional deposits? You would draw a time line:

This problem involves compounding, or the calculation of a future value of a single (lump) sum deposit. The deposit value grows as move to the right on the time line. Each year the account will earn 5% on the balance at the beginning of the year. Assuming that there are no intervening withdrawals, you will accumulate:

FV = 100(1.05)5

FV = 100(1.2763)

FV = 127.63

FV = future value

At the end of 5 years, you will have $127.63

Finding a number raised to an exponential power can be accomplished with a calculator or printed tables. However, the most versatile and easy way is with a financial spreadsheet. EXCEL has functions for future value, FV(rate, nper, pmt, pv, type), and present value, PV(rate, nper, pmt, fv, type). You must, however, know what you are looking for. A time line can be helpful.

MULTI-YEAR ANNUITIES AND OTHER CASH FLOWS

In addition to lump sum cash flow problems, many involve cash flows over the course of a time period. If all of the periodic cash flows are equal, the cash flow is called an annuity. For example, installment loan payments are annuities, since the amount paid is the same for each payment. Other investments may provide equal amounts of expected income every year. Present and future values for annuities can be determined using an annuity formula, which can be found on a financial calculator or, again, in Excel. Your textbook disk has problems worked for each chapter. Review these problems to help you learn to use Excel.

Calculating the present and future values for cash flow strings in which the periodic cash flows unequal involves discounting or compounding each individual cash flow to the desired point of measurement. The point of measurement is T=0 for a present value problem and T=N for a future value problem. While these types of problems are more realistic, especially in capital budgeting for corporations, the procedure can be time consuming. A spreadsheet is almost essential for these problems, as you will see in Chapter 13 and 14.

As you might expect, there are four types of TVM problems: present value of a lump sum and an annuity and future value of a lump sum and an annuity. Every time you need to use the TVM, the first question you need to ask is whether you are looking to find a present or future value and whether it is a lump sum or annuity. As you work the problems in the book, set up a time line to help you answer this question.

RATES AND PERIODS

The same formulas that are used to determine future and present values can be used to determine growth rates. For example, I might want to know what rate of return I had earned on a stock that I purchased 5 years ago for $40 if it was worth $70 today. These numbers would appear on a time line as:

The question is what rate would make 40 grow to 70 in 5 years. This is a future value problem. We would set up the formula as:

FV = PV (1 + r)5

70 = 40 (1 + r)5

The only unknown in the formula is r, the interest rate. Again we can use tables, financial calculators or the EXCEL function RATE(nper, pmt, pv, fv, type, guess). The answer is 11.84%. The annual growth rate was 11.84%. Dividends are not included.

Similarly, we could ask how long we would need to hold the stock for it to grow from $40 to $80 at a growth rate of 15% per year. In this case, we know the growth rate, but not the number of years. The answer is 5 years.

In both these problems we assume beginning and end values and ask about time and rates. Calculating rates of return is important for financial experts. There are many different types of return, as you will see in this course. A few are Internal Rate of Return, Yield to Maturity, Compound Rate of Return, Dividend Yield, and Current Yield. Some of these calculations require the TVM, some do not. You must know not only how to calculate the returns, but also which is most applicable and what it tells investors and lenders. Understanding calculations is just as important as using the spreadsheet.

You should stop now and begin working problems from the book and the disk to make sure that you can derive the correct answers.

BOND VALUATION ASSUMPTIONS

Bond valuation is one of the most common applications of the time value of money. Bond investments are straightforward in the sense that you make a purchase up front and then are paid back either over time (interest payments and principle) or in one lump sum (as in a pure discount bond). What are needed to value a bond are the expected cash flows and a discount factor. Since the discount factor determines the price of the bond, the discount factor is also referred to as the yield. This is so because once you pay a price for the bond, the rate of return is fixed as long as the cash flows materialize as expected. The yield is the discount rate that provides the bond investor with the return s/he requires as compensation for risk.

Bond yields are a function of the degree of risk perceived in a bond investment. There are several factors that influence this perceived risk. Each corresponds to an element that contributes to the addition of a risk premium to the required rate of return. The factors are as follows:

The risk free rate of return: no risky investment should earn a return that is less than the risk free rate. This is the starting point for calculating the bond yield.

The term to maturity: Typically, the longer the term to maturity, the greater the risk and therefore the greater the bond’s yield, ceteris paribus (remember that term from Econ 101? It means “all other things remaining the same). The longer the time period, the more difficult it is to predict what will happen in the future, making the situation more risky. The relationship between the term to maturity and bond yields is referred to as a yield curve. Typically the yield curve slopes upward, meaning that the longer the term, the higher the yield. On some occasions in the past, when short-term interest rates were very high, the yield curve was actually inverted, or sloped downward. This was due to the expectation that interest rates would be lower in the future. Because of the yield curve, the yield of a bond will change over time. As it approaches maturity, the yield typically falls, as the term risk premium falls.

Interest rate risk: Interest rate risk is related to the term to maturity in some ways. Interest rate risk is the degree to which the bond investor is exposed to the loss of value from rising interest rates in the market. As indicated before, when an investor buys a bond, the yield is locked in by the price paid and the expected cash flow to be earned (unless the bond has a floating interest rate). If market rates rise, the investor is losing out on higher interest earning opportunities. At the same time, the yield of the bond in the market rises, and the value of the bond itself falls (as you will see). The impact of changing yields is greater the longer the term of the bond, therefore the term risk premium is higher in an environment in which interest rates are considered to be more volatile.

Default risk: Default risk represents the likelihood that the borrower will fail to pay back either the interest or the principle amount borrowed. Default risk premia are usually based on an assessment of both the business and financial risk of the borrower. Business risk is characterized by potential volatility in the operating income of a borrower, and is related to its industry, product mix, marketing and production strategy, management quality, and other factors fundamental to the performance of the firm. Financial risk is based on the relative degree of indebtedness of a company. Some companies, such as public utilities, can afford higher amounts of debt than other companies, such as those in competitive industries such as telecommunications. When assessing financial risk, therefore, industry benchmarks are important considerations.

Formation of a bond yield is a very subjective exercise on the part of investors, since the perception of each may differ, as well as his or her subjective need to be compensated for risk. As the forces of supply and demand interact in the marketplace (where lenders are suppliers of funds and borrowers form the demand), interest rates (yields) adjust to the some measure of the average risk/return preferences of those involved.

BOND VALUATION

To value bonds in this way you need to know the following:

• The face value, or par value, of the bond (specified in the bond covenant),
• The coupon interest rate of the bond (specified in the bond covenant),
• The maturity date of the bond (specified in the bond covenant), and
• The discount rate (in our case a given).

For example, assuming the discount rate of 7%, we could value a 4-year bond with a face value of $1,000, a coupon rate of 12% as follows:

Note that the annual coupon payment is the coupon rate multiplied by the face value (here 0.12 x $1,000 = $120).

While this approach will work for any bond, it becomes inconvenient for bonds with long maturities. Instead, it is better to value the coupon payments and principal payment separately. The coupon payments are an annuity, and the principal is a one-time payment:

We have assumed so far that the coupon payments are annual, but it is more usual to find bonds that pay a semi-annual coupon, i.e., twice a year. In the semi-annual case, there is a simple way to adapt the formula above. If the bond is semi-annual:

• Divide both the coupon interest rate and discount rate in half, and
• Double the time periods.

To return to our example, assuming the discount rate of 7%, we could value a 4-year, semi-annual bond with a face value of $1,000, a coupon rate of 12% as follows:

Yield to Maturity

The yield to maturity (YTM) of a bond is expected rate of return for the bond, or the internal rate of return (IRR) on the bond (see Chapter 13 for more details). It is the return you expect to receive when you buy a bond, and the return that is assumed in pricing a bond. Practically speaking, it is the discount rate that makes the present value of the cash flows from the bond equal to the current price of the bond. It is not, however, necessarily the actual return you will get when you sell the bond or it comes to maturity. The actual, or realized, rate of return, depends on how interest rates change between the time you buy the bond and the time you sell the bond or it comes to maturity.

STOCK VALUATION

In the last session (Session 3) you learned about the capital asset pricing model (CAPM) which provided one method for calculating the required rate of return for an equity (stock) investment. The required rate of return resulting from the CAPM will be used as a discount factor when valuing an equity security much as the bond yield is used to as a discount factor in valuing the bond. There are some significant differences, however between a stock and bond valuation. These include:

• Equity securities represent ownership as opposed to debt, which represents a legal liability against the borrower;
• The return to equity is a residual claim, meaning that stockholders get what is left after all other claims have been satisfied. Bondholders, on the other hand, are entitled to the claim that is fixed by the loan contract;
• Equity securities may return dividends and/or capital gains, but are not required to distribute a cash return to investors;

As a result, the concept of how to value an equity security is a lot more elusive than how to value a bond. The following provides a representation of the context of equity valuation and presents a general model that can be used to apply the principles of valuation to equity issues.

A Conceptual Model for Valuing Equity

As with any other asset, an equity security should be valued using a process of discounting the net cash flows earned by the investor. As you learned in the prior session, returns to stockholders can accrue in the form of cash income (i.e., dividends) or increases in value (capital gains). Ultimately, the source of dividends and capital gains must be the earnings of the firm. The earnings (net income) of the firm represent the residual value after other claims have been satisfied. It is out of the earnings that the equity investors are compensated, and any capital gain accruing to the stock is typically based on the expectation of earnings in the future (often in response to current earnings levels).

In valuing an asset, the net cash flows are discounted over the expected life of the asset. An equity security is assumed to have an infinite life because the firm is considered to be a going concern. As a result, when setting up the discounting operation to value the stock, the net cash flow will be some function of the company earnings (eps), and the discount factor (k) will be based on a model similar to the CAPM, if not the CAPM itself. The general model looks like this: