Assignment
"Time Value of Money and Bond Valuation" Please respond to the following:
• Examine the concept of time value of money in relation to corporate managers. Propose two methods in which time value of money can help corporate managers in general.
• Examine the pros and cons of a sinking fund from the viewpoint of both a firm and its bondholders. Determine the fundamental manner in which this knowledge could be helpful to a financial manager. Provide a rationale for your response.
• Part 1: Time Value of Money
Slide 1 Introduction Welcome to Financial Management. In this lesson we will discuss the time value of money.
Slide 2
Topics
The following topics will be covered in this lesson:
Timelines;
Future values;
Present values;
Finding the interest rate, I;
Finding the number of years, N;
Annuities;
Future value of an ordinary annuity;
Future value of an annuity due;
Present value of ordinary annuities and annuities due;
Finding annuity payments, periods, and interest rates;
Perpetuities;
Uneven, or irregular, cash flows;
Future value of an uneven cash flow stream;
Solving for I with irregular cash flows;
Semiannual and other compounding periods;
Fractional time periods;
Amortized loans; and,
Growing annuities.
Slide 3 Timelines Recall, the primary objective of financial management is to maximize the value of the firm's stock. Moreover, the value of the firm's stock depends in part on the timing of the cash flows investors expect to receive from investing in the firm. Hence, it is very important that the financial manager have an understanding of the time value of money and how it impacts the firm's stock price. Time value of money is also referred to as discounted cash flow, or DCF, analysis. As we study this concept it is important to remember that there is no other concept in finance that is more important than time value of money or DCF.
When we analyze time value of money it is important to draw a timeline because this helps us visualize what is happening in a particular problem and helps us solve the problem. Consider the timeline shown on the slide.
Time zero is today;
Time one is one from today, or the end of period one;
Time two is two time periods from today, or the end of period two and so on.
Many times the periods are measured in years, but that is not a requirement. Time can be measured in semiannual periods, quarters, months, or days. Look that time period one. The tick mark at time one represents the end of period one and it also represents the beginning of time two since time one has just passed. Cash flows are placed directly underneath the tick marks. Suppose a lump sum or single amount of cash outflow in the amount of one hundred dollars is invested at time zero. The five percent is the interest rate for each of the three time periods. Look at time period three. At time three the cash flow is unknown. Note that in time periods one and two there are no cash flows and the interest rate is constant for all three time periods.
Next slide
Slide 4 Future Values A dollar today is worth more than a dollar in the future primarily because of inflation. We refer to the value of a dollar today as present value or PV. If we invest money today at some interest rate we refer to the value received in the future as the future value or FV. The process of going from present value to future value is referred to as compounding. I is the interest rate the bank pays on the account each year. INT is the dollar amount of interest earned during the year. We calculate this amount by multiplying the beginning amount by I. Therefore, INT equals PV times I. FV sub N is the future value, or ending amount, in the account at the end of N years. PV is the value today but FV sub N is the value N years in the future after the interest earned is added to the account.
There are four ways we can solve this problem. First we can use a step by step approach. This method requires that we calculate the future value for each year and then sum the results.
The second method we can use is called the formula approach. The formula approach uses a mathematical equation to solve time value of money problems. In general, FV sub N equals PV times one plus I raised to the Nth power.
Next we can use a financial calculator to solve time valued money problems. Financial calculators have five keys corresponding to the five variables in the time value equations:
Specifically, N is the number of time periods;
I divided by YR is the interest rate per period;
PV is the present value and since we began by making a deposit this number is an outflow and must have a minus sign in front of it;
PMT is the payment -this key is used only if there is a series of equal payments; in our example this value should be entered as a zero; and
FV is future value which is automatically determined by the calculator.
The last method we can use to solve a time value of money problem is an Excel spreadsheet. To calculate future value we locate the FV function which is given by FV. This formula calculates the FV. We can set up this formula by using either numbers or cell references from an Excel spreadsheet and the results are the same. Using spreadsheets to solve a time value money problems has two advantages over the other methods. First, it is easy to verify the inputs. Second the analysis is more transparent.
Recall when interest is earned on interest earned in prior periods it is referred to as compound interest. If instead interest is earned solely on the principal it is referred to as simple interest. Mathematically with simple interest is total interest is given by PV times I times N or principal times interest times the number of time periods. Then the future value equals PV plus PV times I times N.
Slide 5 Present Values Present value is the opposite of future value. To see that this is true consider the following example. Assume we have money to invest and a broker offers to sell us a bond that pays one hundred fifteen dollars and seventy-six cents in three years. Assume that banks offer a three year certificate of deposit, or CD, at five percent and if you don't purchase the bond you'll purchase the CD. The five percent paid on the CD is called the opportunity cost or the rate of return we would earn on a different investment of similar risk. We want to know how much we should pay for the bond today. To determine this amount we must calculate the present value which means we are discounting a future sum.
Recall, to find FV we use the formula FV sub N equals PV times the quantity one plus I raised to the Nth power. To find the PV we rearrange the formula and find that PV equals FV sub N divided by the quantity one plus I raised to the Nth power. We know FV sub N equals one hundred fifteen dollars and seventy-six cents and I equals five percent.
Then PV equals one hundred fifteen dollars and seventy-six cents divided by one point zero five cubed which equals one hundred dollars.
This amount is referred to as the fair price of the bond. If we could purchase the bond for less than one hundred dollars we should buy the bond instead of the CD. If we must pay more than one hundred dollars for the bond we should purchase the CD. If the price of the bond is exactly one hundred dollars we are in different between the bond and the CD.
The one hundred dollars is the present value of one hundred fifteen dollars and seventy-six cents due in three years when the interest rate is five percent. It is important to remember that time value of money problems can be solved using more than one method. Additionally always keep in mind that the goal of financial management is to maximize the company's intrinsic or fundamental value. This value is the present value of the firm's expected future cash flows.
Slide 6 Finding the Interest Rate, I So far we've calculated FV and PV. But notice that the equation has four variables. If we know the values of three of the variables we can easily calculate the fourth. Note that the variables are PV, FV, I, and N. Suppose we know the values for TV, FV and N and we want to find I. How do we do this?
Now we know PV, FV, N, and must determine I. To calculate I we solve the following equation: FV equals PV times the quantity one plus I raised to the Nth power. We should use either a financial calculator or the RATE function in Excel to solve the problem since any other method would prove to be very difficult and very time-consuming.
Slide 7 Finding the Number of Years, N Now suppose we have five hundred thousand dollars to invest when the interest is four point five percent. We want to calculate how long it will take five hundred thousand dollars to accumulate to one million dollars. To determine N we solve the following the equation:
One million dollars equals five hundred thousand dollars times the quantity one plus zero point zero four five raised to the Nth power. We can solve for N by using a financial calculator, the NPER function in Excel or by working with natural logarithms using natural logarithms. Regardless of the method used, the result is the same.
Slide 8 Annuities An annuity is a series of equal payments made at fixed intervals. If the payments are made at the end of each period the annuity is called an ordinary annuity or deferred annuity. If the payments are made at the beginning of each period the annuity is called an annuity due. In finance ordinary annuities are more common than annuities due. It is important to observe that in the case of an annuity due, each payment is shifted back one time period.
Slide 9 Check Your Understanding
Slide 10 Future Value of an Ordinary Annuity Suppose we have an ordinary annuity where we deposit one hundred dollars at the end of each year for three years and earn five percent per year. We want to calculate the future value of the annuity or FVA sub N. To solve for FVA sub N we can use a step by step formula approach, a financial calculator or the FV function in Excel. If we use the step by step approach we set up the problem in the following way:
FVA sub N equals PMT times the quantity one plus I raised to the N minus one power plus PMT times the quantity one plus I raise to the N minus two power plus PMT times a quantity one plus I raised to the N minus three power.
This equation tells us that the first payment earns interest for two periods, the second for one period, and the third earns no interest because the payment is made at the end of the annuity's life. It follows that FVA sub N equals one hundred dollars times one point zero five squared plus one hundred dollars times one point zero five plus one hundred dollars which equals three hundred fifteen dollars and twenty-five cents. In general, the future value of an annuity is given by FVA sub N equals PMT times the quantity one plus I raised to Nth power divided by I minus one divided by I.
Slide 11 Future Value of an Annuity Due The future value of an annuity due is larger than that of an ordinary annuity because in the case of an annuity due payments are made at the beginning of each time period and for this reason each payment occurs one period earlier and therefore the payment earns interest for one additional period. These types of problems are solved by using either a financial calculator where we set the calculator to begin mode or the FV function in Excel where we set Type equal to one.
Additionally, FVA sub due equals FVA sub ordinary times the quantity one plus I.
Slide 12 Present Value of Ordinary Annuities and Annuities Due To calculate the present value of an annuity, with PVA sub N we can use the step by step approach, the formula approach, a financial calculator, or the spreadsheet method. Let's look at the present value of an ordinary annuity. The PV of an ordinary annuity can be written as PVA sub N equals PMT times the quantity one divided by I minus I divided by I times the quantity one plus I raised to the Nth power. Additionally, we can use a financial calculator or the PV function in Excel to solve this problem.
If instead we want to calculate PVA sub due we can use the following formula:
PVA sub due equals PVA sub ordinary times one plus I. We use this formula because each payment occurs one period earlier.
Slide 13 Finding Annuity Payments, Periods, and Interest Rates Assume we need ten thousand dollars in five years. If we earn six percent interest per year on our money. How much must we deposit to earn this amount? In other words, we need to calculate PMT. We know that FV equals ten thousand dollars, PV equals zero, N equals five and I equals six percent. We can use either a financial calculator or the PMT function in Excel to solve this problem. In the case of an ordinary annuity we would need to deposit seventeen hundred seventy three dollars and ninety-six cents per year. In the case of an annuity due we would need to deposit sixteen hundred seventy-three dollars and fifty-five cents at the beginning of each year.
Continuing with our example, assume we need ten thousand dollars and decide to make end of year deposits but can only deposit twelve hundred dollars per year. Assuming we earn six percent per year how long would it take to accumulate ten thousand dollars?
In this case it is not advisable to use the step by step approach since it would require a trial and error procedure to determine N or I for that matter. Hence, we should use either a financial calculator or the NPER function in Excel. It turns out that N equals six point nine six years. If instead we make deposits at the beginning of each time period and equals six point six three years.
Now assume we save twelve hundred dollars annually but need ten thousand dollars in five years. We need to calculate the rate of return we have to earn in order to achieve our goal. In this case we should use either a financial calculator or the RATE function in Excel. It turns out we would have to earn twenty-five point seventy-eight percent on our deposits to accumulate ten thousand dollars by the end of five years!
Slide 14 Perpetuities A perpetuity is a bond that promises to pay interest forever. Sometimes perpetuities are called consols. To find the PV of a perpetuity we use the following formula: PV for a perpetuity equals PMT divided by I. Assume a consol or perpetuity pays twenty-five dollars per year and the going interest rate is two point five percent. In this case the original value or present value of the consol is given by twenty-five dollars divided by zero point zero two five which equals one thousand dollars. What happens to the original value of the console if the interest rate increases to five point two percent?
Now the PV of the perpetuity equals twenty-five dollars divided by zero point zero five two which equals four hundred eighty dollars and seventy-seven cents. If instead, the interest rate drops to two percent the present value of the consol is twenty-five dollars divided by zero point zero two which equals one thousand two hundred fifty dollars. These examples illustrate a very important point about the relationship between bonds and interest rates. Specifically there is an inverse relationship between the price of outstanding bonds and interest rates. Hence if interest rates rise, the price of outstanding bonds decline and if interest rates decline the price of outstanding bonds increases. This rule holds true for both consols and bonds with finite maturities.
Slide 15 Uneven, or Irregular, Cash Flows Recall the definition of an annuity requires that the payments are identical over a given number periods. Many times financial decisions involve uneven or irregular cash flows. When we work with uneven or irregular cash flows we label them CF sub t where t denotes the period in which the cash flow occurs. There are two types of uneven cash flows that are important in finance. The first is one in which the cash flows stream is composed of a series of annuity payments plus a lump sum paid in year N. A bond is an example of this type of uneven cash flow. The second is one in which all the cash flows are uneven. Stocks and capital investments are examples of this type of uneven cash flow.
To solve problems in which we have an annuity payment plus a lump sum we use the following formula:
PV equals summation t equal one to T CF sub T divided by the quantity one plus I raised to the tthpower.
Solving problem like these is a two-step process. First we calculate the present value of the annuity. Then, we calculate the present value of the final payment. Last we add these numbers together to find a present value of the income stream. To calculate these values we can use a financial calculator or the PV function in Excel.
In cases where the cash flows are all uneven, we can use a step by step approach, a financial calculator, or the NPV function in Excel. If we use a financial calculator we must remember that the cash flows must be entered into the cash flow register in order to solve the problem.
Slide 16 Future Value of an Uneven Cash Flow Stream Now let's look at how to calculate the future value of stream of uneven cash flows. Sometimes this value is referred to as the terminal or horizon value. We calculate it by compounding each payment to the end of the term and then adding them together. The mathematical equation we use to calculate the future value has the form FV equals summation from t equals zero to N CF sub t times the quantity one plus I raised to the N minus t power.
Alternatively, we can use a financial calculator or Excel.
If we use Excel, calculating the FV is a two-step process. First we use the NPV function to calculate NPV. Second, we use the FV function to compound the NPV to obtain the future value.
Slide 17 Solving for I with Irregular Cash Flows Now let's look at how to determine I if we know the values of the other inputs. If we have an annuity plus a lump sum it's easy to determine I. However it is considerably more difficult to determine I if we have irregular or uneven cash flows. When all cash flows are irregular or uneven we use a financial calculator the internal rate of return, or IRR function in Excel to solve this problem. Using a financial calculator requires that we enter the cash flows into the cash flow register and press the IRR key to obtain the value for I. This is also called the rate of return on the investment. Additionally it is important to remember that the initial investment at t equals zero must be entered as a negative number since it is a cash outflow.
Slide 18 Semiannual and Other Compounding Periods Up to this point we assumed that interest has compounded annually. This is referred to as annual compounding. Assume we deposit one hundred thousand dollars into a bank account.
The interest paid on the deposit is six percent but it is paid every six months. This is referred to as semiannual compounding. If we leave the funds in the account how much will we have at the end of year one?
Since the bank pays six percent interest we receive sixty dollars at the end of one year. We receive thirty dollars at the end of six months and another thirty dollars at the end of the year. With semiannual compounding we earn interest on the first thirty dollars during the second six month period. For this reason, the total amount of interest earned is more than sixty dollars. Interest can also be paid quarterly, monthly, weekly or daily. It is very important to understand nonannual compounding because many financial instruments pay or charge interest on a nonannual basis.
If interest is not compounded on an annual basis we must deal with four types of interest rates, namely, nominal annual rates, I sub NOM, annual percentage rates, APR, periodic rates, I sub per, and effective annual rates, EAR or EFF percent. The nominal or quoted rate, I sub NOM is the rate quoted by bankers, brokers, and other financial institutions. Additionally, when the nominal rate is quoted it must include the number of confounding periods per year. The nominal rate is never shown on a timeline, nor is it entered into a financial calculator unless compounding occurs only once per year.
The periodic rate, I sub PER is the rate charged by a lender or paid by a borrower each period. We calculate the periodic rate using the formula I sub PER equals I sub NOM divided by M where I sub NOM is the nominal annual rate and M is the number of compounding periods per year. Hence, a six percent nominal rate with semiannual payments yields a periodic rate of I sub PER equals zero point zero six divided by two which equals zero point zero three. The periodic rate is the rate shown on timelines and used in calculations.
The effective annual rate EAR or EFF percent is the annual rate that yields the same result as compounding at the periodic rate for M times per year. This rate is determined using the following equation: EAR equals EFF percent equals the quantity one plus I sub NOM divided by M raised to the M power minus one where I sub NOM divided by M is the periodic rate and M is the number of periods per year. The EFF percent is used to compare the effect of costs on loans or rates of return on investments when the payment periods are different. They're rarely used in calculations.
Slide 19 Fractional Time Periods So far we've assumed that payments occur either at the beginning or at the end of the time periods but not within the time periods.
Solving these types of problems is three-step process.
First, we calculate the periodic rate which yields the interest rate paid per day.
Second, we calculate the number of days the money will be invested.
Last, we calculate the final value.
Slide 20 Amortized Loans A very important application of compound interest is in the case of installment loans which are paid overtime. These loans are repaid in equal amounts on a monthly quarterly or annual basis and are referred to as amortized loans. Problems like these require that we determine PMT and we solve them by using either a financial calculator or the PMT function in Excel. Each payment is broken into two parts, that part which is interest and the second part which is a repayment of principal. This breakdown is typically shown in an amortization schedule.
Slide 21 Growing Annuities A growing annuity is a series of payments that grows a constant rate. One example of a growing annuity is a situation, in which an individual wants to determine the maximum constant real or inflation-adjusted withdrawals he or she can make over a given number of years. There are two ways in which to solve this problem. First we can set up a spreadsheet in Excel's Goal Seek function which is found under the What If tab in the program. Second, we can use a financial calculator. If we use a financial calculator we must first calculate the expected real rate of interest. Using the real rate of interest we can solve an annuity due problem. There's a third method which we could use to solve this problem however it is very complicated and time consuming to use. The preferred method is either Excel or a financial calculator.
Suppose instead we want to accumulate a certain sum over given time period. We plan to make a deposit at time zero and then made nine more payments at the beginning of each of the next nine years. If we know the interest rate earned on the deposit and the expected inflation rate, we can calculate the real rate of interest and the amount of the initial deposit. In this case it is easier to use a financial calculator to solve the problem. The key is to remember that all variables must be expressed in real not nominal terms.
Slide 22 Check Your Understanding
Slide 23 Summary We have now reached the end of this lesson. Let's review what we've covered.
First, we identified that the time value of money is an extremely important concept in the field of financed. This principle was demonstrated in the concept of timelines.
Next, we continued with time value of money through an discussion on future values. Because of inflation, a dollar today is worth more than a dollar in the future. This lead to presenting four methods we can use to solve time value of money problems.
Then, we defined present value as the opposite of future value. To demonstrate we determined the possible amount on a bon by calculating the present value and allowing for discounting the future sum. This followed with identifying the interest rate and number of years and how they affect the future value of money.
Also, we defined an annuity as a series of equal payments made at fixed intervals. If the payments are made at the end of each period the annuity is called an ordinary annuity or deferred annuity.
Next, we discussed the future value and present value of ordinary and an annuity due. As an example, the future value of an annuity due is larger than that of an ordinary annuity because in the case of an annuity due payments are made at the beginning of each time period and for this reason each payments occurs one period earlier and therefore the payment earns interest for one additional period.
Then, we defined perpetuities as a bond that promises to pay interest forever. At times, perpetuities are called consols. We examined some examples that illustrate a very important point about the relationship between bonds and interest rates.
Also, we covered uneven or irregular cash flows. Many times financial decisions involve uneven or irregular cash flows. Solving problems related to cash flows we utilize several possible formulas. This included calculations for future value and solving for I with irregular cash flows.
Next, we learned about semiannual and other compounding periods. This followed with solving problems in fractional time periods.
Finally, we discussed amortized loans and growing annuities. Amortized loans are a very important application of compound interest is in the case of installment loans which are paid overtime. These loans are repaid in equal amounts on a monthly quarterly or annual basis. Growing annuities are a series of payments that grows a constant rate. Excel provides a Goal Seek function as way to work through problems related to growing annuities.
• Part 2: Bonds, Bond Valuation, and Interest Rates
Slide 1 Introduction Welcome to Financial Management. In this lesson we will discuss the bonds, bond valuation, and interest rates.
Slide 2 Topics The following topics will be covered in this lesson:
Who issues bonds;
Key characteristics of bonds;
Bond valuation;
Changes in bond values over time;
Bonds with semiannual coupons;
Bond yields;
The pre-tax cost of debt: determinants of market interest rates;
The real risk-free rate of interest;
The inflation premium;
The nominal, or quoted, risk-free rate of interest;
The default risk premium;
The liquidity premium;
The maturity risk premium;
The term structure of interest rates;
Financing with junk bonds; and
Bankruptcy and reorganization
Slide 3 Who issues bonds By definition a bond is a long-term contract in which a borrower makes payments of interest and principal on specific dates to bondholders. In general, there are four types of bonds. Treasury bonds or government bonds are issued by the U.S. government and they have almost no default risk. Bonds are also issued by Federal agencies like Fannie Mae and Freddie Mac. Federal agencies and government sponsored entities, or GSE, like the Tennessee Valley authority and the small business administration issue bonds and their debt is referred to as GSE debt. Federal agency debt and GSE debt are not backed by the full faith and credit of the U. S. government.
Corporations issue corporate bonds which are unlike Treasury bonds because they are exposed to default risk which is sometimes refer to as credit risk. If the issuing company were to have financial problems they may not be able to pay the interest and principal on the bonds. Depending upon the issuing company's characteristics and terms of the specific bond, different corporates bonds have different levels of default risk.
Municipal bonds or munis are issued by state and local governments. While munis are subject to default risk, they have an advantage over corporate bonds because the interest earned on municipal bonds is exempt from Federal taxation and from state taxes if the bondholder is a resident of the issuing state.
Foreign bonds are issued by foreign governments and foreign corporations. These types of bonds are subject to default risk if the bond is denominated in a currency other than that of the investor's home currency.
Slide 4 Key characteristics of bonds All bonds have several key characteristics. A bond's par value is the face value of the bond which is usually one thousand dollars. Bonds issued by a company usually require the company to pay a fixed number of dollars of interest usually every six months. This is called the coupon payment and when divided by the par value yields a coupon interest rate. Sometimes a bond coupon rate varies over time. When this happens the bonds are referred to as floating-rate bonds. In the case of a floating rate bond the coupon rate is set for about a six month. Subsequently, the coupon rate is adjusted every six months based upon some market rate which could be the U.S. Treasury bond rate, the London Interbank Offered Rate or another rate.
Bonds that don't pay a coupon are called zero coupon bonds. They are issued at a substantial discount from par value and offer the investor capital appreciation. Usually zero coupon bonds are issued in the form of Treasury bonds. Any bond originally offered at a price substantially below its par value is referred to as an original issue discount or OID bond.
Payment-in-kind bonds or PIK bonds don't pay cash coupons. Instead their coupons consist of additional bonds or a percentage of an additional bond. Typically, PIK bonds are issued by companies with cash flow problems and for this reason PIK bonds are usually risky investments.
If a bond includes a step-up provision it means that if a firm's bond rating is downgraded, the firm must increase the bond's coupon rate. From the company's viewpoint this can be dangerous because the downgrade means that the firm is having difficulty servicing its debt.
The maturity date of the bond is the specific date on which the par value must be paid. Many times, especially in the case of corporate bonds, there is a call provision attached to the bond issue. This gives the company the right to pay the bondholders an amount greater than the par value if the bonds are called. The additional amount paid to the bondholders is called a call premium. If the bonds are not callable for five to ten years this is referred to as a deferred call and the bonds are said to have call protection. Sometimes companies issue bonds during periods of high interest rates. When this happens there is usually a call provision attached to the bond issue. Should interest rates drop, the firm can sell a new issue at a lower yield and use the proceeds to retire the high rate issue and therefore reduce its interest expense. This is referred to as a refunding process.
Bonds are subject to event risk. Event risk occurs when something happens to change the firm's credit risk which lowers the firm's bond rating and the value of its outstanding bonds. Therefore, firms perceived by investors to be subject to event risk must pay very high interest rates to their bondholders. To reduce the interest rate, the firm can include a covenant, called a super poison put, which allows bondholders to turn in their bonds to the company at par in the event of a takeover, merger or major recapitalization.
Sometimes a bond issue will include a make-whole call provision. This allows a company to call the bond issue but the company must pay a call price that equals the market value of a similar noncallable bond. A make-whole call provision gives the company an easy method with which to repurchase bonds as part of a financial restructuring.
When a bond issue includes a sinking fund it enables the orderly retirement of the bond issue. The firm can administer the sinking fund in one of two ways. The firm can call in for redemption at par value a percentage of the bond issue each year or the firm can purchase the required number of bonds in the open market. The firm will choose the least cost method to administer the sinking fund. Unlike a refunding call, a sinking fund does not require a call provision.
Convertible bondholders have the option to convert their bond holdings into a fixed number of shares of common stock. Convertibles typically have a lower coupon rate than comparable nonconvertible bonds because convertible bonds give the investor the chance to share in the upside if a company does well.
Warrants give the holder the option to purchase stock at a fixed price. Therefore, if the stock price increases the holder gains. Like convertible bonds, warrants are issued with lower coupon rates.
Income bonds are riskier than regular bonds because they are required to pay interest only if the firm's earnings are high enough to cover the interest expense. Additionally bondholders do not have the right to force the firm into bankruptcy if the interest is not paid.
When interest payments and principal payments of bonds increase with inflation they are referred to as indexed bonds or purchasing power bonds. In January nineteen ninety-seven the U.S. Treasury began issuing indexed bonds called Treasury inflation protected securities or TIPS. TIPS can be used to approximate the risk-free rate of interest.
Corporate bonds are typically traded in electronic and or telephone markets instead of organized exchanges. The market for them is small because they are owned and traded by a small number of very large financial institutions and investors.
Slide 5 Bond valuation The value of a financial asset is the present value of the cash flows the asset is expected to generate over time. In the case of a regular bond, the cash flows are composed of interest payments during the life of the bond and a lump sum payment at maturity.
To calculate the present value of any bond we can use one of the following the equation:
V sub B equals INT times the total quantity of one divided by r sub d minus one divided the quantity of r sub d times the quantity one plus r sub d plus M divided by the quantity one plus r sub d raised to the Nth powerwhere r sub d is the required rate of return or the market rate of interest for that type of bond;
N is the number of years before the bond matures;
INT are the dollars of interest paid each year which is equal to the coupon rate times the par value; and
M is the par value or maturity value of the bond usually one thousand dollars.
When a bond is first issued the coupon rate is set at the going rate. However, after the bond is issued the coupon rate remains fixed but market interest rates fluctuate. Hence, when r sub d increases the price of an outstanding bond falls and when r sub d decreases the price of an outstanding bond increases. Whenever r sub d increases above the coupon rate a fixed rate bond's price falls below which par value and it is referred as a discounted bond. If on the other hand, r sub d falls below the coupon rate a fixed rate bond's price is above par value and it is referred to as a premium bond.
Slide 6 Changes in bond values over time The market value of a bond changes over time. When a bond is first issue, the coupon is usually set at a rate that causes the market price of the bond too equal to its par value. At this time the bond is referred to as a new issue. A bond is usually classified as an outstanding issue or a seasoned issue about one month after the first issue. While a newly issued bond usually sells for close to par, the prices of seasoned issues vary substantially. Except for floating-rate bonds, coupon payments are fixed. Hence, when economic conditions change a ten percent coupon bond with a one hundred dollars coupon that sold at par when it was first issued sells for more or less than one thousand dollars subsequent to the first issue.
There are other yields or returns attached to bond issues. The first is the current yield which is defined as the annual coupon rate divided by the current market price. This is also referred to as the rate of return due to the interest payment.
The second yield is the capital gains yield and is defined as the profit or loss from the sale of the capital asset for more or less than its purchase price.
The third yield is a total yield or bond yield and is determined by adding the current yield to the capital gains yield. If interest rates increase, the market value of the bond decreases below one thousand dollars and therefore the bond sells at a discount.
There is an important relationship between interest rates and the market value of bonds:
First, when r sub d equals the coupon rate a fixed rate bond sells at par;
Second, when interest rates change over time the coupon rate remains fixed. If r sub d rises above the coupon rate a fixed rate bond sells at a discount because its price falls below par value;
Third, when r sub d falls below the coupon rate a fixed rate bonds sells at a premium because its price rises above par value;
Fourth, an increase in interest rates results in the prices of outstanding bonds falling and a decrease interest rates results in the prices of outstanding bonds increasing; and
Fifth the market value of a bond always approaches par value as its maturity date approaches assuming the firm remains solvent.
Slide 7 Bonds with semiannual coupons Most bonds pay interest semiannually. In order to evaluate bonds with semiannual payments we must modify our valuation model. To do this, first divide the annual coupon interest by two. Doing this expresses the interest payments on a semiannual basis. Second multiply the years to maturity, N, by two to express it on a semiannual basis. Third divide the nominal interest rate r sub d by two to express it on a semiannual basis. Making these changes to the valuation model gives us a mathematical equation that has the following form:
V sub B equals summation t equals one through two times N INT divided by two divided by the quantity one plus r sub d divided by two raised to the t power plus M divided by the quantity one plus r sub d divided by two raised to the two times N power.
Under semiannual compounding the bond's value is somewhat higher because interest payments are received sooner.
Slide 8 Bond yields While a bond's coupon interest rate remains fixed its yield varies on a daily basis depending on market conditions. All bonds have three yields attached to them. The first is the yield to maturity or YTM. If we hold the bond to maturity what is rate of return on our investment? This rate is called the YTM and we calculate the YTM by solving the following equation for r sub d by using either a financial calculator or the RATE function in Excel:
Bond price equals summation t equals one through N INT divided by the quantity one plus YTM raised to the tth power plus M divided by the quantity one plus YTM raise to the Nth power.
The YTM is the bond's promised rate of return and is the return the investor earns provided all promised payments are made. The YTM is the expected rate of return only if the probability of the firm's default to zero and there is no call feature attached to the bond.
The yield to call or YTC is the rate of interest on a bond if it is called. To calculate the YTC we solve the following the equation for r sub d using either a financial calculator:
Price of a callable bond equals summation t equals one through N INT divided by the quantity one plus r sub d raised to the tth power plus the call price divided by quantity one plus r sub d raised to the Nth power;
Where N is the number of years until the bond is called, r sub d is the YTC and the call price is the price the firm must pay to call the bond. Usually, the call price is set equal to the par value plus one year's interest.
The current yield of a bond equals the annual interest payment divided by the bond's current price. If we hold a bond with a ten percent coupon that currently sells for nine hundred eighty-five dollars the bond's current yield is equal to one hundred dollars divided by nine hundred eighty-five dollars which equals ten point one five percent. The current yield is not equal to the return investor should expect on the bond. Instead, it provides information about the cash income the bond generates in any given year. In general, the yield to maturity equals current yield plus capital gains yield.
Slide 9 Check your understanding
Slide 10 The pre-tax cost of debt: determinants of market interest rates The pretax cost of debt equals either the YTM or the YTC if it is likely the firm will call the bond. Additionally, the cost of debt impacts the firm's weighted average cost of capital, or WACC the cause from the company's perspective the cost of debt is due required return from the debtholders' perspective.
Since different debt securities have different market rates, it follows that the nominal rate of interest on a debt security, r sub d decomposes into several factors. The formula for:
R sub d equals r star plus IP plus DRP plus LP plus MRP equals r sub RF plus DRP plus LP plus MRP.
Let's look briefly at each of these variables.
Slide 11 The Real Risk-Free Rate of Interest, R* The risk free rate of interest, r star, equals the interest rate that exists on a riskless security if no inflation is expected. We think of r star as the rate on short term U.S. Treasury securities in an inflation free world. However, this rate is not static it changes over time with economic conditions that depend on the rate of return corporations and other borrowers expect to earn on productive assets and people's time preferences for current vs. future consumption.
Slide 12 The Inflation Premium (IP) Since every investor is aware of the impact inflation has on interest rates, when they lend money they add an inflation premium or IP to r star. The IP is equal to the average expected inflation rate over the life of the security. In the case of a short term default free U.S. Treasury bill the actual interest rate charged:
R sub T-bill, equals the real risk - free rate, r star plus the inflation premium so that r sub T-bill equals r sub RF which equals r star plus IP.
Slide 13 The Nominal, or Quoted, Risk-free Rate of Interest The nominal, or quoted, risk free rate, r sub RF equals the real risk, free rate plus a premium for expected inflation. It is important to understand that there is no security that is truly risk free. Therefore, we use a proxy for the risk-free rate. We use the T-bill rate to approximate the short-term risk free rate. To approximate the long-term risk-free rate we use the T bond rate. Hence, we can express the risk-free rate :
R sub d equals r sub RF plus DRP plus LP plus MRP where r sub d equals r star plus IP.
Slide 14 The Default Risk Premium (DRP) The default risk premium, DRP, represents the possibility that the bond issuer will not pay the interest or principal and in the stated an amount and at the stated time. The greater the perceived risk of default on the part of the firm, the greater the DRP and the higher the bond's yield to maturity. For U.S. Treasuries the DRP is virtually zero. Default risk is affected both by the financial strength of the issuer and the terms of the bond contract. Let's look at several types of contract provisions.
A bond indenture is a legal document that details the rights of the bondholders and the issuing corporation. It includes the provision for a trustee who represents the bondholders and ensures that the terms of the indenture are satisfied. An indenture includes restrictive covenants that covers the conditions under which the issuer may pay off the bonds before maturity, the levels at which certain ratios must be maintained, and restrictions against the payment of dividends unless earnings meet certain specifications.
If a firm issues a mortgage bond it must pledge assets as security. A mortgage bond may be a senior or first mortgage or a junior or second mortgage bond. If a firm issues a second or junior mortgage, it is paid only after the first mortgage bondholders are paid.
A debenture is an unsecured bond and hence has no lien against any specific asset of the firm as collateral for the obligation. For this reason, debenture holders are referred to as general creditors and their claims are protected by property not pledged to other obligations. If a debenture is a subordinate debenture its claim cannot be paid until all senior debt has been paid.
Development bonds or pollution control bonds are issued by state and local governments. Under certain circumstances, state and local governing agencies are permitted to sell tax - exempt bonds with the proceeds made available to firms for specific uses. The bonds are guaranteed by the firm that uses the funds and since they are tax-exempt, these types of bonds have a relatively low interest rate.
Many times municipalities purchase insurance to guarantee coupon and principal payments of their bonds. This, in turn, reduces the risk to the investors who are willing to accept a lower coupon rate because the bond issue is insured.
There are three major bond rating agencies. These are Moody's Investors Service, Standard and Poor's Corporation, and Fitch ratings. So long as a bond issue is rated BBB or better it is considered an investment grade bond but if the bond issue is rated below BBB it is considered a speculative or junk bond.
Bond ratings are based on quantitative and qualitative factors. First, financial ratios are important. Specifically, the return on assets, debt ratio, and the interest coverage ratio are very important in predicting financial distress. Second, the bond contract terms includes information regarding issues such as whether the bond is secured by specific assets, whether the bond is considered subordinate debt, and any sinking fund provisions. Some of the qualitative factors that should be considered are the sensitivity of the firm's earnings to the strength of the economy, the impact of inflation on the firm, whether the firm has or may have labor problems, and potential antitrust problems.
Bond ratings are important for three reasons. First, most bonds are purchased by large institutional investors and most are restricted to investment grade securities. Second, many times bond covenants include a provision that stipulates the coupon rate must be increased if the bond rating falls below a certain level. Last, a bond's rating is an indicator of default risk which influences the bond's yield since lower-rated bonds have higher yields.
Slide 15 The liquidity premium (LP) Recall, a liquid asset is one that can be converted to cash quickly and at fair market value. Financial assets are usually more liquid than real assets. For this reason investors include a liquidity premium or LP when the market rate of the security is set. Corporate bonds issued by small firms tended to be less liquid than those issued by large corporations and therefore have a higher liquidity premium.
Slide 16 The maturity risk premium (MRP) The maturity risk premium, known as MRP, affects all bonds including Treasury bonds and is the net effect of interest rate risk and reinvestment risk. Interest rate risk arises because bond prices decline when interest rates increase. The longer the maturity of the bond the greater the price change in response to a given change in the interest rate. The risk of a reduction in income because of a decrease in interest rates is called reinvestment risk. How does this happen? Assume a retiree as a bond portfolio and lives off the income it produces. The bonds, on average, have a coupon rate of ten percent and suppose interest rates dropped from ten percent to five percent. When short-term bonds mature they must be replaced with lower yielding bonds. To the extent that long-term bonds are callable these too must be replaced with the lower-yielding five percent bonds. In this way the retiree suffers a reduction in income.
Slide 17 The Term Structure of Interest Rates The relationship between long term and short term rates is called the term structure of interest rates. It is important to both corporate treasurers who must decide whether to borrow by issuing short-term or long-term securities and investors who must decide whether to invest short-term or long-term. We can obtain interest rates for bonds with different maturities from sources like The Wall Street Journal and Bloomberg. When we plot a graph with interest rates on the Y-axis and maturity dates on the X-axis, this set of data for a given date is called the yield curve for that date. An example of the yield curve for U.S. Treasury bonds is show on this slide. A normal yield curve is upward sloping because historically longer term rates are usually higher than short term rates because of the maturity risk premium. The yield curve for March two thousand nine is an example of a normal yield curve. Any other shape for the yield curve is considered abnormal. Over time the yield curve can change. An inverted or downward sloping yield curve existed in March nineteen eighty and was downward sloping because the IP was larger for short term bonds than four long term bonds. In March two thousand ninethe yield curve was humped because medium term rates were higher than either short term or long term rates.
Slide 18 Financing with junk bonds Junk bonds are rated less then BBB and are considered noninvestment grade debt. A bond can become a junk bond in one of two ways. First, the bond may have been investment grade when it was first issued but its rating declined because the firm suffered financial difficulties. Second, some bonds have junk status when they are issued. In the nineteen eighties using junk bonds as part of leveraged buyouts was a popular method of financing the purchase of companies.
Slide 19 Bankruptcy and Reorganization A firm becomes insolvent when it does not have sufficient cash to meet its interest and principal payments. In this case it is necessary to decide whether to dissolve the company through either liquidation or chapter seven bankruptcy, or reorganization or chapter eleven bankruptcy. If the firm is reorganized its debt is usually restructured so that the firm's financial charges are reduced to a level that can be covered by the firm's cash flow. Liquidation occurs when the firm's financial situation is so dire it is better to sell off the assets. In this case there is a priority of claims stipulated by the Bankruptcy Act.
Slide 20 Check your understanding
Slide 21 Summary We have now reached the end of this lesson. Let's review what we've covered.
First, we learned about bonds, bond valuation, and interest rates. Bonds are issued by various government entities and corporations. At any point in time the value of a bond is given by the present value of the cash flows an asset backing the bond is expected to generate over time. While most bonds pay a coupon rate, zero coupon bonds and PIK bonds are exceptions.
Also, we learned that there is an inverse relationship between the market value of bonds and interest rates. Even though a bond's coupon rate is fixed, its yield varies with market conditions. Bond rating agencies use quantitative and qualitative factors to assign a rating to a bond issue. The higher the rating the lower the perceived risk associated with the bond issue. Because of perceived risk, investors attach premiums to the rate of return they require.
This concludes this lesson.
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