Bond Tutor

7/31/2019 Bond Tutor

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CHAPTER 1: The Time Value of Money

1.1Introductionould you lend someone $10,000 today in return for $10,000 in five years? Probably not. Infact, you should not, because $10,000 in five years is not worth $10,000 today. Moneyreceived at different times in the future will have different values today. For example,$10,000 in 20 years is probably worth less to you than $10,000 in 10 years. The principlebehind this is called the time value of money.So what is $10,000 in the future worth today? The answer depends on the interest rate. Thefollowing interactive calculator shows you what happens when the interest rate is 6%.

Time is on the X-axis. The red chart is $10,000 at times in the futures, the yellow chartshows you the value today. By placing your mouse over the dot on the top of the yellowchart, you can see that $10,000 in six years is worth $7,049 today. Similarly, $10,000 in 10years is worth just about half that amount today. You can change the numbers and click OK.For example, you can verify that if the interest rate is 8%, the $10,000 in five years is worth$6,805 today. Click the numeric button to see all the values.

Now try increasing the number of periods to 30, and click OK. This lets you see how thecurve changes with time to maturity. You will see the yellow chart take on a pronouncedconvex shape. This convexity results from discounting because the present value decreasesat an increasing rate with the time to maturity. You can also explore how the curve itself

changes with interest rates. Scroll the interest rate to see how the whole curve moves as youincrease or decrease interest rates. You can see how interest rates capture the time value ofmoney: the higher the interest rate, the lower the value today of money in the future.

The Time Value of Money is a central concept of finance because it affects everything thatinvolves borrowing or lending. This includes car loans, mortgages, student loans, bonds,savings accounts, and so on.

For example, if you want to borrow money, you can go to a financial institution such as a bank.If you satisfy the bank that you have the ability to repay the loan, plus interest, it will lend youthe money. This creates a fixed-income security, which is a contract specifying the timing and

amounts of cash flows over time. The "timing" is how often you make payments, and the"amount" is the dollar amount you pay each time. If you take out a car loan, you have the sametype of security: a fixed amount that has to be paid over a period of time by you to the bank.

For these fixed-income securities, you are the "seller" and the bank is the "buyer" (it ownsthe loan). That is, you sell a legal obligation to repay the loan plus interest, and in return, youreceive the amount of the loan. The interest rate is what the bank charges you for use of the


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money. It compensates the bank for not being able to lend the money to anyone else for theterm of your loan.Bonds are another large class of fixed-income securities. When an economic unit, such as theU.S. Treasury, borrows cash from the general investing public, it sells bonds. The public isnow the buyer that lends money to the Treasury. This fixed-income security specifies the

timing and amounts of cash payments by the U.S. Treasury (the seller) to the owner of thefixed-income security.Fixed-income securities are traded in primary and secondary markets. The primary market isthe market in which the security is first issued. For example, the U.S. Treasury initiallyauctions Treasury bills. Once issued in the primary market, securities are re-traded in thesecondary market. These markets determine a price for each fixed-income security. Since afixed-income security specifies a series of future payments (or cash flows), the price of thesecurity represents the value today of future cash flows.

1.2 PRESENT VALUE

A One-Period Example

uppose you have the opportunity to invest $100 for one year at 10% interest. At the endof the year, your investment is worth $110: the $100 you invested plus (0.1)(100)interest:

$110 = $100(1 + 0.10)

Here, $110 is called the future value of $100 invested at 10% today.

Now, what is the value today of $110 in one year? The answer is called the presentvalue of $110 in one year discounted at 10%. It is calculated by rearranging the futurevalue equation:

Next, ask the following question: If the present value of $110 in one year is $100, whatis the interest rate? The answer, of course, is 10%, again calculated by rearranging thefuture value equation:

To describe these valuation principles more generally, let FV= future value, PV=present value, and r= interest rate. Then, the relationship among these variables is:


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Therefore, if you know thefuture value (FV) and the interest rate (r), you can calculatethe present value (PV). Similarly, you can calculate the future value from the presentvalue and the interest rate, or the interest rate from the other two variables.

A Two-Period Example

Now, suppose an investment opportunity will pay you $121 at the end of two years.What is the value today (i.e., the present value) of this opportunity?

To work out the answer, we work backward by solving two one-period problems.

First, suppose you are at the end of Year 1. What is the value of $121 in one year? Asyou know, the answer depends on the interest rate between Year 1 and Year 2. If this is10%, then the value at the end of Year 1 is

Second, what is the value today of $110 in one year? You already know that the answeris $100:

Note that you have to discount the $121 twice, once for each period. If the interest rateis rper year, then the relationship between the future value and the present value is

You can easily deduce the general principle. If the interest rate is rper year, and youreceive FVin tyears, then the present value is

An example of a fixed-income security that has only one future payment is a zero-coupon bond. You will encounter these in much more detail throughout this text.

A Three-Period Example

In the two-period example, you started with the future value and worked back to thepresent value. You can also start with the present value and work forward.

Suppose you can invest $100 for three years at 10% per year. This means that eachyear, you will earn 10% on whatever you invest that year. You already know that your


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$100 will be worth $110 in one year. If you reinvest the $110 at the end of the year, thevalue at the end of Year 2 will be $110(1 + 0.10) = $121. Equivalently, this equals:

If you reinvest the $121, then at the end of Year 3 you will have:

The value of your investment over time is shown in Figure 1.1.

Figure 1.1 Value over Time

The relationship between present and future values in this example can be summarizedby noting that as an investor, you are indifferent among:

1. $100 now,

2. $110 in one years time,

3. $121 in two years time, or

4. $133.10 in three years time.

We state this as:

More generally, if you invest PVnow at interest rate rper year, then the value at the endof year tis

A Sequence of Cash Flows: A Two-Period Example


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So far, we have valued single payments in the future. We now examine a sequence offuture payments. Suppose there is an investment opportunity that pays $50 at the endof Year 1 and $100 at the end of Year 2. If the interest rate is 8%, what is the present

value of the future cash flows from the investment?

The answer is obtained by discounting each part separately and then adding presentvalues, a principle known as value additivity:

The important point to remember is that money received at different times cannot beadded together; a dollar today is very different from a dollar to be received one yearlater. However, you can add present values together, as we have done. In fact, you can

add the value of dollars together whenever they are expressed with respect to anycommontime, present or future.

When the price of a fixed-income security is determined in the market, investors are inthis same position; they are valuing cash flows that occur at different times. From valueadditivity, the price of the security is the sum of the present values of the future cashflows.

Given the price at which investors buy or sell a security, we can ask the question: Whatis the interest rate that investors are implicitly using to discount cash flows?

In our current example, suppose that the market price is $138.32. You can calculate theimplied interest rate, r, from the following:

You can verify that r= 0.05, which is the rate that equates the present value to the sumof the discounted future cash flows.

In the next topic, we apply time value of money concepts to value a particular type offixed-income security called a bond.


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1.2.1 EXAMPLE: ANNUITY

Lotto Winnings: Lump Sum or an Annuity

n March 2007, the Mega Millions lotteryhad a $390 million jackpot, the largest in the world atthe time. The lottery winners had the choice of taking the payments over time (an AnnualPayout) or taking a single lump sum payment (the Cash Option). The annual payout was $15million every year for 26 years. The Cash Option was to receive an immediate payment of $233million.

If you were the winner, what would you do, take the Annual Payout or the Cash Option?

Answering this question requires understanding the time value of money and taxes. Inthis topic we will ignore taxes and instead focus on the time value of money as capturedby interest rates and the annuities. An annuity is a series of fixed payments over time, such asthe Annual Payout from a lottery.

There are two types of annuities: ordinary annuities and annuities in-advance. In an ordinaryannuity, you receive the payments at the end of the period, e.g. at the end of the year. With anannuity in-advance, you receive the payment at the beginning of the period. Lottery payouts areannuities in-advance; if you choose the Annual Payout option, you get the first payment rightaway, the next payment one year later, and so on.

The present value of the Annual Payout is given by the annuity formula (a derivation is in theappendix to this chapter). The formula says if you receive a constant payment C for t periodsand the interest rate is r, then the present value P(C,t), of an ordinary annuity is

The present value of an annuity-in-advance is C+P(C,t-1) since you get C immediately and then

for t-1 more periods.
http://megamillions.com/faqs/#16http://megamillions.com/faqs/#16http://megamillions.com/faqs/#16http://megamillions.com/faqs/#16


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Refer now to the interactive calculator below and assume that the interest rate is 6%. Now youcan compute the present value of $15m for 26 years, for an in-advance annuity.

Online, using the calculator above you can check how changing the interest rate from6% to a rate that is lower and then higher than 6% changes the present value of theprize.

1.3 BONDS

In the fixed-income security markets of the world, prices (or values) are beingdetermined constantly. The largest and most liquid security market in the world is themarket for U.S. Treasury securities. In this market, traders around the world trade

Treasury obligations to pay cash over many different times -- up to 30 years into thefuture. In particular, simple Treasury obligations called strips (or zero-coupon bonds)are traded for a wide range of different maturities. These maturities can range from afew days to 30 years. If you buy a Treasury strip, you receive a fixed payment (the facevalue) at a fixed date in the future (the maturity date). As you will see, zero-couponbonds such as strips are fundamentally important because they allow us to deduce thevalues of many other types of fixed-income securities.

A common feature of any fixed-income security is that it can be represented on atimeline. This is a graphing concept illustrated by Figure 1.2. This timeline displays boththe timing and the amount of cash flows that the buyer of a fixed-income security will

receive.

For example, often a fixed-income security is defined in terms of some face value plusthe promised interest rate, or coupon rate, that is paid for the amount of the facevalue. In this case, the security is called a "coupon bond," and its cash flows are definedby the coupon rate times the face value over time plus the face value at the end of itslife.

Consider a 30-year bond with a face value equal to $10,000. Suppose that it payssimple interest on the face value two times a year (i.e., semiannually) at a couponinterest rate equal to 10% per year. Because interest is paid semiannually, we divide

10% per year by two to express it relative to six months. This means that every sixmonths, the bond pays $500 = $10,000(0.10/2) in interest payments. Finally, at the timeof maturity, this bond pays the $10,000 face value in addition to the final interestpayment. The cash flows from this security can be represented on a timeline as shownin Figure 1.2.

Figure 1.2 Coupon Bond


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This 30-year example is easily solved using the Time Value of Money subject in BondTutor. Suppose the market rate of interest equals 8% per year. Online, you enter thisproblem as follows:

Graphically, both the present value at the end of every six months and the nominal totaldollar value are graphed as follows:

The top bar chart represents the total cash flows accruing from the current couponperiod until the time to maturity. At Coupon Period 1, the cumulative, undiscounted cashflows are $40,000; at Coupon Period 2, these cash flows are $39,500 and so on.

Coupon Period 1 is 60 times $500, plus $10,000. This equals $40,000. Online, theremaining undiscounted and present values are provided by clicking on the Numericbutton at the bottom of the Bond Tutor screen.

You should click on Numeric in the above live screen and the popup grid withundiscounted and discounted values will appear.

The bottom bar chart (yellow in default colors) graphs the present value at each couponperiod. This value declines from above the face value. The reason for this is that thepromised or coupon rate is greater than the market's required rate of return. Therefore,

the market will bid the price up above the face value, which is equivalent to lowering theexpected return from the bond.

Now consider this same example using a market rate of interest equal to 12% per year.The market rate of interest exceeds the coupon rate for the bond. This implies thatinvestors want a higher return than is provided by coupon payments alone. As a result,the market will let the price fall below the face value, to increase the return from thisissue.

Online, you can repeat this example by entering 12% into the Interest Rate cell asfollows:


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The nominal values remain the same because the cash flows specified by the Bondindenture have not changed. What haschanged, however, is the discount rate. As aresult, the present value declines relative to the 12% market interest rate example. Thefollowing graph shows both the nominal and the present values over time.

Because the coupon rate is less than the market's required rate of interest, the presentvalue of the bond increases over time from below $10,000, the face amount. Click onthe Numeric button to see the undiscounted and discounted values per coupon period.

Cash Flows and Present Values

In the current example, the cash flows accruing to the bondholder can be viewed onlineby clicking on the Cash Flows option:

Graphically, this is displayed as follows:

Click on the Numeric button above to verify that the discounted and undiscounted cashflows for each Coupon Period.

As you can see, the coupon bond is made up of two types of cash flows. The first is astream of coupon payments (or cash flows) of equal size which occurs at the end ofeach time period. This type of cash flow is known as an ordinary annuity or an annuityin-arrears. The second component is a one-time cash flow, equal to the face value,which occurs at the end of the bonds life.

The cash flows from most fixed-income securities can be described by these two typesof cash flows. For example, home mortgage agreements, car loans, and leases are


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often specified in terms of an equal series of cash flows (i.e., an annuity). If the firstcomponent of the cash flow series occurs at the beginning of the period (rather than theend of the period), then the annuity is called an annuity in-advance.

Other types of fixed-income securities are defined only in terms of the second

component of a coupon bond, the face amount. This special case of a coupon bond iscalled a "pure discount" or "zero-coupon bond." This is a fixed-income security thatmakes no promised coupon payments over its life. Instead, its issuer is obliged to paythe face value of the bond at the time of its maturity. This is the most primitive form of afixed-income security, in the sense that all other types can be viewed as some bundle ofzero-coupon bonds that vary by face amount and maturity.

The U.S. Treasury issues a zero-coupon bond, called a Treasury bill. It is sold as a purediscount bond with face values ranging from $10,000 to $1,000,000. Suppose you buy aTreasury bill with a face value equal to $10,000 at a Treasury auction. Your time-line isrepresented in Figure 1.3.

Figure 1.3 Zero-Coupon Bond

Suppose, as in the previous example, that the time to maturity is 30 years and themarket interest rate is 8%. The difference now is that there are zero-coupon payments(i.e., Coupon Frequency and Coupon Rate are set to zero). Online, the undiscountedand discounted values for this example are computed as follows:

You can see that, over time, the value of the zero-coupon bond increases from belowthe face amount because the payment of the face amount is drawing closer. This isdisplayed as follows:


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Again click on the Numeric button to see the Undiscounted and Discounted Values percoupon per coupon payment period.

Observe that the undiscounted value remains the same over the zero-coupon bond's lifeand the present value increases.

Long-term zero-coupon bonds are very sensitive to shifts in interest rates. For example,at a market rate of interest equal to 8%, you can see that the present value at thebeginning of Coupon Period 1 equals $993.773. Suppose that interest rates jumped by75 basis points (where a basis point is 0.01 of 1%). The present value of the zerocoupon bond will decline to $807.462.

This can be computed online by entering the numbers as above and clicking on OK andthen Numeric to see the undiscounted and discounted values.

The initial outlay on the timeline of any fixed-income security is its price. In Topic 1.5,The Lotto Case we will discuss what an arbitrage-free price is. Then, in Chapter 2 we

discuss how it is determined. Before moving on, however, you may find it useful toreview the distinction between simple and compound interest in the topic titled TheCalculation of Interest.

1.4 THE CALCULATION OF INTEREST

nterest rates are quoted two ways: as simple interest and compound interest.

Simple Interest

Suppose you borrow $1,000 for 1 year, and the bank charges you simple interest of

10%. At the end of the year, you will have to repay $1,100.

If you borrow $1,000 for three years, your total repayment will be $1,000 + $100 x 3.Here, the simple interest is the principal ($1,000) times the interest rate (10%) times thenumber of periods (3). If you repay the loan in equal annual installments, you will makethree payments of $433.33.


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Given the principal P(the amount borrowed), the interest rate, r, and the term of theloan, T, simple interest on the loan is

If you repay the loan in ninstallments, your payment per installment is

Note that every period, the interest is calculated relative to the original amountborrowed.

Simple Interest Example

The simple interest on $10,000 for six months at 8% per year is calculated in the

following way. In one year, the simple interest is $10,000 x 0.08 = $800. As a result, in 6months, the simple interest is $10,000 x 0.08/2 = $400.

In some cases, simple interest is computed for some specified number of days.Suppose you want to find out the simple interest for 137 days. In this case, you need toknow the number of days in one year (or the "day count"). A bankers year typically has360 days. In some cases, the day count is 365 days or 364 days. Once the conventionis known, the computation of simple interest is straightforward. With a day count of 360,the simple interest for 137 days is:

Simple interest = $10,000 x 0.08 x (137/360)

Compound Interest

Suppose you borrow $1,000 for 1 year and the bank charges you interest of 10%. At theend of the year, you will have to pay $1,100. This is no different from simple interest.

The difference between simple and compound interest arises with multiple periods.Consider a two-year loan in which you only pay back the loan at the end of two years.Then, you would have to pay $1,000 (1 + 0.10)2,which is $1,210. What happens here isthat at the end of one year, your new principalis $1,100 = $1,000 (1 + 0.10). Theinterest for the second year is calculated on this new principal.

Table 1.1 summarizes the repayment if you make only one payment on the loan. As youcan see, you pay more if interest is compounded. This is because in Period 2, interest ischarged on both the original principal and on the Period 1 interest.

Simple Interest CompoundInterest


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Principal $1,000 $1,000

Year 2 Payment $1,200 $1,210

Table 1.1 Simple versus Compound Interest

Therefore, compounding works against you if you pay off a loan only at the end.Compounding works in your favor, however, if you make intermediate payments. Withsimple interest, you would pay $600 at the end of Year 1 and $600 at the end of Year 2.

With compound interest, the story is more complicated, and involves the concept ofamortization. Suppose you pay Cat the end of each year. The interest on Pis rP, sothe amount C - rPis applied toward reducing the principal. Therefore, your principalremaining after one payment is P1 = P - (C - rP).The second years interest is nowcalculated on P1 , and is rP1, which is less than rP.

Here, you are paying less interest in the second year than you would with simpleinterest if C> rP.

You may be asking, what is C? It is determined by applying the principle of discountingfuture cash flows.

For the bank, Ptoday must equal the present value of Cin one year and Cin two years:

For our example, P= $1000 and r= 0.10, so C= $576.19. You can see immediatelythat Cis less than the $600 you would pay with simple interest.

Let us check our calculations at the end of Period 1. At this point, we have a one-periodloan with principal amount P1 = P- (C - rP) = $523.81. At the end of Period 2, Cmustcover this principal plus the interest on this principal (i.e. it must be the case that C= P1(1 + r) = $523.81(1.01) = $576.19).

The process of recalculating the principal every time you make a payment is calledamortization.

The Compounding Frequency

With compound interest, you have to be careful to consider not only the quoted interestbut also the compounding frequency. Interest is usually quoted on an annualizedbasis with the compounding frequency stated separately (e.g. daily, monthly, quarterly,semi-annually, or annually).


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If the quoted interest rate is r, compounded ntimes a year, interest per period iscomputed using the rate r/n. Therefore, if you invested $1,000 for one year at 12%compounded monthly, at the end of the first month your investment would be worth$1,000 x (1 + 0.01). At the end of two months, it would be $1,000 x (1 + 0.01) 2 . At theend of the year, it would be worth $1,000 x (1 + 0.01)12 = $1,126.825.

Table 1.2 shows the effect of the compounding frequency on your investment.

Compounding Frequency Value

Daily $1127.48

Weekly $1127.34

Monthly $1126.82

Semi-Annually $1123.60

Annually $1120.00

Table 1.2 Effect of Compounding Frequency

As you can see, the greater the compounding frequency, the greater the final value,even though the quoted interest rate is the same. The moral of the story is: If you areinvesting money, then you want frequent compounding. If you are borrowing money,you want the frequency to be as small as possible.

The General Form

If interest is compounded ttimes per year, the investment lasts for nyears, then ingeneral, if Pis the principal, then at the end of nyears, the value is

In the limit, as tapproaches infinity, the compounding interval approaches zero. This iscalled continuous compounding. This limit is:

where eis the transcendental number = 2.718... .

A $1,000 investment continuously compounded at 12% leads to a value of $1,127.50 atthe end of one year, which is only slightly greater than the value with dailycompounding.


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For most bonds, interest is calculated using compound interest, rather than continuouslycompounded interest.

The Annuity Formula

Many financial transactions involve a constant stream of payments over time. Anordinary annuity is a sequence of equal cash flows occurring regularly, say, everymonth or every year. Coupon payments from a bond and mortgage payments on afixed-interest loan are examples of ordinary annuities.

The present value of an annuity is given by the annuity formula. This formula says thatif you receive Cevery period for tperiods, and the interest rate is r, then the presentvalue of the annuity, P, is

You can see the derivation of the annuity formula in the Appendix to this chapter.

Ordinary annuity tables are often constructed for the case where C = $1. These tablespresent the numbers pre-computed for this case in a table of interest rates r, by life t.

To value a t-period annuity in-advance, you add C and subtract one period from t inthe ordinary annuity formula. To see why, consider the following example.

In Pennsylvania, the major winner in a Lotto game receives the cash prize as an annuityin-advance. For marketing purposes, the prize is quoted in total nominal dollars (e.g.,

$21 million), but this prize is paid over 21 years. Quoting prizes in nominal dollarsclearly sounds more favorable than quoting prizes in present dollars.

Suppose you are lucky enough to win $21 million dollars paid at the rate of $1,000,000per year starting from the present. What is the present value of this prize (before tax)?

You can value your win by breaking the prize into two parts: $1,000,000 today and anordinary annuity of $1,000,000 for 20 years. The present value of the first part is$1,000,000, because $1 today is worth $1. If the market rate of interest is 8.15%, thenby applying the annuity formula, the present value of this ordinary annuity component isworth $9,709,515. Therefore, the present value of this annuity in-advance is the sum of

the two parts, which equals $10,709,515.

This is a lot less than the nominal dollar quote of $21,000,000, but you might argue that"If I won the lottery I wouldnt care about nominal versus present values!" After you workthrough The Lotto Casepresented in the next topic, however, you may want toreevaluate this argument.


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1.5 THE LOTTO CASE

The following story was reported by The Pittsburgh PressWednesday, June 14, 1989.

Willa T. shared in the Lotto jackpot on February 10, 1986. She won $454,389.63. At first

she found winning thrilling, but at her age, she found that "the thrill wears off when theprize comes in 21 installments."

Mrs. T. was entitled to receive $21,638.03, or $17,310.43 after federal withholding tax.After collecting four installments, Mrs. T. decided to sell her remaining 17 installmentsfor an immediate payment of $111,000.

Mrs. T. hired an attorney to sell the remaining installments. At a dinner party, theattorney ran into the financial consultant. The financial consultant was reported to havesaid that buying the annuity "fits right into my portfolio."

How do you think each of the parties fared in this transaction? The answer to thisdepends upon how you would value Mrs. T.'s lotto annuity.

We will now apply Bond Tutor to work through this example. First we make severalsimplifying assumptions:

Taxes: For incremental cash income less than $30,000 per year, let the assumed taxrate be 20%. For incremental cash income greater than $100,000 let the tax rate be35%.

Market Information: At the time of this story, the yield to maturity on a long-term

Government bond was 8.40%.

Annuity Assumption: Remaining annuity is an ordinary annuity from the present time ofJune 14, 1989.

Mrs. T.'s annuity of $21,638.03 for 17 years is virtually a riskless sequence of cashflows. The state of Pennsylvania has invested in a trust account an endowment to covercash payments. As a result, in a perfect market, with no taxes, transaction costs, orliquidity problems, Mrs. T. would receive the present value discounted at 8.40 usinginterest rates in 1989.

Online in the calculator below set up the problem as follows.


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The annuity value of the remaining 17 payments, measured in both nominal and presentvalues, declines over time as follows:

Numerically this equals where the undiscounted value in period 1 equals$21,638.03*17=$367846.51. Online click on the Numeric button to see theundiscounted and discounted values per year.

Thus, in a perfect market the present value is $192,216.20 when discounted at 8.40%compounded annually.

Unfortunately, Mrs. T. faces several types of market imperfections: taxes, no organizedmarket for Lotto payments, and legal fees.

Ignoring legal fees, Mrs. T. received $111,000 from the financial analyst. Now supposeMrs. T. pays 35% in tax. This leaves $72,150 -- and even less after legal fees.

As a result, Mrs. T. has lost a lot due to market imperfections.

With your acquired knowledge from Chapter 1 to date about the time value of money,can you improve on the reported result?

The attorney was quoted as saying that he first contacted several financial institutions.These institutions were not interested because the prize was not large enough.However, by framing the question in terms of the bank's regular business, the followingdeal could potentially be struck.

Deal I: A Simple Solution

Mrs. T. takes out a standard 5-year personal loan for $62,400. In return, Mrs. T. assigns(recall that the courts approved the sale) the after-tax lotto revenues ($17,310.42) to thebank for 5 years. Structuring the settlement this way requires no additional cash outlayby Mrs. T. and lets Mrs. T. keep the lotto annuity for Years 6 - 17. She could repeat thetransaction on February 10, 1994.

At the time of the article, the personal loan rate was 12%. The bank is strictly better offwith this loan because the cash flows are riskless, whereas personal loans are risky. Asa result, a better-than 12% rate should be negotiated by Mrs. T.


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If the bank booked a 6-year personal loan at 12%, then Mrs. T. could borrow$71,170.19 in the same way. This amount approximately equals her after-taxsettlement, but once again, Mrs. T. can repeat the transaction on February, 10, 1995!

In fact, Mrs. T. would have received $138,856.50 if the bank had booked a 17-year loan

at 10%. The bank is strictly better off in this case because the repayments have zerodefault risk and at the time of the article, the 17-year Government bond rate was 8.40%.Mrs. T. is clearly better off she receives $138,856.50 now and faces no future liabilities.That is, her loan repayments are covered by assigning the after-tax Lotto receipts to thebank.

In the solution reported by the newspaper, an arbitrage opportunity exists. That is, anopportunity exists for an investor with zero cash outlay to generate strictly positive cashflows.

Deal II: Highlighting a Pure Arbitrage Opportunity

Suppose some outside party had sufficient equity in his/her home or other real estate totake out either a first or second mortgage to cover the $111,000. At the reported settledamounts, a pure arbitrage opportunity exists by paying Mrs. T. $111,000 for the Lottoannuity.

To see why, suppose the third party buys the Lotto annuity from Mrs. T. and financesthis investment by borrowing the entire amount of $111,000 against some property.Assume that the mortgage loan is booked at a fixed rate equal to 10%.

Online, we can work out the arbitrage opportunity.

The first part of the problem is to compute the cash outflows from borrowing $111,000over a ten-year period at 10%. To simplify the problem, assume that all cash paymentsare annual.

Click on Amortization and Cash Flows as follows:


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Online enter the above numbers and click on OK. Then click on the Numeric button andyou will see for this mortgage loan the cash repayments as well as the part of theserepayments that service the interest expense:

You can open a spreadsheet and cut and paste these numbers from Bond Tutor in

order to complete this analysis.

First, online reset the calculator above as follows:

That is, enter the Lotto payments. Click on the Numeric button to get the Lotto unitypayments and then click on the Copy button.

Second, open your spreadsheet and paste in the raw Lotto numbers from above:


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Online change the inputs to the calculator to the parameters for Deal II (funding from aequity in the home mortgage at the time of this case). Select Amortization and CashFlows and then click on OK and Numeric as illustrated below.

Again from the popup grid containing the numbers for Constant Payment and InterestExpense. Click on Copy and then paste into cell D1 in the above spreadsheet.

Now the above cash flows can be matched to illustrate the arbitrage:

First, the interest expense on the mortgage loan is tax deductible and serves to reducethe Lotto receipts. In the spreadsheet below the taxable lotto payments are in column G(Column B - Column F) and the cash tax expense equals the numbers in column H.This is illustrated below:


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Finally we need to account for the cash Mortgage repayments which are in the abovespreadsheet in Column E. Thus the after tax cash flows can be calculated as follows inColumn I as Col B - Col E - Col H:


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The pure arbitrage opportunity is revealed from the realized net positive cash flows inevery year except Years 9 and 10 in Column G. The positive cash flows from earlieryears easily eliminate these two negative years.

This Lotto case illustrates that with zero cash outlay, a strictly positive return is earned.

This is an arbitrage opportunity. Clearly, the arbitrage-free value of the Lotto winningsis significantly higher than what Mrs. T. received.

In Chapter 2, we further develop the no-arbitrage arguments to the problem of valuingbonds.

Appendix 1A DERIVATION OF THE ANNUITY FORMULA

he present value of a constant cash flow, C, for tperiods is:

Define the discount factor:

Substituting ainto the formula, we get

To simplify the present value formula, we need to simplify the expression in thebrackets:

To simplify this formula, we first add at+1,at+2, and so on, and then subtract all the termswe added:

We can rewrite this as:

Note that the infinite number of terms in each of the brackets is the same. Define:


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+

Now, observe that V =1 + aV, which means that

Therefore, the expression

Now, replace awith the discount factor 1/(1 + r) and simplify to get:

We can now simplify the present value formula as follows:

Replacing the expression in square brackets with what we derived, we get:

which is the annuity formula.

Given the interest rate, r, this formula can be used to compute the present value of thefuture cash flows. Given the present value, it can be used to compute the interest rateor yield. Finally, given the present value and the interest rate, it can be used todetermine the cash flow.

You can use the formula in different ways as you go through Bond Tutor. The topicTime Value of Money in this chapter lets you calculate the present value of each cashflow for an annuity and also lets you see the annuity value through time. In The TermStructure of Interest Rates topic, you can see how the present value is affected by theinterest rate and also calculate the interest rate given the present value.

http://www.bondtutor.com/btchp1/topic7/topic7.htm

CHAPTER 2: Bond Prices and Interest Rates
http://www.bondtutor.com/btchp1/topic7/topic7.htmhttp://www.bondtutor.com/btchp1/topic7/topic7.htmhttp://www.bondtutor.com/btchp1/topic7/topic7.htm


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2.1 Introduction

any of the principles for valuing fixed-income securities depend on arbitrage arguments.For there to be no arbitrage, it must be the case that no investor can receive a positive(risk-free) return without investing any money. In this chapter, we will identify the

arbitrage-free value of bonds. The argument is first developed in Topic 2.2, Arbitrage-Free Prices: Single-Period Cases. In Topic 2.3, this is extended to Arbitrage-FreePrices: Multi-Period Case.

Generally, when fixed-income securities are traded in the marketplace, the buyer paysthe price in cash and receives a legal claim to the future cash flows from the security.However, it is possible to trade fixed-income securities where this is not the case. Thatis, no cash is exchanged at the time of the trade. In this type of trade, a forward contractis entered into where both parties are obligated to settle at some time in the future for aprice agreed upon now. Therefore, in the case of forward contracts, investors areconcerned not only with the arbitrage-free value of securities today, but also with the

arbitrage-free value of the position at some time in the future. In Topic 2.4, FutureValue, we consider develop this concept further.

2.2 Bond Prices: Single-Period Case

he present value of a bond can be derived, given interest rates, by applying the principleof discounting and assuming that no arbitrage opportunity exists. We argue here thatthis value must equal the price of the bond. Otherwise, there is an arbitrage opportunity.

Let us apply these arguments first to a zero-coupon, or pure-discount, bond. Supposethe bond pays its face value, F, in one period's time. Let the bond's price equal Pand

suppose that ris the one-period interest rate at which investors can borrow or lend inthe market. If there is no arbitrage, then the price of the bond must satisfy:

P(1+r) = F

or

To see why, suppose P(1 + r) < C. Now you should borrow Pat interest rate rand buy

the bond. In the next period, you will owe P(1 + r) but will get F > P(1 + r) and will havemade a pure arbitrage profit. Therefore, at least P(1 + r) > F. However, if P(1 + r) > F,you should sell the bond and invest the proceeds at interest rate r. In the next period,you will owe Fbut your investment will be worth P(1 + r) > F. This implies P(1 + r) < F.Taken together,


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Example 1: Determining the Arbitrage-Free Price of a Bond

If the face value of a zero-coupon bond equals $100, and the interest rate is 5%, thenthe arbitrage-free price of the bond is:

Alternatively, suppose you observe that the price of a one-period bond is P. Now, youcan use the formula to determine the interest rate implicit in the price. This interest rateis the return (or yield) you would get if you bought the bond today at price P and held itto maturity.

In one period, the price of the bond must equal its face value, F. To determine yourreturn, you have to ask: At what interest rate would I have to invest Pin order to get Finone period ?

The answer is the number y, such that P(1 + y) = F. If you rearrange this, the yield is

which simplifies to

Of course, because the one-period borrowing and lending rate is r, the yield must equalr, otherwise there is an arbitrage opportunity. If the yield is y > r, then this canimmediately be exploited by borrowing at rand buying the discount bond. If y < r, youshould sell the discount bond and lend the proceeds at rate r.

Example II: Determining the Arbitrage-Free Price of a Bond

If the face value of a bond equals $100 and the price is $95.24, then the yield from thisinvestment is approximately (rounding to two decimal places):

In both of the above examples, it is straightforward to construct the arbitrage-free, purediscount bond price given the one-year interest rate, r.

The next topic, Arbitrage-Free Prices: Multi-Period Case, extends the above analysis.

2.3 Bond Prices: Multi-Period Case


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he cash flows from most fixed-income securities extend beyond one period. Thevaluation of these cash flows uses two principles: that of discounting and that of valueadditivity. Value additivity says that you can simply add up the different discounted cash

flows. The discounting principles you have learned say that you must be careful to addtogether only present values at the same time.

With these principles in mind, it is easy to move to multiple periods and to multiple cashflows. One complication that arises is that different interest rates may be used todiscount cash flows at different periods. For example, you may be willing to lend moneyfor one year at 7%, but may require a higher interest rate, say, 8% per year, to lendmoney for two years. The extra return would compensate you for not being able to usethe money for that extra year.

The interest rate used to discount cash flows in period t, rt , is called the spot interest

rate for tperiods. We will quote interest rates relative to one period (which is generallyone year). That is, rtper period means that interest accrues (or is paid) at the rate of rt%per period for tperiods.

For example, consider a zero-coupon bond that pays its face value, F, in two years. Youcan value this bond given the two-year spot interest rate. The arbitrage-free price of thispure discount bond is the present value of the face amount Fdiscounted at the two-yearspot interest rate.

Example: Present Value using Spot Rates of Interest

The present value of $100 at the end of two years when two-year spot interest rates are6% per year is:

As you can see, the only differences between a cash flow in one period and a cash flowin two periods are (1) we use the two-period interest rate, and (2) we discount the cashflows "twice." If the cash flow occurs in tperiods, we use the t-period spot interest rateand discount the cash flow for tperiods.

With many periods, however, you can also receive intermediate payments. Forexample, a coupon bond makes a payment every period in addition to the face value,which is paid at maturity.


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These more complex cash flows can also be valued using what you have learned inChapter 1. First, you know how to discount each cash flow. Second, you know that youcan add up the discounted cash flows, as in the derivation of the derivation of theannuity formula (see appendix A, in chapter 1).

For example, consider the timeline in Figure 2.1 for a newly issued 30-year Treasurybond with a face value of $10,000 and a coupon rate equal to 10% compoundedsemiannually.

Figure 2.1 30-Year Coupon Bond

Here, we know the present value of each cash flow. Value additivity says that the valueof the bond must equal the sum of all the present values, so we also know the value ofthe bond.

There is another way to see why value additivity must hold, and this comes fromarbitrage. The cash flows from the 30-year Treasury bond are exactly the same as aportfolio of 59 zero-coupon bonds with face values equal to $500 plus one zero-couponbond with a face value equal to $10,500. The first zero-coupon bond has maturity equalto 6 months, the second 12 months, and so on.

Since it is possible to reconstruct the Treasury bond from the zero-coupon bonds, thevalue of the Treasury bond must equal the value of the collection (or portfolio) of zero-coupon bonds. If the Treasury bond had a higher price, you would sell it and buy all thenecessary zero-coupon bonds, giving you a pure arbitrage profit.

You can calculate the value of this bond using the procedure in Chapter 1, in the TimeValue of Money topic 1.2. You should be able to verify that if the interest rate is 7%, thevalue of the bond is $13,741.71.

Example: Coupon Bond

Suppose you want to value a 10-year U.S. Treasury bond with a face value of $10,000and a coupon rate equal to 6% payable semiannually. The bond is issued today.

With a face value of $10,000 and a coupon rate equal to 6% per year, the bond will pay$300 ( = 10,000 x 0.06 x 1/2 ) every six months for ten years and then pay the facevalue of $10,000 at the end of ten years. The cash flows are shown in the time-line inFigure 2.2.


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Figure 2.2 10-Year Coupon Bond

In other words, the holder of this bond receives 20 separate cash payments of $300each and one cash payment of $10,000 over the ten years. The arbitrage-free price ofeach coupon payment (denoted by C =$300for the tth coupon payment) is:

The arbitrage-free price of the face value is:

Value additivity (or lack of arbitrage) then says that the price of this bond is:

in the above each rt is the six-month interest rate.

The stream of coupon payments makes up an ordinary annuity, so the value of the bondequals the value of an ordinary annuity plus the present value of the face value.

However, you cannot apply the annuity formula directly because it assumes that all thespot interest rates are the same. In Chapter 3, (see Overview (Topic 3.1)), you will seehow to value bonds when interest rates for maturities differ.

For now, if we make the simplifying assumption that all interest rates are constant, we

can apply the annuity formula. You can use the software in Bond Tutor to calculate thevalue of any annuity.

Example: Present Value of a Coupon Bond

Assume for the bond in Figure 2.2 that each spot interest rate is 6%. What is the valueof this coupon bond?


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To compute this value you calculate the present value of the coupon payments and addto this number the present value of the bonds face value. The coupon payments forman ordinary annuity. Using Bond Tutor, you can calculate that the present value of $1 forten periods at 6% compounded twice per period is 14.8775:

The present value of all future coupon payments for this bond (to the nearest dollar) isthen:

$4,463 = $300 x 14.8775

The present value of $1 at the end of 20 periods at 3% per period is $0.5537, which iscomputed as follows:

Thus, the present value of the face amount is:

$5,537 = $10,000 x 0.5537

As a result, the value of the bond is $10,000. Note that this equals the face value. If abonds price equals its face value, it is said to be trading "at par." O therwise, it maytrade at a "discount from par" or at a "premium to par."

What happens if the market interest rate falls below or rises above 6%?

In the Bond Tutor subject two titled, Bond Values and Interest Rates, you can see whathappens in each of these cases. For example, suppose we want to see how the valueof the bond is altered when exposed to a range of interest rates from 1% to 10%, with astep size of 1%.

In the interactive calculator below, enter the following values:


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You want to look at a sensitivity analysis in step sizes of 1% (i.e., 10 steps between 1%to 10%). The exposure profile is the following:

Click on numeric button and verify the following bond values:

You can see that the bond value declines as interest rates increase. When the interestrate equals the coupon rate, the value equals the face amount of $10,000.

You can also see the principle of value additivity at work, by selecting to plot CashFlows

and viewing the components both numerically and graphically. For example, for the firstten coupon payments, the present value of each component is:


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To see the relative weighting of coupon versus face value, click on the Compositionbutton to view Period 0 (i.e., present time) :

The display appears as follows:

The numbers supporting the above graph for Time 0 are:

That is, the present value of the face value is $5,536.76 and the present value of thefuture coupon payments is $4,463.24.

The valuation principles developed thus far apply to any point in time. In the next topic,we calculate the future value of a fixed-income security.


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2.4 Future Value

any cases, investors are concerned not only with the value of securities today but alsowith the value of a position at some time in the future. For example, you may want toknow: If I invest $1,000 today at 7% for five years (compounded annually), what will myinvestment be worth after five years? This is the future value of your investment.

The future value of an investment also provides important "risk management"information. It lets you answer "what if" questions such as: Whatwill be the value of myinvestment six months from now ifspot interest rates increase?

To calculate future values, we continue with the assumption that all spot interest rates

are equal. In Chapter 3, you will see how to perform similar calculations with differentspot rates for each maturity.

Calculating a future value requires us to replace "discounting" with "compounding." Thatis, instead of discounting future values to the present, we compound present values tothe future. If P0is the arbitrage-free price relative to the current set of spot interestrates, r, then the future value at time tis:

Computation of this future value actually requires some cash flows to be discounted and

some to be compounded, but all aspects are always measured at a common time. Tosee this, consider Figure 2.3, where we want to determine the value of the bond at theend of Year 1.

Figure 2.3 Computation of a Future Value

If you want to compute the future value of the coupon bond as of the end of Year 1, youmust first compute the future value of the coupon payment at the end of six months, addthis to the coupon value as of the end of Year 1, and then add the present value of eachof the subsequent coupon payments and the face value. Of course, you neither discountnor compound cash flows occurring at the end of Year 1.


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As a result, in Figure 2.3 the future value at the end of Year 1 equals the sum of:

1. The future value of the $300 paid at the end of six months.

2. The $300 paid at the end of Year 1.

3. The present value of an ordinary annuity of $300 per period for 18 six-month periods.

4. The present value of the $10,000 face value paid at the end of Year 10 (18 periodsfrom the end of Year 1).

If the interest rate is 8% per year compounded semiannually, then you can verify thatthe future value at the end of Year 1 for this ten-year bond is:

$9,349.70 = ($300 x 1.04) + $300 + ($300 x 12.659) + (0.494 x $10,000)

You can use Bond Tutor to verify these figures and step through time using theinteractive calculator below. You can enter the current problem by changing theInterest Rate to 8% and scrolling the View Period to 2, as follows:

In this form, we have first chosen to consume (i.e., not to reinvest) the realized couponpayments. Equivalently, this is consistent with the perspective of investors in the marketat the end of Period 2 (immediately prior to the realization of the current couponpayment). These investors are trading claims to current and future cash flows only.

Online, selecting Composition lets you choose the period you want to view. For thecurrent example, this is one year's time (i.e., View Period 2).

Your display screen appears as follows:

At this point in time, the face value makes up 54.6% of the bonds value. Numerically,these figures are:


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Alternatively, you can repeat the exercise assuming that realized coupon payments arereinvested at the market interest rate. To do this, click on the option Reinvest Couponsin the above interactive calculator.

Numerically, this is:

By scrolling the view period, you can take a journey through bond value time. Forexample, scroll to after 16 periods, above

and verify that the bond value with reinvestment breaks down as follows:


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In Chapter 3, The Term Structure of Interest Rates, you will learn how to value fixed-income securities for the case that relaxes the assumption of a constant market interestrate over different time to maturities.

WEITERMACHEN chapter 3 - 5

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