Fixed-Income Securities and Interest Rates

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1 Chapter 2 Fixed-Incoe Securities and Interest Rates We now begin a systeatic study of fixed-incoe securities and interest rates. The literal definition of a fixed-incoe security is a financial instruent that proises fixed (or definite) payents at prescribed future dates. In soe cases only one payent is ade at aturity; in other cases, periodic payents are ade. Such a security is really a loan ade by the the purchaser of the secuirty to the issuer of the security. The purchase price of the security is the principal of the loan. In order for such a security to be desirable to investors, the total aount of the future payents to be received ust be greater than the purchase price, so that these payents reflect repayent of the original principal plus interest. The interest represents the price paid for the use of the borrowed oney. The payent aounts of the securities are typically described in ters of an interest rate. Roughly speaking, by a debt security we ean a financial security that is characterized by a principal aount (the aount borrowed), together with a plan for repayent of the principal plus interest. Soe debt securities have variable interest rates that depend on the future values of certain econoic variables and are unknown when the security is issued. In practice, there is soeties abiguity regarding what is or is not a fixedincoe security. For soe securities that are broadly classified as being fixed-incoe securities, the payents are not known with certainty at the tie that the security is issued either because the payents are tied to a variable interest rate that fluctuates or because the security has certain features of optionality built in. For our purposes, a reasonable working definition of fixed-incoe security is a security that is either a debt security (possible with a variable interest rate) or a derivative written on debt securities. In this chapter we focus ostly on situations where the payent aounts and payent dates are known with certainty at the tie that the security is issued. We refer to such securities as securities with fixed payents, deterinistic payents, or deterinistic cash flows. 37

2 2.1 Zero-Coupon Bonds and Discount Factors The ost basic type of fixed-incoe security is a zero-coupon bond. This type of security ake a single payent of aount F (called the face value) at a prescribed future date T (called the aturity date). Every security with deterinistic payents can be expressed as a portfolio of zero-coupon bonds. Most people have soe failiarity with interest rates because they are used in the descriptions of any coon financial transactions. One tricky point concerning interest rates is that they cannot be used to deterine payent aounts unless a copounding convention is specified. In other words, to copute payent aounts, one needs not only an interest rate, but also an additional convention (that is often hidden in the fine print ). There is a ore intrinsic concept, known as a discount factor, that does not rely on any additional conventions. The discount factor for tie T > 0, denoted D(T), is defined to be the tie-zero price of a zero-coupon bond having aturity T and face value F = 1. The no-arbitrage principle iplies that the tie-0 price of a zero-coupon bond with face value F and aturity is siply FD(T). Discount factors for various aturities can be inferred fro the prices of traded bonds. Various kinds of interest rates can then be coputed fro discount factors. 2.2 Basic Interest Rate Mechanics Suppose that at tie 0, an aount A > 0 is invested in zero-coupon bonds aturing at tie T > 0. The value of the investent at tie T will be A D(T) ; in other words, the capital will grow by a factor of 1/D(T). An interest rate is a way to describe the growth of capital in ters of a rate per unit of tie. Unless stated otherwise, we easure tie in years and interest rates will be expressed as annualized rates. There is ore than one convention for associating interest rates with discount factors. Suppose that soe oney is deposited in the bank at tie 0 and is left in the bank to be withdrawn at a future date T. At the withdrawl date the custoer will receive receive ore than the initial deposit. The difference between the final account value and the initial deposit is interest. The aount of interest to be paid will be based on the aount deposited, an interest rate, and a copounding convention. (The interest rate offered by the bank will generally be different for deposits of different lengths.) Unless otherwise stated, we assue that the interest rate for deposits is the sae as the rate for loans. Interest ay be either siple or copound. In order to describe the basic echanics of interest calculations, we shall assue that an initial aount A > 0 is invested at tie 0, that no additional investents (or withdrawals) are ade, and that the annual interest rate is r 0. We are interested in the value V T of the investent at the aturity date T. Analogous forulas apply to the case of a loan. (Indeed, a loan can be considered as an investent of an aount A < 0.) 38

3 Reark 2.1. Unless stated otherwise, all interest rates should be assued to be nonnegative. Under extree econoic conditions it can happen that interest rates becoe slightly negative. (This ay see strange because with a negative interest a deposit would lose oney, so it would be better to just hold on to the oney. However, it would not be wise to store $100,000,000 under your attress or even in your office safe.) It is quite rare for this to happen and if it does the rates generally are only negative by a few one hundredths of a percentage point, and becoe positive again after a short period of tie Siple Interest The interest earned up to tie T is siply equal to rta. (This is just the failiar forula Interest = (Principal) (rate) (tie).) Therefore, we have V T = A + rta = A(1 + rt). In practice, the siple interest convention is used only for investents or loans of relatively short aturity (one year or less). Exaple 2.2. (a) Suppose that you invest $100 for 3 onths in an account paying 4% siple interest. At the end of three onths, you will have in the account V.25 = $100(1 + (.04).25) = $101 (b) Suppose that you invest $2,500 for 9 onths in account paying 6.6% siple interest. At the end of 9 onths you will have in the account Annual Copounding V.75 = $2, 500(1 + (.066).75) = $2, Suppose that an aount A is to be left on deposit for N years, where N is a positive integer. If interest is coputed according to the annual copounding convention at rate r, the account value is updated to be A(1 + r) at the end of year one. The interest that will be credited to the account at the end of year two is ra(1 + r) and consequently the account value is updated to A(1 + r) + ra(1 + r) = A(1 + r) 2. 39

4 For each k = 1, 2,, n the account balance will be updated to A(1 + r) k at the end of year k. In particular, at the end of N years, the account balance will be V N = A(1 + r) N. (For an N year certificate of deposit (CD) at a bank, an early withdrawl penalty is generally charged if the custoer wishes to withdraw the oney before the end of year N. For exaple, if you invest $1,000 in a 5-year CD at 6% and you want to withdraw the oney at the end of 4 years, you should expect to receive less than $1, 000(1.06) 4.) Exaple 2.3. Suppose that you invest $100 for 2 years in an account paying 8% interest copounded annually. Then, after two years the value of the account will be V 2 = $100(1 +.08) 2 = $ Exaple 2.4. Suppose that $1 were invested 217 years ago at the interest rate r with annual copounding. (We take t = 0 to correspond to 217 years ago, so that today corresponds to t = 217.) (a) If r = 3.3% then the value of the investent today would be ( ) 217 $1, (b) If r = 6.6% then the value of the investent today would be ( ) 217 $1, 055, (c) If r = 9.9% then the value of the investent today would be Fractional Copounding ( ) 217 $787, 951, With annual copounding, interest is credited to the account once a year. In practice, the interest is usually credited ore frequently (e.g. quarterly, onthly, or daily). Let be a positive integer and let T > 0 be an integer ultiple of 1. Suppose that an aount A is deposited for T years at the rate r. Suppose further that interest is credited to the account ties per year at equally spaced ties according to the following rule: For each k {0, 1, 2, } the account value at tie k+1 V k+1 = V k 40 ( 1 + r ), is updated to

5 where V k is the account value at tie k. After n periods, the value of the account will be V n (1 = A + r ) n. To find the value after T years, we siply put n = T and arrive at ( V T = A 1 + ) r T. In order to copare situations with different frequencies of copounding, it is useful to introduce the notion of an effective interest rate R. The effective interest rate R is the rate under which annual copounding yields the sae result at the end of each year as the rate r does with fractional copounding. More precisely, R is the unique solution of the equation ( 1 + R = 1 + ) r, i.e. R = ( 1 + r ) 1. It is also straightforward to check that R r. In this context we refer to r as the noinal interest rate. Exaple 2.5. Suppose that the noinal interest rate is r = 8% and that $100 is deposited for 2 years at t = 0. The following table gives the effective rate and the value of the account after 2 years under various frequencies of copounding R 8% 8.16% % % % % % V Continuous Copounding The above exaple suggests that if the noinal rate is held fixed and the nuber of copoundings tends to infinity, then the effective rate will approach a liit. This is indeed the case, by virtue of the well-known fact li ( 1 + r ) = e r fro basic calculus. We refer to the situation where the value of the account balance for a T-year investent is deterined by the rule V T = Ae rt as continuous copounding at the noinal interest rate r. The corresponding effective interest rate R for continuous copounding is given by R = e r 1. 41

6 Although soe financial institutions advertise that they use continuous copounding, the priary value of this concept is that it leads to uch cleaner atheatical forulations in any situations. Exaple 2.6. Suppose that $100 is deposited at t = 0 for 2 years and that interest is copounded continuously at the noinal rate r = 8%. We find that and that R = e.08 1 = % V 2 = $100 exp (.08 2) = $100e.16 = $ The following reark addresses the iportant issue of what happens if different banks are using different frequencies of copounding. Reark 2.7. Let 1 and 2 be positive integers and assue that T > 0 is an integer 1 ultiple of 1 and 1 2. Suppose that there are two banks - Bank 1 and Bank 2. At Bank 1, interest is copounded 1 ties per year and we can borrow or invest for T years at the noinal rate r 1. At Bank 2, interest is copounded 2 ties per year and we can borrow or invest for T years at the noinal rate r 2. If there is no arbitrage, then the effective rates at the two banks ust be the sae; in other words, we ust have ( 1 + r ) 1 ( 1 = 1 + r ) Indeed, if the effective rates were not the sae then we could borrow oney at t = 0 fro the bank with the lower effective rate and invest the oney in the bank with the higher effective rate. At tie T we could withdraw the oney we deposited (together with the interest), pay back the loan, and have soe oney left over. The sae analysis applies if one of the banks copounds continuously. In order to avoid repeating soe standard assuptions over and over, it is useful to describe soething called an ideal oney arket. In an ideal oney arket with constant effective rate R we can, at any tie we choose, borrow or invest any aount of oney for any length of tie at the effective rate R. It is possible to choose any nuber of copoundings per year we wish, including continuous copounding. No fees or coissions are charged. Separate transactions in an account are additive in their effect on future balances. Notice that in an ideal oney arket with constant effective rate, the rate is assued to be independent of the date at which the transaction is initiated and also the length of the loan or deposit. Notice that the effective rate R iplies a corresponding noinal rate for each possible nuber of annual copoundings. It is convenient to denote the noinal rate for copoundings per year by r[], so that ( 1 + r[] ) = 1 + R. The noinal rate corresponding to continuous copounding will be denoted by r[ ], so that e r[ ] = 1 + R. 42

7 Observe that in an ideal oney arket with constant effective rate R, if V 0 is deposited at t = 0 then the value of the account at tie T will be V T = V 0 (1 + R) T, even if T is not an integer. If an aount V τ is deposited at tie τ, then for each t τ the value of the account at tie t will be V t = V τ (1 + R) t τ, assuing that no additional deposits or withdrawls were ade between tie τ and tie t. The situation is analogous for loans. In fact, when we can borrow or invest at the sae rate, a loan can be treated as an investent in which which the aount invested is negative. For this reason, any forulas will be stated only for the case of an investent (or only for a loan). Reark 2.8. It will be very convenient for us to use effective rates ost of the tie so we can avoid having to worry about copounding conventions. However, it is very iportant to realize that other conventions are frequently used to describe interest rates in practice (and in books on atheatical finance). It is coon for a single bank to use different conventions to describe their different products so as to ake the rates sound ore attractive to potential custoers. For exaple, bank CD rates are generally quoted as effective rates whereas loan rates are usually quoted as noinal rates. When coparing investent opportunities, it is crucial to ascertain the precise conventions used for coputing the interest. If interest is actually being copounded ties per year, one ust keep in ind that forulas involving effective interest rates should be applied only at ties that are consistent with the copounding schedule. (In practice, a nuber of different conventions are used to copute the interest that is accrued for a fraction of an investent period. This issue will be discussed briefly in Reark 2.18.) Reark 2.9. In an ideal oney arket with constant effective rate R, the account balance V t at tie t satisfies d dt V t = (ln (1 + R)) V t = r[ ]V t (at ties when no oney is being deposited or withdrawn). This eans that if an aount α is deposited for a sall tie interval of length t then the interest over this tie interval will be approxiately α (ln (1 + R)) t = αr[ ] t. In such a situation, the description in ters of the noinal rate r[ ] is certainly cleaner. This is iportant to keep in ind during arguents in which a liit is taken as the length of an investent tends to zero. In such arguents the standard practice in the literature is to use the noinal rate r[ ] rather the effective rate. 43

8 2.3 Ter Structure of Interest Rates Anyone who has studied bank CD rates or bond prices (or who has looked into getting a loan) knows that in reality, interest rates are not constant, but depend on factors such as the risk of default, the aount of oney to be invested, the length of tie (or ter) of the investent, and the tie at which the investent will be ade. We now consider situations where the interest rate depends on tie. We still assue that there is no risk of default, that the sae rate is used for loans and investents, and that the sae rate applies to transactions of any size. Throughout this discussion, we ake the convention that the present tie is t = 0, unless stated otherwise. It is very iportant to understand that the interest rates that will prevail at future ties are unknown at the present tie. Suppose that we can borrow or invest any aount of oney we want at t = 0 (the present tie) up to any given aturity T > 0 at the effective annual rate R (T). In other words, the interest rate is allowed to be a function of the aturity. It is assued here that the full aount invested or borrowed is transferred at t = 0 and the only withdrawal or payent takes place at aturity (i.e. t = T). Early withdrawals and early payents are not allowed. If V 0 is invested at t = 0 (for a ter of T years), then the terinal capital is given by V T = V 0 (1 + R (T)) T. Reark If early withdrawals and early payents on loans were allowed without penalty, then a no-arbitrage arguent could be used to show that R (T) ust be constant in T. We refer to R (T) as the effective (annual) spot rate corresponding to aturity T. Here the ter spot eans current or present. It refers to the fact that the transaction will take place on the spot. It is not known at the present tie what the spot rates will be at future dates. The graph of R (T) versus T is called the effective spot-rate curve or the 1 effective yield curve. The dependence of interest rates on the length of the investent (or loan) is referred to as the ter structure of interest rates. It is typically the case that longer aturities correspond to higher rates, although it soeties does happen that short-ter rates are higher than long-ter rates. This latter situation occurs when investors expect interest rates to drop significantly in the future. Reark Spot interest rates are frequently described in ters of noinal rates. (This will occur in several exercises at the end of the chapter.) Throughout the text, spot rates will always be given as effective rates. In practice, it is essential to be sure what convention is being eployed to describe the interest rates in question. It should also be kept in ind that the notion of copounding can be isleading when discussing spot rates, because interest is not actually added to an account balance at interediate ties the principal and all interest are paid at tie T. 1 There is ore than one type of yield curve. The graph of R (T) versus T is the effective zero-coupon yield curve. 44

9 Consequently, the capital is well defined only at ties 0 and T. The notion of a copounding convention siply tells us how to copute the terinal capital fro the initial capital. In fact, the ratio V 0 /V T for a T year investent or loan is siply the discount factor D(T) for aturity T. In particular, we have D(T) = ( ) (1 + R (T)), R T (T) = 1. T D(T) For any purposes, it is uch ore convenient to use discount factors instead of spot rates. Indeed, discount factors are ore intrinsic than spot rates because they are independent of any kind of copounding convention. It is convenient to generalize the notion of an ideal oney arket to allow for ter structure. We shall refer to this generalized notion as an ideal bank (because when there is ter structure, the phrase oney arket is usually associated with short ter investents). In an ideal bank with effective spot rate function R we can choose any tie T > 0 and borrow or invest any aount of oney at tie 0 (the present tie) until aturity T at the effective rate R (T). The full aount invested or borrowed is transferred at tie 0 and settleent is ade at tie T with a single lup-su payent (principal and interest). We can ake ultiple transactions at t = 0, each having different aturities. The effects of the individual transactions on future balances are additive. No fees are charged for aking transactions. Exaple Assue that there is an ideal bank with R (.5) = 4% and R (8) = 5.2%. (a) If an agent invests $10,000 at t = 0 for six onths, the value of this investent at aturity will be $10, 000(1 +.04).5 = $10, $10, (b) If an agent borrows $5,000 between t = 0 and t = 8, then at tie 8 the agent will owe the bank $5, 000( ) 8 $7, It is iportant to understand that an ideal oney arket with constant effective rate R is not exactly the sae an ideal bank with R (T) = R for all T > 0 because in the ideal oney arket with constant effective rate R it is assued that the effective spot rates that prevail in the future will also be equal to R for all aturities 2. We shall use the ter ideal bank to ean an ideal bank having soe effective rate function R, possibly constant. Unless stated otherwise, fro now on, we shall always assue the presence of an ideal bank. 2 It is possible to have arbitrage-free odels for interest rates in which the yield curve is flat at t = 0 but does not reain flat in the future. In such a odel, interest rates for investents ade at t = 0 will be independent of aturity, but interest rates for investents initiated at other ties can depend on the aturity of the investent. 45

10 Reark In order to study ter structure in a serious way, it is necessary to copare the spot rates that prevail on different dates. Therefore, it is necessary to use two tie variables to describe interest rates. One of the variables will be the date on which the investent (or loan) will be initiated. The other variable is soeties taken to be the length of the investent, and soeties taken to be the date at which settleent is ade. Of course, once the initiation date has been specified, one can deterine the settleent date fro the length of the investent, and vice-versa, so the two different notational conventions provide the sae inforation. In practice, one ust be very careful to check which convention is used to describe a particular interest rate. In this course, we use the notation R τ,t to denote the effective spot rate at tie τ for investents (or loans) to be settled with a single lup-su payent at tie T > τ. (In other words this rate corresponds to investents of duration T τ that are initiated at tie τ.) The rates R τ,t are known at tie τ, but unknown prior to tie τ. Notice that R (T) = R 0,T for all T > 0. Nevertheless, we shall continue to use the notation R (T) even though it is not really needed any ore partly in the hope that the asterisk will serve as a reinder that we are aking the convention that the present tie is t = 0. (Indeed, there are situations in which one needs to talk about R 0,T at ties other than t = 0.) 2.4 Forward Interest Rates It is not known today what the spot rates will be at future dates. However, if we want to agree today on an interest rate for a loan that will be initiated 6 onths fro today and repaid with a single lup-su payent two years later (i.e., 2.5 years fro today), we can use the no-arbitrage principle to deterine an appropriate interest rate in ters of the spot rates R (.5) and R (2.5). Such an interest rate is known as a forward rate. It is evident that the idea of locking a rate now for an investent or loan to be ade in the future could be extreely useful. Moreover, forward rates will play a key role in the developent of a full theory of ter structure. For these reasons, we give the definition here in a bit ore generality than is required for iediate use of this concept, naely we allow for the interest rate to be agreed on at an arbitrary tie τ rather than only at tie 0. Let τ, η, T with τ η < T be given. Let us denote by R for τ,η,t an effective interest rate agreed upon at tie τ for an investent or loan (of a prescribed aount) initiated at tie η and settled with a single lup-su payent at tie T. In particular, if V η is invested at tie η under the ters of an agreeent ade at tie τ then the value of the investent at tie T will be We refer to R for τ,η,t seen fro tie τ. V T = V η (1 + R for τ,η,t )T η. as the effective forward interest rate for the tie interval [η, T] as 46

11 Given η 0 and T > η, let us deterine a general expression for R for 0,η,T using the no-arbitrage principle. The easiest way to do this is to replicate a forward loan (or deposit) using zero-coupon bonds, expressing the bond prices in ters of discount factors, and then expressing the discount factors in ters of spot rates. Let us replicate a forward loan in which it is agreed at tie 0 to borrow $1 at tie η and repay the loan with a single payent of $(1 + R for 0,η,T )T η at tie T. We replicate the position of the agent borrowing the oney. The agent needs to (i) recieve $1 at tie η and (ii) owe $(1 + R for 0,η,T )T η at tie T. To acheive (i) a zero-coupon bond with face value 1 and aturity η should purchased at tie 0 and (ii) can be acheived by shortselling a zero-coupon bond with face value (1 + R for 0,η,T )T η and aturity T at tie 0. The initial capital of this strategy is X 0 = D(η) (1 + R for 0,η,T )T η D(T). Since nothing is paid at tie 0 to enter the agreeent we ust have X 0 = 0 which gives (1 + R for 0,η,T )T η = D(η) D(T) = (1 + R (T)) T (1 + R (η)). η After a bit of algebra we find that ( ) (1 + R for 0,η,T = R (T)) T 1 T η 1. (2.1) (1 + R (η)) η It is not necessary (nor is it even a good idea) to eorize equation (2.1). When you need a forward interest rate in practice, the copounding convention will likely be different. What you should do is be sure that you know how to replicate a forward loan using zero-coupon bonds. For any calculations with forward loans, it will be siplier to use zero-coupon bonds and discount factors than it will be to introduce forward rates. Reark The sae arguent shows that for each tie τ we have ( ) (1 + R for τ,η,t = Rτ,T ) T τ 1 T η 1. (1 + R τ,η ) η τ Exaple Today s date is t = 0. Suppose that we want to borrow $10,000 six onths fro now (at t =.5) and repay the loan with a single lup-su payent 2 years later (at t = 2.5). Suppose further that we go to a bank today to agree on an interest that will be used for this loan. Given that R (.5) = 3% and R (2.5) = 4%, let us deterine the appropriate effective interest rate and the aount that will be needed at t = 2.5 to repay the loan. Putting η =.5 and T = 2.5, we find that 47

12 R for 0,.5,2.5 = ( (1 +.04) 2.5 (1 +.03).5 )1 2 1 = %. The aount we will need to pay at t = 2.5 is given by $10, 000 ( ) 2 = $10, Reark It is iportant to understand that even though forward interest rates do not depend on the aount of oney to be borrowed or invested, when entering into an agreeent to borrow or invest oney at a future date it is essential that the aount of oney to be borrowed or invest is agreed upon at the sae tie that the interest rate is agreed upon. It is also very iportant to realize that the spot rate R η,t that will prevail at tie η will be different fro the forward interest rate R for τ,η,t, except by chance. 2.5 Basic Types of Securities with Deterinistic Payents We now describe soe iportant securities having fixed payents (or deterinistic cash flows). Such a security is copletely described by a payent schedule (i.e. payent aounts and dates). We assue that the payent aounts and dates are known with certainty at the tie that the security is issued. In practice, such securities are often described using interest rates or yields. We shall talk about yields or rates of return of securities with deterinistic payents later. Unless stated otherwise, we assue that the present tie is t = 0, that there is no risk of default, and that all securities can be bought or sold short in any desired quantities (including fractional nubers of securities) Zero-Coupon Bonds Although we have already introduced zero-coupon bonds, we repeat soe inforation here for the sake of copleteness. A zero-coupon bond (also called a pure discount bond) is characterized by a face value F > 0 and a aturity date T > 0. The holder of the bond receives a single payent of aount F at tie T. Zero-coupon bonds are extreely siple securities. However, they are building blocks that can be used to replicate any security with fixed payents. In other words, any given security with fixed payents can be expressed as a portfolio of zero coupon bonds. The face value of a zero-coupon bond is also referred to as the par value General Security with Fixed Payents A general security with fixed payents will be characterized by a list (F i, T i ) 1 i N where the T i are ties with 0 < T 1 < T 2 < < T N and the F i are payents received by the holder of the security at the ties T i. It is convenient to allow 48

13 soe (or all) of the F i to be negative, but then we need to be careful about a sign convention: (a) If F i > 0 then the holder of the security receives F i at tie T i. (b) If F i < 0 then the holder of the security ust pay F i at tie T i. (c) If F i = 0, then no oney changes hands at tie T i. Although it is convenient to allow the F i to be negative, soe caution ust be exercised. This can lead to situations in which the phrase holder of the security as used above ay conflict with the usage of this phrase in ordinary parlance. If all of the F i have the sae sign, then it is clear which party ust pay the other initially. If the F i change sign soe interesting and surprising situations can occur. In the vast ajority of cases encountered in this course, the securities will be such that the F i all have the sae sign. Of course, payents of aount zero can be ignored. We are allowing for the possibility that soe of the F i are zero siply because this can soeties siplify the indexing of the payent ties. In order to avoid cubersoe language we shall usually not distinguish between a general security with fixed payents and the list (F i, T i ) 1 i N of payent ties and aounts Coupon Bond A coupon bond is characterized by a face value F > 0 (also called the par value), a aturity date T > 0, a positive integer giving the nuber of coupon payents per year, and a noinal annual coupon rate q[]. The holder of the security will receive payents ties per year with equal spacing between payent ties. To siplify the indexing, we assue that the aturity date is T = N, for soe positive integer N, and that the payents will be received at the ties T i = i, i = 1, 2,..., N. The holder of the bond receives coupon payents C = F q[] at the ties T i = i, i = 1, 2,..., N plus the face value F at aturity. Notice that holder receives F + C at tie N. The coupon rate is usually chosen so that at the tie the bond is originally issued, the price of the bond will be very close to the face value. Consequently the coupon payents are effectively interest payents on the principal F. The ter coupon payents coes fro the fact that in the past, physical coupons were attached to the bond docuentation, and the holder of the bond would send in the coupons by ail in order to receive payents. Today, bonds generally do not include actual paper coupons, but the terinology persists. Coupon bonds issued by the US Treasury pay coupons twice per year, i.e. = 2. 49

14 2.5.4 Annuity An annuity is a security that pays the holder the sae aount periodically (at equally spaced ties) over soe interval of tie. An annuity is characterized by a payent size A > 0, the nuber of payents per year, and a aturity date T > 0. Again, to siplify the indexing, we assue that the aturity date is T = N, for soe positive integer N and that the payents will be received at the ties T i = i, i = 1, 2,..., N. The holder of the annuity receives the aount A at each T i, i = 1, 2,..., N. Reark In practice, soe coupon bonds are callable. This eans that at certain ties prior to aturity, the issuer of the bond can decide to pay the face value F (or soe other contractually specified aount) plus any coupon that is due at that date to the holder thereby relieving the issuer fro the obligation to ake any further payents to the bond holder. Callable bonds will not be treated here. Reark In our description of coupon bonds, the assuption that T is a positive integer could easily be replaced by the assuption that T is an integer ultiple of 1, necessitating only inor changes in forulas. However, the assuption that the first payent will be received at tie 1 is uch ore significant. This indicates that we are considering a newly issued bond or that we are considering a previously issued bond iediately after a coupon payent has been ade. As soon as a coupon is paid the value of a bond ust drop by the aount of the coupon payent. For several reasons, it is inconvenient to quote prices that are discontinuous. In practice, even when one ignores the bid-ask spread, there will be two relevant prices for coupon bonds at ties after their initial issue a so-called flat price and a so-called full price. The flat price is the one that is quoted in places such as The Wall Street Journal. The full price (or invoice price) is the one that a purchaser of the bond ust actually pay. These two prices are the sae iediately after a coupon payent has been ade. At ties between coupon payents, the full price is higher. The difference between the two prices is the interest I that has accrued on the face value since the last coupon payent. The forula for coputing the accrued interest is I = q[] T, where T is the tie in years since the last coupon payent. In other words, the accrued interest is coputed using the siple interest convention. It should noted that the forula used to coputed the accrued interest is just a arket convention designed to ake quoted prices of bonds continuous in tie. One could use a different convention if desired and this would slightly change the values of the flat price and the accrued interest. This is not an issue because it is the full price that is deterined by arket conditions; by definition the flat price price plus the accrued interest ust add up to the full price. We know that I < C because T < 1. Since ost years have an odd nuber of days, and because payents generally are not ade on weekends and holidays, the calculation of T is not copletely clearcut. The anner in which the interval between two calendar dates is interpreted as a fraction of a year is called a day count convention. In practice, different day count conventions are used for different types 50

15 of fixed-incoe securities. Although such arket details fall outside the scope of the course, and we will not pursue these issue further, it is iportant to be aware that one ust be careful when talking about prices of coupon bonds at ties between the coupon payents. The flat price is soeties called the clean price and the full price is soeties called the dirty price. Reark In practice, there are lifetie annuities that pay the holder the sae aount periodically for the reainder of his or her life, and annuities in which the aount of the payents is variable. Annuities involving variable aturity or variable payents will not be treated here. Unless stated otherwise, when we use the ter annuity it is to be understood tacitly that the payent aounts are constant and that the aturity date is finite and specified at tie 0. Reark A ortgage (or any other loan with constant payents at equally spaced ties) can be viewed as an annuity with the roles of the custoer and the bank being interchanged. Consequently the forulas we obtain for annuities can also be used to analyze loans of this type. However, there are often iportant practical differences between ortgage and annuity contracts. In particular, ost ortgages (and other types of loans) have pre-payent provisions that allow the client to pay the outstanding balance of the loan early and avoid interest charges. Annuities typically do not have such provisions. Most ortgages and consuer loans involve onthly payents. The standard day count convention for such loans is that all onths are treated as 1 of a year Discount Factors and Present Value of Future Payents Suppose that we have access to an ideal bank with effective spot rate function R. If we invest an aount A between tie 0 (the present tie) and tie T, we can easily copute the value of this investent at tie T. This process can also be carried out in reverse. If we want the value of our investent at soe given future tie T to be B, we can easily deterine an aount A that if invested now will have value B at tie T. In this case, we say that A is the present value of a payent B that will be ade at tie T. For exaple if R (1) = 5% and you are going to receive a rebate of $210 fro a copany one year fro now, the present value of the rebate is $200, because an investent of $200 now will be worth $200 (1.05) = $210 one year fro now. It ay see that receiving $210 one year fro now is different than receiving $200 now, because in one case you have to wait a year until you receive any oney and in the other case you get soe oney right away. However, since borrowing fro the bank is allowed, the two situations are copletely equivalent. Indeed, you can borrow $200 fro the bank (between now and tie 1) and do with it as you please. One year fro now, when you receive $210 you can repay the loan. Although the notion of present value is very siple, it provides us with a powerful tool for evaluating investents involving payents at different ties. In particular, if an investor truly has access to an ideal bank then it is sensible for the investor to decide 51

16 between two investent possibilities having fixed payents by siply coputing the net present values (i.e., the su of the present values of all payents) for each alternative and choosing the one whose net present value is ost favorable to the investor. Indeed, if two payent streas have the sae net present value, then one can be converted to the other via transactions with the bank. Recall that the discount factor D(T) for tie T is the price at tie 0 to receive $1 at tie T and it is given by D(T) = 1 (1 + R (T)) T. (2.2) Observe that if A is the present value of a payent of size B that will occur at tie T then A = BD(T). Reark A very iportant property of discount factors is that they reove the need to worry about the convention that is being used to describe interest rates. The forula to copute D(T) will look different if the interest is coputed using a rate other than the effective rate, but the nuerical value of D(T) will be the sae no atter what convention is used to describe interest rates. The discount factor D(T) is siply the ratio of the initial capital to the capital at tie T for a T-year investent in the oney arket. For this reason, discount factors are ore fundaental than interest rates. 2.7 Arbitrage-Free Prices of Securities with Fixed Payents in the Presence of an Ideal Bank Suppose that there is an ideal bank with effective spot rate function R. It is straightforward to deterine the arbitrage-free prices of the securities discussed above. Securities with fixed payents can be (and frequently are) traded at ties after the issue date. In coputing prices for such trades we ust be careful to ake sure that the payent ties are easured relative to the present tie (rather than the issue date) and we ust be sure to realize that the new owner of the security receives only those payents ade after the trade is executed. We generally use the sybol P (possibly ebellished) to denote the current price of a security with fixed payents. If a security is being traded on a date at which the security is to ake a payent, we assue that the trade takes place iediately after the party who is selling the security has received the payent in question. We ake a siilar convention in pricing (or valuing) securities that were issued in the past and are currently being held as part of a portfolio the price on a payent date is understood to be the price just after the payent has been ade. 52

17 2.7.1 Zero-coupon Bond Consider a zero-coupon bond with face value F and aturity T. In order to deterine the arbitrage-free price of the bond at t = 0 we can construct a replicating strategy using the bank account. If we invest X 0 at t = 0 until t = T, then the value of the investent at tie T will be X T = X 0 (1 + R (T)) T. Since the terinal capital ust satisfy X T = F, we ust have X 0 = F (1 + R (T)) T = FD(T). Therefore, the arbitrage-free price P of the bond at t = 0 is P = FD(T) = F (1 + R (T)) T. (2.3) Since zero-coupon bonds can always be replicated by siple bank deposits, we now consider zero-coupon bonds of all possible face values and aturities to be basic (or traded) securities General Security with Fixed Payents Consider a general security with fixed payents (F i, T i ) 1 i N. We have not yet defined exactly what a replicating strategy is for securities that ake ultiple payents. (We shall do so shortly.)however, it sees pretty clear what a replicating strategy should be for this security. Let X be a portfolio consisting of zero-coupon bonds with aturities T i and face values F i, for each i with 1 i N; if F i > 0 then one bond with aturity T i and face value F i is purchased in the portfolio, while if F i < 0 then one bond with aturity T i and face value F i is sold short in the portfolio. (If F i = 0, the there is nothing to worry about.) Notice that the strategy X will not be self-financing. We shall refer to this strategy as a replicating strategy for the general security with fixed payents (F i, T i ) 1 i N. The initial capital of the replicating strategy is X 0 = N F i D(T i ). i=1 (Notice that if the i th bond has been sold short then F i < 0 and the corresponding contribution to the initial capital of the portfolio is F i = F i.) The arbitrage-free price of the security at t = 0 is therefore given by P = N F i D(T i ) = i=1 N i=1 F i. (2.4) (1 + R (T i )) T i Our official definition of a replicating strategy for a security aking ultiple payents is as follows: A strategy X is a replicating strategy for a security aking 53

18 ultiple payents provided that the strategy Y obtained by ipleenting X and selling short the security is a self-financing strategy with 0 terinal caplital. Notice that Y is self-financing and has terinal capital Y T = 0. If we let Z be the zero strategy, (i.e. the strategy in which no oney is ever invested in anything) then Z is also self-financing and has terinal capital Z T = 0. Consequently, Y 0 = Z 0 = 0, which iplies that initial price of the security ust be the sae as the initial capital of a replicating strategy. We note that P (as defined by (2.4)) is also called the net present value of the security. This is very appropriate terinology because F i D(T i ) is the present value of a payent of F i received at tie T i. The su of these payents is siply the total or net present value of the future payents. If P > 0 then one ust pay oney initially to hold the security, while if P < 0 then the holder of the security should initially receive oney fro the counterparty. (Of course, if P = 0 then no oney changes hands at t = 0.) Notice that the sign convention for P is opposite to the sign convention for the F i ; P > 0 eans that the holder pays oney, while F i > 0 eans that the holder receives oney. The reason for this is that the aount P paid by the holder at t = 0 is supposed to balance out the net present value of the payents that the holder will receive in the future. When there is an ideal bank, we ay treat general securities with fixed payents as basic (or traded) instruents because they can easily be replicated in ters of transactions with the bank Coupon Bond Consider a coupon bond with face value F > 0, and aturity T > 0, aking coupon payents ties per year at the noinal annual coupon rate q[]. As discussed previously, we assue that T = N, for soe positive integer N, and that the holder of the bond receives coupon payents C = F q[] (2.5) at the ties T i = i, i = 1, 2,...,N plus an additional payent of aount F at tie N. The arbitrage-free price at t = 0 (net present value) of the bond is therefore P = FD(N) + N i=1 CD( i ) = F N (1 + R (n)) + N i=1 C (1 + R ( i ))i/. (2.6) Annuity Consider an annuity that akes payents of size A > 0 ties per year, and has aturity date T > 0. We assue that T = N, for soe positive integer N and that the holder of the annuity receives payents of size A at each of the ties T i = i, i = 1, 2,..., N. Therefore, the arbitrage-free price at t = 0 (net present value) is given by 54

19 N P = i=1 AD( i ) = N i=1 A (1 + R ( i ))i/. (2.7) Reark When we analyze ortgages (or other loans with constant payents at equally spaced ties), we shall usually replicate the position of the creditor (i.e., the lender) in order to cut down on the nuber of inus signs needed. At t = 0, the creditor will pay a positive aount P to the debtor (i.e., the borrower) in exchange for the debtor s proise to pay the creditor a positive aount A at each of the ties T i = i, i = 1, 2,..., N. Exaple Assue that R (3) = 5% and consider a zero coupon bond with aturity T = 3 and face value F = $10, 000. Suppose that we have an opportunity to purchase one of these bonds at the price ˆP = $8, 500. What should we do? To answer this question, we should copute the arbitrage-free price of the bond. The discount factor D(3) is given by D(3) = which yields an arbitrage-free price of 1 (1 + R (3)) 3 = 1 (1 +.05) 3 = , P = FD(3) = $10, 000 ( ) = $8, Since the price at which we can purchase the bond is lower than the arbitrage-free price, we should buy the bond. This would lead to an iediate profit of $ at t = 0 via the following strategy: (a) Borrow $8, fro the bank between t = 0 and t = 3. (b) Buy the bond for $8,500 at t = 0. The difference, $138.38, constitutes a profit that we can safely consue now (or invest for later). At t = 3 we will receive $10,000 (the face value of the bond) and we ust pay the debt $8, (1+.05) 3 = $10, 000 on the bank account. The debt on the bank account will be perfectly offset by the face value of the bond. We could also ipleent a slightly different strategy: (a ) Borrow $8,500 fro the bank between t = 0 and t = 3. (b ) Buy the bond for $8,500 at t = 0. There is no oney left over at t = 0. However, if we follow this strategy, then at t = 3 we will receive $10,000 (face value of the bond), but we will owe only $8, 500 (1 +.05) 3 = $9, to the bank, leaving us a profit of $10, 000 $9, = $ This is perfectly reasonable because $ is the present value of $ at tie 3. Observe that the opportunity to buy one of these bonds for $8,500 is an arbitrage opportunity. However, the arbitrage is liited in scope since we have the opportunity 55

20 to buy only one bond. Such an opportunity ight very well exist in the real world. However, if a dealer were offering to sell any of these bonds at $8,500 per bond (and if transaction costs were significantly less than $ per bond) investors would flock to purchase the bond and the high deand would drive the price upward. Exaple Assue that R (.5) = 5%, R (1) = 5.25%, R (1.5) = 5.5%, R (2) = 6%. (2.8) Consider a coupon bond with aturity T = 2 and face value F = $10, 000 that pays coupons twice per year at the noinal coupon rate q[2] = 4% Suppose that we have an opportunity to buy one of these bonds at the price ˆP = $9, 750 What should we do? To answer this question, we should copute the arbitrage-free price P of the bond. Before coputing P, let us observe that we should expect P to be less than the face value F because the effective coupon rate is (1 +.02) 2 1 = 4.04% and this is noticeably less than all of the relevant effective spot rates. The coupon payents are given by C = F q[2] 2 = $10, 000 ( ).04 = $ Using (2.2) and (2.8), we find that the discount factors are given by D(.5) = D(1.5) = 1 (1 +.05).05 = , D(1) = 1 ( ) 1 = , 1 ( ) = , D(2) = (1 +.06) = The arbitrage-free price of the bond is given by P = $200 D(.5) + $200 D(1) + $200 D(1.5) + $10, 200 D(2) = $9, We should not purchase a bond at the price ˆP = $9, 750 because this price is higher than the arbitrage-free price. Exaple Assue that R (15) = 10%. Annuities with aturity T = 15 that ake payents of $500 per onth are trading at the current price of $50,000. Consider a coupon bond with aturity T = 15 and face value F = $1, 000 that pays coupons onthly at the noinal coupon rate q[12] = 12%. Deterine the price at t = 0 of the bond. Since we have not been given all of the relevant spot rates, we cannot use (2.6) directly. Instead, we can find a siple replicating strategy. The coupon payents are given by C = F q[] = $1, = $10. 56

21 We can replicate the strea of coupon payents by purchasing 1 50 The payent of F at T = 15 can be replicated by investing annuities at t = 0. $1, 000 $1, 000 $1, 000D(15) = = = $ (1 + R (15)) 15 (1 +.1) 15 in the bank between t = 0 and t = 15. The initial capital of this strategy is $50, $ = $1, , 50 and consequently the arbitrage-free price of the bond at t = 0 is P = $1, Prices of Coupon Bonds and Annuities in the Presence of an Ideal Bank with Constant Effective Spot Rate Function When interest rates are the sae for all aturities, the forula for suing a geoetric series can be used to siplify the expressions (2.6) and (2.7) for the prices of coupon bonds and annuities. In order to pursue this issue, we assue, teporarily, that there is an ideal bank with constant effective spot rate function R (T) = R. Recall that k 1 λ i = 1 + λ + λ λ k 1 = 1 λk 1 λ i=0 fro which it follows iediately that i=1 for λ 1, k ( ) 1 λ λ i = λ + λ 2 + λ λ k k = λ for λ 1. (2.9) 1 λ Observe that the relevant discount factors are given by D( i ( ) i ) = 1 (1 + R) = 1, (2.10) i/ (1 + R) 1/ which suggests that we put λ = 1 (1 + R) = 1 1/ 1 + r[]. (2.11) Here r[] is the noinal rate (for copounding -ties per year) that is consistent with the effective rate R. Notice that the expression for λ in ters of the noinal rate r[] is slightly cleaner because it does not involve an th root. Cobining (2.9), (2.10), and (2.11) we find that N i=1 D( i ) = λ ( 1 λ N 1 λ 57 ). (2.12)

22 Observe also that D(N) = λ N. (2.13) Using (2.12) and (2.13) we obtain the following two very iportant rearks. Reark If R (T) = R, for all T > 0 the forula (2.6) for the arbitrage-free price of a coupon bond siplifies to ( ) 1 λ P = Fλ N N + Cλ, (2.14) 1 λ where λ is given by (2.11). Reark If R (T) = R for all T > 0, the forula (2.7) for the arbitrage-free price of an annuity siplifies to ( ) 1 λ n P = Aλ, (2.15) 1 λ where λ is given by (2.11). 2.9 Internal Rate of Return or Yield to Maturity of a Security with Fixed Payents The assuption that interest rates are independent of aturity is not very realistic especially for securities such as coupon bonds and annuities that have a long period of tie reaining until aturity. Since payents will be received at nuerous ties (with each payent tie possibly corresponding to a different spot rate), it is not evident how to assign a single interest rate to such a security. One way that this can be accoplished is by eans of soething called an internal rate of return. Definition A real nuber R I > 1 is called an effective internal rate of return for the general general security with fixed payents (F i, T i ) 1 i N provided that ˆP = N i=1 F i, (2.16) (1 + R I ) T i where ˆP is the price of the security at t = 0 (the present tie). Reark We have used the sybol ˆP, rather than P to denote the price at t = 0 because there are certain fixed incoe securities (such as corporate bonds) that are traded in the real world and for which it is not appropriate to use equation (2.4) to copute an arbitrage-free price. The reason for this is that (2.4) was obtained under the assuption that there is no risk of default. Although bonds issued by copanies having strong credit ratings are usually pretty safe investents, it does soeties happen that such bonds default. It is standard practice to use (2.16) (with ˆP equal to the arket price) to deterine effective internal rates of return for corporate (and 58

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