# Notes on Present Value, Discounting, and Financial Transactions

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1 Notes on Present Value, Discounting, and Financial Transactions Professor John Yinger The Maxwell School Sracuse Universit Version 2.0 Introduction These notes introduce the concepts of present value and discounting and explain how these concepts can be applied to mortgages, annuities and pensions, and bonds. The concepts are illustrated with numerous examples. The concepts of inflation, uncertaint, and risk are briefl considered at the end. Compound Interest Suppose I put \$100 in a savings account on Januar 1 and it earns interest at a 3% rate per ear. For now, suppose that this interest is paid all at once at the end of the ear. Then if I do not withdraw an mone, at the end of the ear m account will contain (\$ \$100) = \$100 (1.03) = \$103. (The smbol indicates multiplication.) Now consider what happens if another ear is added to the analsis. At a 3% interest rate, \$100 in m account turns into \$103 at the end of the first ear. This \$103 turn into \$100 (1.03) (1.03) = \$100 (1.03) 2 = \$100 (1.0609) = \$ at the end of the second ear. This bank account involves a concept known as compound interest. In the second ear, interest is paid not onl on the original \$100, but also on the \$3 of interest received in the second ear a concept known as compounding. These concepts can be expressed in a general wa. Let V 0 be the initial value of an investment (such as m savings account) and let i be the rate of return on this investment (such as the 3% interest on m savings account). Then the value of the account at time is: V V0 (1 i) (1) Note that and i both refer to the interval that determines compounding. If the return is compounded once a ear, then indicates ears and i indicates the annual interest rate. If the return is compounded ever month, then stands for month and i is the monthl interest rate, which is the annual interest rate divided b 12. The frequenc of compounding makes a small difference, which grows over time. Consider \$100 invested in two different savings accounts, both of which have a 6% annual interest rate but one of which compounds once a ear and the other compounds monthl. At the end of the first ear, the first account contains \$100 (1.06) = \$106, whereas the second contains \$100 (1+.06/12) 12 = \$ (The / indicates divided b. ) A third account with a 6 percent

3 3 Thus, it is appropriate to use equation (1) or (2) to calculate the future values of an investment that ields return i, but to use equation (3) to calculate present value based on the return on one s best investment, sa r, which ma not be the same as i. In smbols, PV FV (1 r) (4) For example, with an opportunit cost rate of r, the present value of V 0 invested at i percent for time periods is: PV V0 (1 i) (1 r) (5) Equation (4) is enormousl useful because it allow us to observe returns received in different ears in the future and translate them into a common metric, namel present value. For example, we can compare on investment with returns at annual rate of 6.25% for 3 ears with another investment that has returns at an annual rate of 6% for 4 ears. If the opportunit cost rate, r, equals 5%, then the PV per dollar invested equals (1.0625) 3 /[(1.05) 3 ] = for the first investment and (1.06) 4 /[(1.05) 4 ] = for the second. Even though the second investment has a smaller annual return, the fact that it produces an above-opportunit-cost return for one more ear makes it the better investment of the two. Finall, it is possible to define an individual s opportunit cost rate, also called her discount rate, in an even more general wa as the value an individual places on future consumption relative to current consumption. A person with a high discount rate values current consumption far more than future consumption, whereas a person with a low discount rate receives roughl equal satisfaction from consumption toda and consumption in the future. A high discount rate, that is, a high value for r in equation (5), leads people to reject investments unless the have a high rate of return, but man investments would be worthwhile for a person whose discount rate is low. Mortgages For the purposes of these notes, a mortgage is a contract in which a lender writes a large check to a homeowner in exchange for a promise from the homeowner to make monthl paments for a certain period of time. These paments include interest at a certain rate on the outstanding balance of the loan. In practice, of course, mortgages come in man forms and are used for man purposes. Some mortgages have variable interest rates, for example, homeowners often take out mortgages to fund home improvements, and businesses often take out mortgages to purchase equipment or real estate. Moreover, when mortgage interest rates decline, man homeowners take out a new mortgage (at the new low rates) to pa off their old mortgage (which has a high rate). This is called refinancing. I am not going to consider these possibilities here. Instead, I will analze a standard, fixed-rate, home-purchase mortgage.

4 4 It is also worth pointing out that once a mortgage has been issued, it is an asset that can be bought and sold. A lender who issues a mortgage ma sell it to another lender or to an institution that specializes in buing and repackaging mortgages for investors. Selling a mortgage is equivalent to selling the right to receive the monthl paments that are associated with it. These issues are beond the scope of these notes, which simpl explain the logic of a standard mortgage. To analze a mortgage, we need to define some terms. M 0 is the amount of the mortgage, that is, the size of the check the lender writes to the homeowner; m is the contractual monthl interest rate, which is the annual rate divided b 12; P is the size of the monthl pament; and Y is the term of the loan, that is, the number of months it takes to pa it off. Now the ke to analzing a mortgage is to recognize that the amount that is still owed on the mortgage graduall declines over time, until it reaches zero at the end of ear Y. The amount of the loan can be thought of as the amount that is owed at the initiation of the mortgage, which is wh we call this amount M 0. Now let M be the amount that is still owed at the end of month, where goes from 1 to Y. For a 30-ear mortgage, for example, goes from 1 to = 360. To find a formula for M, we have to recognize (a) that the borrower owes interest at the end of each period based on the outstanding balance at the beginning of the period and (b) that the borrower makes a pament equal to P. With these points in mind, we can write: M M (1 m) P (6) 1 0 The first term on the right side indicates the initial balance and the interest owed on it and the second term indicates the borrower s pament. (To keep the equation simple, the sign indicating that M 0 is multiplied b (1+m) is implicit in (6) and in the equations that follow.) Part of the pament covers the interest that is due at the end of the first period (mm 0 ) and the rest goes to reducing the loan balance. After the first pament, therefore, the balance declines b (P - mm 0 ). Following the same logic, we can write down an expression for the outstanding balance at the end of month 2: M M (1 m) P (7) 2 1 In the second period, therefore, the mortgage balance declines b (P - mm 1 ). Because M 1 is smaller than M 0, the interest portion is smaller and the decline in the balance is larger in the

5 5 second period than in the first. Over the lifetime of a mortgage, the share of the pament that goes to interest graduall declines and the share that goes to reducing principal graduall rises. Now we can substitute equation (6) into equation (7) to obtain: M M (1 m) P [ M (1 m) P](1 m) P M m P m P 2 0(1 ) (1 ) (8) One more time: M M m P M m P m P m P 2 3 2(1 ) [ 0(1 ) (1 ) ](1 ) M m P m P m P 3 2 0(1 ) (1 ) (1 ) (9) This process can easil be generalized to obtain a formula for the balance remaining at the end of an period,. To be specific M M m P m P m P m P (10) (1 ) (1 ) (1 ) (1 )... The series of dots indicates the set of terms that we have not written out all the terms with exponents between -3 and 1. The number of these terms obviousl depends on the value of we are looking at. With = 10, for example, there are 6 terms in this set. Two more steps lead to the formula for a mortgage. The first step is to use a little algebraic magic. We can re-write equation (10) as follows: where M (1 ) M0 m P S (11) S m m m (1 ) (1 ) (1 )... 1 (12) A sum of this tpe can be re-arranged so that onl a few terms remain. To see, how, first multipl S b (1+m) to obtain 1 2 S(1 m) (1 m) (1 m) (1 m)... (1 m) (13) Now if we subtract equation (12) from equation (13) we find that all the middle terms in the string all cancel out and what remains is S(1 m) S (1 m) 1 (14) With just a little bit of algebra, this expression can be solved for S:

6 6 (1 m) 1 S (15) m Substituting equation (15) back into (11) results in: (1 m) 1 M M0(1 m) P m (16) The second step is to recognize that the balance due at the end of month Y has to be zero. In other words, the mortgage has to be full paid off at the end of its term! So when we evaluate equation (16) at the value of = Y, the left side has to equal zero. In smbols: Y Y (1 m) 1 MY 0 M0(1 m) P m or Y (1 m) 1 P M0(1 m) m Y (17) The final steps just involve re-arranging this result to figure out how the pament, P, depends on the mortgage amount, M 0 ; the interest rate, m, and the term, Y. We want to multipl both sides of equation (17) b m and divide both sides b (1+m) Y. Remember that 1/(1+m) Y =(1+m) -Y. These steps lead to: m P M0 1 (1 m) Y (18) In short, in a fixed-rate mortgage contract, the pament is proportional to the mortgage amount, and the proportion, which is shown in equation (18), depends on the mortgage s interest rate and term. This is a useful equation to know. If ou ever borrow mone with a fixed-rate loan, ou can use equation (18) to see if the lender accuratel calculated the monthl pament! Suppose ou are taking out a mortgage of \$200,000 to bu our dream house. If ou have a 30-ear mortgage at a 5% annual interest rate, then our monthl pament will be.05 /12 \$200, 000 \$1, (30 12) 1 [1 (.05 /12)] Finall, note that ou can also use this formula to figure out the mortgage ou can afford (M 0 ) for a given monthl pament, P, and given mortgage characteristics, m and Y. Lenders sometimes offer borrowers a choice between option A, a mortgage with a relativel high interest rate, and option B, a mortgage with a relativel low interest rate

7 7 combined with points paid up front. In this context, a point is 1 percent of the mortgage amount, M 0. Because the points in option B are paid at the beginning of the mortgage, their impact is felt no matter how long one lives in the house. In contrast, the lower interest rates in option B do not ield an benefit unless a person lives in the house for a long time. Let us sa that people who expect to move in the near future have a short time horizon. People with a short time horizon experience the costs of option B (points up front) but not the benefits (a long stream of lower interest rates). The same is true for people with a high discount rate, because the longterm benefits of a lower rate are heavil discounted. Both of these groups should pick option A. For people with a long time horizon and a low discount rate, that is, people who expect to be in the house for the long haul, the benefits from a long stream of lower interest rates outweighs the cost of the up-front points. These people should pick option B. Annuities and Pensions An annuit is a pament stream supported b an initial fund. The most important tpe of annuit is a pension. For man pensions, a person contributes mone into a fund during her working life and then draws a pension pament an annuit from the fund after she retires. Annuities do not have to be pensions, however. It is not correct to calculate the annuit b dividing the fund b the number of pa periods because the fund earns interest along the wa. As a result, the paments are actuall larger than the fund divided b the number of periods. An annuit is designed so that the fund will be exhausted at the end of the expected number of pa periods. We have to use the qualifier expected here because an annuit generall continues even if the expected number of periods is exceeded. In the case of a pension, for example, the pension pament will be calculated on the basis of an expected lifetime, but will continue even if a person lives longer than expected. The people who set up annuities must account for this tpe of uncertaint, but we are not going to do that here. Instead, we will just refer to an expected number of pa periods. As it turns out, except for the uncertaint about timing, a pension is just the inverse of a mortgage. You will be happ to know that we do not have to do an new algebra! An annuit is designed so that the fund is exhausted at the end of the expected number of pa periods just like a mortgage is paid off at the end of its term. Moreover, an annuit must account for interest received from the amount that remains in the fund just like a mortgage must account for interest paments on the amount that is still owed. So let F be the amount of mone in a fund supporting an annuit. This mone is given to a fund administrator b a retiree (or other investor) to invest in exchange for a promise to pa the retiree an annuit of \$A per month for a period of L months. Suppose g is the interest rate that the fund administrator expects to earn on this fund. Then plugging these terms into equation (18) we find that

8 8 g A F 1 (1 g) L (19) In some cases, an investor or potential retiree might want to know how large the fund has to be to support a given annuit. The answer to this question (which is analogous to the question of how large a mortgage one can afford at a given pament) can be found b rearranging equation (19): 1 (1 g) L F A g (20) If I want a pension of \$2,000 per month for 25 ears and I expect the investments from the fund to earn a 6% return per ear, then the required fund is I ll have to save up! (2512) 1 [1 (.06/12)] \$2,000 \$2,185, /12 Pension planners have to conduct calculations such as this one to ensure that a pension fund will have enough mone in it when person retires to cover promised pensions. The pension fund ma receive contributions both from an emploee and an emploer, and the amount in the fund obviousl depends on the size of these contributions, along with the growth in the emploee s wages during her working life. These notes do not pursue the issue of pension planning in this sense, but interested students ma want to consult the Spreadsheet on the Algebra of Pensions that is posted on the Computer Programs tab on m web site. Bonds A bond is an example of a certificate of indebtedness. A person who bus a bond (from a corporation or a municipal government, for example) receives the legal assurance that she will receive a stream of paments from the institution that issued the bond. Thus, a bond is a form of investment with a particular pattern of returns for the person who bus it. A bond is, of course, quite different from a stock. A person who bus a stock becomes one of the owners of a compan, with the right to vote on certain matters and the right to receive dividend paments, which are much less predictable than bond interest paments. The holder of a bond has no ownership rights. A bond is similar to a mortgage; the person who bus the bond is like the lender and the institution issuing the bond is like the borrower. Local governments issue bonds, for example, to raise funds for infrastructure projects, such as bridges or schools. Moreover bonds, like mortgages, come in man different forms, including forms with variable interest rates and bonds that can be redeemed before the maturit date. These notes just examine basic, fixed-rate bonds.

9 9 However, bonds have one characteristic that makes them quite different from a mortgage, namel that the paments do not alter the principal, that is, the do not alter the amount upon which interest is calculated. As a result, bonds retain a principal amount at the end of the contract, the maturit date; this principal amount must be repaid to the investor. More specificall, three characteristics are associated with each bond: (1) the face value, F, (also called the par value or the redemption value or the value at maturit), which is the amount upon which the interest paments are calculated; (2) the coupon rate, c, which indicates the interest to be paid as a percentage of F; and (3) the ears to maturit, N, which is the number of ears during which the investor is entitled to receive interest and also the number of ears until the bond can be redeemed. As an aside, bonds are generall used to cover the cost of a project and the are generall issued in sets with different maturities. More specificall, the are usuall issued in serial form, which means that some of the bonds in a set have a maturit of 1 ear (N=1), others have N=2, and so on all the wa up to the highest selected maturit, sa N*. This approach eases the repament burden on the issuer. If each maturit up to N* has the same number of bonds, then the issuer will onl have to pa back a share equal to 1/N* of its bonds each ear. The ke to understanding bonds is to think about what an investor would pa for a bond that has a certain face value, F, coupon rate, c, and maturit, N. The answer is that an investor will pa the present value of the stream of benefits from owning the bond. An good investor knows about present value! Suppose an investor has an alternative, similar investment, perhaps a U.S. Treasur Bill, that offers an interest rate r. Then r is the opportunit cost of investing in bonds, and the investor s willingness to pa is the present value of the benefits from holding the bond discounted at rate r. The present value of a bond is not the same thing as its face value. In fact, these two values ma be quite different. To see wh, we have to recognize that the present value of benefits from owning a bond consists of an interest pament of cf per ear and a redemption pament of F in ear N. (In practice, interest paments are often made twice a ear; if ou want to account for this ou just need to express everthing in half ears instead of ears, with an interest rate of c/2, where c is the annual interest rate.) Thus, the present value of a bond, and hence its market price, is cf cf cf F P... 2 N (1 r) (1 r) (1 r) (1 r) N (21) Note that the last term in this expression does not have c in it; this term is the present value received b the investor when the issuer of the bond returns the face value to the investor, which is called redemption. Note also that this expression, like an earlier one, has a series of dots, which indicate the terms for interest received between ears 2 and N.

10 10 This expression can be simplified using the same algebraic trick that led to equation (15). Define the sum of all the terms on the right side with cf in the numerator as C, for coupon paments. Then subtract C from C (1+r) and solve the result for C. Replacing all the cf terms with this expression for C leads to: 1 (1 r) P r N F cf (1 r) N 1 (1 r) 1 Fc N r (1 r) N (22) So there is the echo of the mortgage formula in the price of a bond! Again, the difference is that the principal in the bond does not decline in absolute terms (although it does decline in present value!) and is repaid in ear N. The price of a bond thus reflects the present value of the coupon paments (the first term in equation (22)) plus the present value of the redemption amount (the second term). In the second line of equation (22) we can see that P is proportional to F. If this proportion is greater than 1, the bond is said to sell at a premium. If it is less than one, the bond is said to sell at a discount. Consider a bond with F = \$5,000; c = 4%, and N = 20 for investors with an opportunit cost rate of 6%. Then the amount this investor will bid on the bond, P, is \$5, \$3, This bond is priced at a discount (\$3, < \$5,000) because the rate of return it offers is below the opportunit cost rate. Investors will not bu the bond unless it ields a capital gain, that is, unless the redemption value the receive in ear 20 is considerabl greater than the price the have to pa. Sometimes an investor who owns a bond wants to sell it at a given price. In this case, P is known but the rate of return on the bond is not. (This contrasts with the preceding discussion, in which the opportunit cost to investing a bond, r, was known, but the price was not.) This changes the wa we use equation (22), because r can now be interpreted as the rate of return on a bond with price P, face value F, coupon rate C, and maturit N. This rate is called the internal rate of return. When comparing bonds (or other assets) an investor wants to select the bonds with the highest internal rate of return. The problem is that equation (22) is highl nonlinear and cannot easil be solved for r. Fortunatel, however, finding the internal rate of return is such a common problem that most calculators and spreadsheet programs include a function that calculates r once P, F, C, and N have been entered.

11 11 Some municipal bonds push the issuer s paments into the future. This approach lowers the tax burden of repaments in the short run, but raises this burden in the long run. These bonds therefore undermine to some degree the main purpose of issuing bonds to smooth out the tax burden of infrastructure projects but the ma also enhance the appeal of the bonds to certain classes of investors. This higher appeal ma translate into a lower required return, which corresponds to a lower the cost to the municipalit. A zero-coupon bond, for example, sets the coupon rate, c, equal to zero. In this case equation (22) indicates that the price of the bond is: Zero-coupon bond: F P (1 r) N (23) This tpe of bond is designed for investors who want to receive their returns in the form of capital gains, not interest, probabl for income tax reasons. Capital gains are indicated b the difference between F and P, so the price of a zero coupon bond obviousl has to be heavil discounted, that is, P must be far below F, or else no investor would bu it. Another tpe of bond that has been used b some school districts to postpone repaments into the future is a capital appreciation bond (CAB). With this tpe of bond, the interest paments are placed in a fund and reinvested at the coupon rate. The investor claims the mone in this fund, along with the face value, at the maturit date. Thus, all returns to this investment are received in ear N and discounted b (1+r) N. The pament into the fund each ear is cf. The first pament grows to cf(1+c) N-1 b the time the bond matures; the second pament grows to is cf(1+c) N-2 ; the last pament of cf is paid in ear N and does not grow. Let Z be the sum of these accumulated amounts in ear N. Then the same algebraic steps that led to equation (15), impl that Z = cf[(1+c) N 1]/c = F[(1+c) N 1]. The price of the bond equals the present value of (F + Z) or Capital Appreciation Bond : N F F (1 c) 1 N (1 c) P F N N (1 r) (1 r) (24) When c = r, the price of a capital appreciation bond equals its face value. Moreover, a CAB is similar to a zero-coupon bond in that the CAB interest rate, c, is algebraicall equivalent to the implicit annual capital gains rate in a zero coupon bond. (This equivalence ma not be recognized b the Internal Revenue Service, which means that these bonds ma appeal to different investors than zero-coupon bonds). To put it another wa, a zero-coupon bond will sell for the same amount as a CAB if the capital gains reflected in its face value equal the accumulated interest for the CAB; that is, if its face value equals the CAB face value multiplied b (1+c) N. A rarer (and more dubious) tpe of bond is a perpetuit. The holder of this tpe of bond is entitled to receive interest paments indefinitel, which is equivalent to having an infinite number of ears until maturit. Because [1/(1+r) ]=[(1+r) - ] = 0, equation (22) indicates that the price of this tpe of bond is:

13 13 value of \$100 when the nominal discount rate is 8.15%. It is also possible to obtain the same answer using real values. We can assume that the beginning of the ear is the base ear, so a nominal value of \$ equals a real value of \$108.15/1.03 = \$105. Now using a real discount rate of 5%, the present value of \$105 is \$100. Because we observe nominal rates, not real rates, in the market place, we would have to calculate the real discount rate to follow this approach. Using the approximation in the previous paragraph, the real discount rate is 8% - 3% = 5%. To write down a general form for this real/nominal relationship, let i R be the real interest rate, i N be the nominal interest rate, and p be the anticipated rate of inflation. Then and i (1 i )(1 p) 1 i i p p (26) N R R R i R 1 in in p 1 (27) 1 p 1 p Because i R and p are both small numbers, their product is ver small, and these equations are sometimes approximated as i N = (i R + p) and i R = (i N - p). Risk and Uncertaint The financial transactions examined in these notes are all designed with a view toward the future, which cannot, of course, be predicted with perfect accurac. A homebuer ma or ma not default on her mortgage loan. A retiree ma live much longer than expected when her pension was designed. A compan or a public agenc ma go bankrupt and be unable to pa off its bondholders. In common parlance, the future is uncertain and financial transactions involve risks. (Following the work of economist Frank Knight, some scholars define these terms in a different wa: risk arises when the probabilities of various outcomes are known, even if the outcomes themselves are not, whereas uncertaint refers to cases in which even the underling probabilities are not known. In these notes I stick to the more conventional definitions.) The presence of risk and uncertaint lead to two new central concepts in the analsis of financial transactions. The first concept is expected value. The expected value of a financial transaction is the sum across possible outcomes of the probabilit of an outcome multiplied b its present value. In the case of a mortgage, for example, a default results in a loss of interest paments and to a foreclosure process that might be expensive but that eventuall transfers the ownership of the house from the borrower to the lender. The net benefit to the lender could be positive or negative, depending on the magnitude of their costs and the value of the house on which the foreclose. The expected value of mortgage returns in ear is the probabilit of a default multiplied b the present value of the net benefits from default plus the probabilit of no default multiplied b the present value of the mortgage paments. The concept of expected value therefore makes it possible to introduce a range of outcomes with known probabilities into the analsis of financial transactions, but it can lead to some prett complicated algebra! You ma be relieved to learn that this tpe of algebra is beond the scope of these notes.

14 14 The second ke concept is risk aversion, which is said to exist when a person prefers a (relativel) certain set of outcomes with a given expected value to a (relativel) uncertain set of outcomes with the same expected value. Consider an investor who is tring to decide between buing one set if mortgages with a relativel low mortgage rate and a relativel low probabilit of default and buing another set of mortgages with a relativel high mortgage rate and a relativel high probabilit of default. Suppose the expected values of the two investments are the same because the high interest rate on the second investment compensates for the high risk of default. An investor who is risk averse will alwas prefer the first investment to the second. Scholars and financial analsts have developed elaborate mathematical tools to account for risk in financial transactions. These tools go wa beond simple algebra and are wa beond the scope of these notes. The concept of risk aversion also leads to three important ideas about financial markets. First, the widespread presence of risk aversion in a market implies that risk assets will not attract investors unless the offer a higher return than offered b low-risk assets with returns that have the same expected value. In other words, investors have to be compensated for risk. Second, risk-averse investors often seek to minimize their risk b diversifing their portfolio. In this context, a diversified portfolio is one that contains a series of investments with returns that are not highl correlated with each other, so that bad outcomes for one investment are unlikel to be accompanied b bad returns for another. If ou are risk averse, ou should think about diversifing our portfolio! Third, because man investors are risk-averse, the are willing to pa something for a financial agreement that insures them against certain kinds of financial risks. As a result, modern financial markets contain institutions that specialize in accepting risk or, to put it another wa, that specialize in providing this tpe of insurance. The expansion of this tpe of institution in the mortgage market, along with the lack of regulation of such institutions, contributed to the financial crisis of That is a subject for a different set of notes altogether.

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