Time Value of Money. Financial Mathematics for Actuaries Downloaded from by on 01/12/18. For personal use only.

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1 Interest Accuulation and Tie Value of Money Fro tie to tie we are faced with probles of aking financial decisions. These ay involve anything fro borrowing a loan fro a bank to purchase a house or a car; or investing oney in bonds, stocks or other securities. To a large extent, intelligent wealth anageent eans borrowing and investing wisely. Financial decision aking should take into account the tie value of oney. It is not difficult to see that a dollar received today is worth ore than a dollar received one year later. The tie value of oney depends critically on how interest is calculated. For exaple, the frequency at which the interest is copounded ay be an iportant factor in deterining the cost of a loan. In this chapter, we discuss the basic principles in the calculation of interest, including the siple- and copound-interest ethods, the frequencies of copounding, the effective rate of interest and rate of discount, and the present and future values of a single payent.

2 2 CHAPTER Learning Objectives Basic principles in calculation of interest accuulation Siple and copound interest Frequency of copounding Effective rate of interest Rate of discount Present and future values of a single payent. Accuulation Function and Aount Function Many financial transactions involve lending and borrowing. The su of oney borrowed is called the principal. To copensate the lender for the loss of use of the principal during the loan period the borrower pays the lender an aount of interest. At the end of the loan period the borrower pays the lender the accuulated aount, which is equal to the su of the principal plus interest. We denote A(t) as the accuulated aount at tie t, called the aount function. Hence, A(0) is the initial principal and I(t) =A(t) A(t ) (.) is the interest incurred fro tie t to tie t, naely, in the tth period. For the special case of an initial principal of unit, we denote the accuulated aount at tie t by a(t), which is called the accuulation function. Thus, if the initial principal is A(0) = k, then A(t) =k a(t). This assues that the sae accuulation function is used for the aount function irrespective of the initial principal..2 Siple and Copound Interest Equation (.) shows that the growth of the accuulated aount depends on the way the interest is calculated, and vice versa. While theoretically there are

3 Interest Accuulation and Tie Value of Money 3 nuerous ways of calculating the interest, there are two ethods which are coonly used in practice. These are the siple-interest ethod and the copoundinterest ethod. For the siple-interest ethod, the interest earned over a period of tie is proportional to the length of the period. Thus the interest incurred fro tie 0 to tie t, for a principal of unit, is r t, wherer is the constant of proportion called the rate of interest. Hence the accuulation function for the siple-interest ethod is a(t) =+rt, for t 0, (.2) and A(t) =A(0)a(t) =A(0)( + rt), for t 0. (.3) In general the rate of interest ay be quoted for any period of tie (such as a onth or a year). In practice, however, the ost coonly used base is the year, in which case the ter annual rate of interest is used. In what follows we shall aintain this assuption, unless stated otherwise. Exaple.: A person borrows $2,000 for 3 years at siple interest. The rate of interest is 8% per annu. What are the interest charges for year and 2? What is the accuulated aount at the end of year 3? Solution: The interest charges for year and 2 are both equal to 2, = $60. The accuulated aount at the end of year 3 is 2,000 ( ) = $2,480. For the copound-interest ethod, the accuulated aount over a period of tie is the principal for the next period. Thus, a principal of unit accuulates to +r units at the end of the year, which becoes the principal for the second year. Continuing this process, the accuulation function becoes and the aount function is a(t) =(+r) t, for t =0,, 2,, (.4) A(t) =A(0)a(t) =A(0)( + r) t, for t =0,, 2,. (.5) Two rearks are noted. First, for the copound-interest ethod the accuulated aount at the end of a year becoes the principal for the following year.

4 4 CHAPTER This is in contrast to the siple-interest ethod, for which the principal reains unchanged through tie. Second, while (.2) and (.3) apply for t 0, (.4) and (.5) hold only for integral t 0. As we shall see below, there are alternative ways to define the accuulation function for the copound-interest ethod when t is not an integer. Exaple.2: ethod. Solve the proble in Exaple. using the copound-interest Solution: The interest for year is For year 2 the principal is so that the interest for the year is 2, = $60. 2, = $2,60, 2, = $ The accuulated aount at the end of year 3 is 2,000 ( ) 3 =$2, Copounding has the effect of generating a larger accuulated aount. The effect is especially significant when the rate of interest is high. Table. shows two saples of the accuulated aounts under siple- and copound-interest ethods. It can be seen that when the interest rate is high, copounding the interest induces the principal to grow uch faster than the siple-interest ethod. With copound interest at 0%, it takes less than 8 years to double the investent. With siple interest at the sae rate it takes 0 years to get the sae result. Over a 20-year period, an investent with copound interest at 0% will grow 6.73 ties. Over a 50-year period, the principal will grow by a phenoenal 7.39 ties. When the interest rate is higher the effect of copounding will be even ore draatic.

5 Interest Accuulation and Tie Value of Money 5 Table.: Accuulated aount for a principal of $00 5% interest 0% interest Year Siple Copound Siple Copound interest ($) interest ($) interest ($) interest ($) Frequency of Copounding Although the rate of interest is often quoted in annual ter, the interest accrued to an investent is often paid ore frequently than once a year. For exaple, a savings account ay pay interest at 3% per year, where the interest is credited onthly. In this case, 3% is called the noinal rate of interest payable 2 ties a year. As we shall see, the frequency of interest payent (also called the frequency of copounding) akes an iportant difference to the accuulated aount and the total interest earned. Thus, it is iportant to define the rate of interest accurately. To ephasize the iportance of the frequency of copounding we use r () to denote the noinal rate of interest payable ties a year. Thus, is the frequency of copounding per year and year is the copounding period or conversion period. Let t (in years) be an integer ultiple of, i.e., t is an integer representing the nuber of interest-conversion periods over t years. The interest earned over the next year, fro tie t to t +,is a(t) r () = a(t)r(), for t =0,, 2,.

6 6 CHAPTER Thus, the accuulated aount at tie t + is ( a t + ) [ ] = a(t)+ a(t)r() = a(t) + r(), for t =0,, 2,. By recursive substitution, we conclude [ ] t a(t) = + r(), for t =0,, 2,, (.6) and hence [ A(t) =A(0) + r() ] t, for t =0,, 2,. (.7) Exaple.3: A person deposits $,000 into a savings account that earns 3% interest payable onthly. How uch interest will be credited in the first onth? What is the accuulated aount at the end of the first onth? Solution: The rate of interest over one onth is =0.25%, so that the interest earned over one onth is, = $2.50, and the accuulated aount after one onth is, = $, Exaple.4: $,000 is deposited into a savings account that pays 3% interest with onthly copounding. What is the accuulated aount after two and a half years? What is the aount of interest earned over this period? Solution: The investent interval is 30 onths. Thus, using (.7), the accuulated aount is [, ] 30 =$, The aount of interest earned over this period is,077.78,000 = $77.78.

7 Interest Accuulation and Tie Value of Money 7 Exaple.5: paid quarterly. Solve the proble in Exaple.4, assuing that the interest is Solution: The investent interval is now 0 quarters. With =4, the accuulated aount is [, ] 0 =$,077.58, 4 and the aount of interest earned is $ When the loan period is not an integer ultiple of the copounding period (i.e., t is not an integer), care ust be taken to define the way interest is calculated over the fraction of the copounding period. Two ethods ay be considered. First, we ay extend (.6) and (.7) to apply to any t 0 (not necessarily an integer). Second, we ay copute the accuulated value over the largest integral interest-conversion period using (.7) and then apply the siple-interest ethod to the reaining fraction of the conversion period. The exaple below illustrates these two ethods. Exaple.6: What is the accuulated aount for a principal of $00 after 25 onths if the noinal rate of interest is 4% copounded quarterly? Solution: The accuulation period is 25 3 = 8.33 quarters. Using the first ethod, the accuulated aount is [ ] 8.33 = $ Using the second ethod the accuulated aount after 24 onths (8 quarters) is [ ] 8 = $08.29, 4 so that the accuulated aount after 25 onths is [ ] =

8 8 CHAPTER It can be shown that the second ethod provides a larger accuulation function for any non-integer t > 0 (see Exercise.4). As the first ethod is easier to apply, we shall adopt it to calculate the accuulated value over a non-integral copounding period, unless otherwise stated. At the sae noinal rate of interest, the ore frequent the interest is paid, the faster the accuulated aount grows. For exaple, assuing the noinal rate of interest to be 5% and the principal to be $,000, the accuulated aounts after year under several different copounding frequencies are given in Table.2. Note that when the copounding frequency increases, the accuulated aount tends to a liit. Let r denote the noinal rate of interest for which copounding is ade over infinitely sall intervals (i.e., so that r = r ( ) ). We call this copounding schee continuous copounding. For practical purposes, daily copounding is very close to continuous copounding. Fro the well-known liit theore (see Appendix A.) that [ li + r ] = e r (.8) for any constant r, we conclude that, for continuous copounding, the accuulation function (see (.6)) is [ a(t) = li + r ] t [ = li ( + r ) ] t = e rt. (.9) We call r the continuously copounded rate of interest. Equation (.9) provides the accuulation function of the continuously copounding schee at noinal rate of interest r. Table.2: Accuulated aount for a principal of $,000 with noinal interest rate of 5% per annu Frequency of Accuulated interest payent aount ($) Yearly, Quarterly 4, Monthly 2,05.6 Daily 365,05.27

9 Interest Accuulation and Tie Value of Money 9.4 Effective Rate of Interest As Table.2 shows, the accuulated aount depends on the copounding frequency. Hence, coparing two investent schees by just referring to their noinal rates of interest without taking into account their copounding frequencies ay be isleading. Different investent schees ust be copared on a coon basis. To this end, the easure called the effective rate of interest is often used. The annual effective rate of interest for year t, denoted by i(t), is the ratio of the aount of interest earned in a year, fro tie t to tie t, to the accuulated aount at the beginning of the year (i.e., at tie t ). It can be calculated by the following forula i(t) = I(t) A(t) A(t ) = = A(t ) A(t ) For the siple-interest ethod, we have i(t) = ( + rt) ( + r(t )) +r(t ) = a(t) a(t ). (.0) a(t ) r +r(t ), which decreases when t increases. For the copound-interest ethod with annual copounding (i.e., =), we have (denoting r () = r) i(t) = ( + r)t ( + r) t ( + r) t = r, which is the noinal rate of interest and does not vary with t. -copounding is used, the effective rate of interest is i(t) = [ + r() ] t [ ] (t ) + r() When [ ] [ ] (t ) = + r(), (.) + r() which again does not vary with t. Note that when >, [ ] + r() >r (), so that the effective rate of interest is larger than the noinal rate of interest.

10 0 CHAPTER For continuous copounding, we have exp( rt) exp[ r(t )] i(t) = = e r, (.2) exp[ r(t )] which again does not vary with t. As the effective rate of interest for the copound-interest ethod does not vary with t, we shall siplify the notation and denote i i(t), which is given by (.) or (.2). The effective rate of interest shows how uch interest is earned in one year with a -unit investent. It is an appropriate standardized easure to copare interest-accuulation schees with different copounding frequencies and noinal rates of interest. Exaple.7: Consider two investent schees A and B. Schee A offers 2% interest with annual copounding. Schee B offers.5% interest with onthly copounding. Calculate the effective rates of interest of the two investents. Which schee would you choose? Solution: The effective rate of interest of Schee A is equal to its noinal rate of interest, i.e., 2%. The effective rate of interest of Schee B is [ ] 2 =2.3%. 2 Although Schee A has a higher noinal rate of interest, Schee B offers a higher effective rate of interest. Hence, while an investent of $00 in Schee A will generate an interest of $2 after one year, a siilar investent in Schee B will generate an interest of $2.3 over the sae period. Thus, Schee B is preferred. Another advantage of the effective rate of interest is that, for investents that extend beyond one year, the calculation of the accuulated aount can be based on the effective rate without reference to the noinal rate. Exaple.8: For the investent schees in Exaple.7, calculate the accuulated aount after 0 years on a principal of $,000. Solution: The accuulated aount after 0 years for Schee A is,000 ( + 0.2) 0 =$3,05.85, and that for Schee B is,000 ( ) 0 =$3, Note that in the above exaple, the accuulated aount of Schee B is calculated without aking use of its noinal rate.

11 Interest Accuulation and Tie Value of Money Exaple.9: Solve the proble in Exaple.8 for an investent horizon of 2 years and 3 onths. Solution: The tie frae of the investent is 2.25 years. Thus, the accuulated aount for Schee A is and that for Schee B is,000 (.2) 2.25 =$4,007.94,,000 (.23) 2.25 =$4, Note that the calculation of the accuulated aount of Schee B is done without using the fact that its copounding frequency is onthly. Indeed, the use of the effective rate of interest enables us to treat all investents as if the interest is credited only once a year and the copounding forula (.4) is used for any t 0, with the noinal rate r replaced by the effective rate i. While i(t) defined in (.0) is a -period effective rate, the concept can be generalized to a n-period effective rate. Let us denote i(t, t + n) as the annual effective rate of interest in the period t to t + n, for an integer n >. Thus, the aount a(t) at tie t copounded annually at the rate of i(t, t + n) per year accuulates to a(t + n) at tie t + n,where a(t + n) =a(t)[ + i(t, t + n)] n, (.3) fro which we obtain [ ] a(t + n) n i(t, t + n) =. (.4) a(t) We now consider investent intervals shorter than one year. Suppose t <, the effective rate of interest in the period t to t + t, denoted by i(t, t + t) can be defined as a(t + t) a(t) i(t, t + t) =. (.5) a(t) Note that this is an effective rate over a period of t < and is not annualized. Exaple.0: Let the accuulation function be a(t) = 0.0t 2 +0.t +. Copute the annual effective rate of interest for the first 2 years and for the next 3 years. What is the effective rate of interest for the second half of year 3? We ay use this forula to copute the annualized effective rate of interest over any investent horizon of n>, which is not necessarily an integer.

12 2 CHAPTER Solution: We have a(2) = 0.0(2) 2 +0.(2) + =.24, and siilarly a(5) =.75. Thus, fro (.4) the annual effective rate of interest over the first 2 years is [ ] a(2) 2 i(0, 2) = =(.24) 0.5 =.36%, a(0) and the annual effective rate of interest over the next 3 years is i(2, 5) = [ ] a(5) a(2) 3 = [ ] 3 =2.7%. To copute the effective rate of interest for the second half of year 3, we note that a(2.5) =.325 and a(3) =.39, so that the answer is i(2.5, 3) = a(3) a(2.5) a(2.5) = =5.90%. Note that this is an effective rate of interest over a half-year period. In contrast, i(0, 2) and i(2, 5) are annualized effective rates over a period of 2 years and 3 years, respectively..5 Rates of Discount So far we have assued that the interest of a loan or investent is paid at the end of the period. There are, however, any financial transactions for which the interest is paid or deducted up-front. Indeed, a popular way of raising a short-ter loan is to sell a financial security at a price less than the face value. Upon the aturity of the loan, the face value is repaid. For exaple, a Treasury Bill is a discount instruent. For a discount security, the shortfall between the sale price and the face value is called the discount and it represents the interest of the loan. In such cases the noinal principal (face value) has to be adjusted to take account of the interest deducted. This adjustent will affect the rate of interest, as opposed to the rate of discount that is quoted. Thus, if the lender of a loan requires the interest, which is calculated at the quoted annual rate of discount, to be deducted fro the principal at the tie the proceed is released, and if the loan period is one year, the effective principal of the loan after the interest is deducted is Effective principal = Noinal principal ( Rate of discount).

13 Interest Accuulation and Tie Value of Money 3 Note that the effective principal is A(0) and the noinal principal is A(). If the quoted rate of discount is d, wehave A(0) = A()( d), (.6) and I() = A() A(0) = A()d. While the rate of discount is often quoted for such securities, it is iportant to copare it against the equivalent effective rate of interest i over the period of the discount instruent, which is given by i = A() A(0) A(0) = A()d A()( d) = fro which we have d = i +i. Cobining (.4) and (.7), we can see that a(t) = (+i) t [ = + d d = ( d) t, ] t d d, (.7) which is the accuulated value of at tie t at the rate of discount d. It should be noted that the effective rate of interest i is always higher than the rate of discount d. For exaple, if the rate of discount is 6% and the redeption of the security is to be ade after year, the effective rate of interest is =6.38%. When the loan period is less than year, we should first calculate the rate of interest over the period of the loan and then calculate the effective rate of interest using the principle of copounding. Thus, suppose the period of loan is of a year, we denote the noinal rate of discount, which is the discount quoted in annual ter for the -year instruent, by d(). As the face value is to be repaid of a year later, the noinal principal is A ( and the interest deducted is ( ) ( ) I = A d (),

14 4 CHAPTER so that the effective principal is ( ) ( ) ( ) [ ] A(0) = A I = A d(), fro which we obtain the rate of interest charged over the -year period as ( ) A A(0) d () = = d() A(0) d() d (). Hence, the annualized equivalent noinal rate of interest is r () = d() d () = and the annual effective rate of interest is [ ] i = + r() =[ + d() d () d() d() ] =[, (.8) d() ]. (.9) As Treasury Bills that ature in less than a year are discount instruents, their effective rates should be coputed in the above anner. To copute the accuulation function a(t) for a discount instruent with aturity of year ( >), we observe, fro (.9), that a(t) = (+i) t [ ] t = d(), for t =0,, 2,. (.20) Furtherore, fro (.9) we have d() =(+i), (.2) so that fro (.8) we conclude r () =(+i) d (). (.22) For a discount security with a loan period longer than one year, (.6) can be odified to A(0) = A(t)( dt), <t,

15 Interest Accuulation and Tie Value of Money 5 provided dt <. Hence, we have A(t) = A(0), dt < <t, dt and a(t) =, dt < <t, (.23) dt which is the accuulation function for siple discount for dt < < t. Exaple.: The discount rate of a 3-onth Treasury Bill is 6% per annu. What is the annual effective rate of interest? What is the accuulated value of in 2 years? Solution: The rate of interest charged for the 3-onth period is =.52%. Therefore, the equivalent noinal rate of interest copounded quarterly is and the annual effective rate of interest is = 6.08%, (.052) 4 = 6.22%. The accuulated value of in 2 years is, using (.20), a(2) = [ 0.06 ] 8 =.3. 4 Note that this can also be calculated as ( + i) 2 =(.0622) 2 =.3. Exaple.2: The discount rate of a 6-onth Treasury Bill is 8% per annu, what is the annual effective rate of interest? What is the accuulated value of in 3 years?

16 6 CHAPTER Solution: The rate of interest charged for the 6-onth period is =4.7%. Therefore, the annualized equivalent noinal rate of interest copounded seiannually is = 8.34%, and the annual effective rate of interest is (.047) 2 =8.5%. The accuulated value of in 3 years is [ a(3) = 0.08 ] 6 =.28, 2 which can also be coputed as.6 Force of Interest ( + i) 3 =(.085) 3 =.28. Given any accuulation function a(t), the effective rate of interest over the tie interval (t, t + t) can be coputed as i(t, t + t) using (.5), which easures how fast the investent accuulates in value at tie t. The rate coputed by (.5), however, is not annualized. But if we divide i(t, t + t) by t, we obtain the rate of interest of the investent per unit tie in the interval (t, t + t) (if t is in years, the rate of interest is per annu). A particularly interesting question is the instantaneous rate, which is obtained when t is infinitesially sall. Thus, we consider i(t,t+ t) t when t tends to zero, which is given by [ ] a(t + t) a(t) = li t 0 t a(t) i(t, t + t) li t 0 t = a(t) li t 0 [ ] a(t + t) a(t) t = a (t) a(t), (.24)

17 Interest Accuulation and Tie Value of Money 7 where a (t) is the derivative of a(t) with respect to t. Thus, we define δ(t) = a (t) a(t), (.25) which is called the force of interest. As (.24) shows, the force of interest is the instantaneous rate of increase of the accuulated aount, a (t), as a percentage of the accuulated aount at tie t, a(t). Given a(t), the force of interest δ(t) can be coputed using (.25). Now we show that the coputation can be reversed, i.e., given δ(t) we can copute a(t). First, we note that (.25) can be written as fro which we have t 0 δ(t) = δ(s) ds = d ln a(t), dt t 0 d ln a(s) = lna(s) ] t 0 = lna(t) ln a(0) = lna(t), as a(0) =. Hence, we conclude ( t ) a(t) =exp δ(s) ds. (.26) 0 Equation (.26) provides the ethod to calculate the accuulation function given the force of interest. In the case when the force of interest is constant (not varying with t), we denote δ(t) δ, and the integral in (.26) becoes δt. Thus, we have a(t) =e δt. (.27) Coparing (.27) with (.9) we can see that if the force of interest is constant, it is equal to the continuously copounded rate of interest, i.e., r = δ. We now derive the force of interest for the siple- and copound-interest ethods. For the siple-interest ethod, we obtain, fro (.2), a (t) =r, so that δ(t) = r, for t 0. (.28) +rt

18 8 CHAPTER Hence, δ(t) decreases as t increases. In other words, the instantaneous rate of interest as a percentage of the accuulated aount drops with tie. For the copound-interest ethod, we have, fro (.4) and Appendix A.6 (with i replacing r irrespective of the copounding frequency), a (t) =(+i) t ln( + i), so that δ(t) = ( + i)t ln( + i) ( + i) t =ln(+i). (.29) Thus, δ(t) does not vary with tie and we write δ(t) δ, fro which equation (.29) iplies e δ =+i. Coparing this with (.2), we again conclude that the force of interest is the continuously copounded rate of interest. Exaple.3: A fund accuulates at a siple-interest rate of 5%. Another fund accuulates at a copound-interest rate of 4%, payable yearly. When will the force of interest be the sae for the two funds? After this tie, which fund will have a higher force of interest? Solution: Fro (.28), the force of interest of the siple-interest fund at tie t is δ(t) = t. Fro (.29), the force of interest of the copound-interest fund is ln(.04) at any tie. The two funds have the sae force of interest when t =ln(.04), i.e., 0.05 ln(.04) t = =5.4967, 0.05 ln(.04) after which the force of interest of the siple-interest fund reains lower than that of the copound-interest fund. Exaple.4: If a fund accuulates at force of interest δ(t) =0.02t, find the annual effective rate of interest over 2 years and 5 years.

19 Interest Accuulation and Tie Value of Money 9 Solution: Fro (.26), we have ( 2 ) a(2) = exp 0.02sds 0 ( =exp 0.0s 2] 2 0 ) = e We solve the annual effective rate of interest i over the 2-year period fro the equation (copare this with (.3)) ( + i) 2 = e 0.04 to obtain i = e 0.02 =2.02%. Siilarly, ( a(5) = exp 0.0s 2] ) 5 = e 0.25, 0 so that the annual effective rate of interest i over the 5-year period satisfies and we obtain.7 Present and Future Values ( + i) 5 = e 0.25 i = e 0.05 =5.3%. At the effective rate of interest i, a -unit investent today will accuulate to ( + i) units at the end of the year. In this respect, the accuulated aount ( + i) is also called the future value of at the end of the year. Siilarly, the future value of at the end of year t is ( + i) t. This is the aount of oney you can get t years later if you invest unit today at the effective rate i. Soeties it ay be desirable to find the initial investent that will accuulate to a targeted aount after a certain period of tie. For exaple, a +i-unit payent invested today will accuulate to unit at the end of the year. Thus, +i is called the present value of to be paid at the end of year. Extending the tie frae to t years, the present value of due at the end of year t is (+i). t Exaple.5: Given i =6%, calculate the present value of to be paid at (a) the end of year, (b) the end of year 5 and (c) 6.5 years. Solution: (a) The present value of to be paid at the end of year is =

20 20 CHAPTER The answers to (b) and (c) are, respectively, ( ) 5 = and ( ) 6.5 = Exaple.6: An insurance agent offers a policy that pays a lup su of $50,000 five years later. If the effective rate of interest is 8%, how uch would you pay for the plan? Solution: The fair aount to pay for this policy is the present value of the lup su, which is equal to 50,000 (.08) 5 = $34, Exaple.7: A person wants to accuulate $00,000 eight years fro today to sponsor his son s education. If an investent plan offers hi 8% copounded onthly, what aount ust he invest today? Solution: We first calculate the effective rate of interest, which is [ ] 2 =8.30%. 2 The aount required today is 00,000 (.083) 8 = $52,84.6. We now denote v = +i, (.30) which is the present value of due year later. It is also called the discount factor, as ultiplying a payent at the end of the year by v gives its present value. Cobining (.7) and (.30), we obtain d = iv, (.3)

21 Interest Accuulation and Tie Value of Money 2 so that the present value of i is d. Also, v + d = +i + i =, (.32) +i which says that a unit payent at tie is the su of its present value and discount. Since a() = + i, (.30) can also be written as v = a(), which says that the present value of due year later is the reciprocal of the accuulation function evaluated at tie t =. For a general tie t, we denote v(t) as the present value of to be paid at tie t. Then v(t) = ( + i) t = a(t), (.33) which is the discount factor for payents at tie t. The above calculations apply to the case of the copound-interest ethod with a constant effective rate of interest. For a general accuulation function a( ), the discount factor for payents at tie t is v(t) = a(t). Thus, when the siple-interest ethod is used, the present value of due t years later at the noinal rate of interest r is v(t) = a(t) = +rt. (.34) Exaple.8: Find the su of the present values of two payents of $00 each to be paid at the end of year 4 and 9, if (a) interest is copounded seiannually at the noinal rate of 8% per year, and (b) the siple-interest ethod at 8% per year is used. Solution: We first calculate the discount factors v(4) and v(9). For case (a), the effective rate of interest is (.04) 2 =0.086, so that v(4) = (.086) 4 =0.7307

22 22 CHAPTER and v(9) = (.086) 9 = Hence, the present value of the two payents is For case (b), we have 00( ) = $ v(4) = = and v(9) = =0.584, so that the present value of the two payents is 00( ) = $ Note that as the accuulation function for siple interest grows slower than that for copound interest, the present value of the siple-interest ethod is higher. We now consider a payent of at a future tie τ. What is the future value of this payent at tie t>τ? The answer to this question depends on how a payent at a future tie accuulates with interest. Let us assue that any future payent starts to accuulate interest following the sae accuulation function as a payent ade at tie 0. 2 As the -unit payent at tie τ earns interest over a period of t τ until tie t, its accuulated value at tie t is a(t τ). However, if we consider a different scenario in which the -unit aount at tie τ has been accuulated fro tie 0 and is not a new investent, what is the future value of this aount at tie t? To answer this question, we first deterine the invested aount at tie 0, which is the present value of due at tie τ, i.e., a(τ). The future value of this investent at tie t is then given by a(t) a(t) = a(τ) a(τ). 2 We adopt this assuption for the purpose of defining future values for future payents. It is not claied that the current accuulation function applies to all future investents in practice.

23 Interest Accuulation and Tie Value of Money 23 We can see that the future values of the two investents, i.e., a -unit investent at tie τ versus a -unit aount at tie τ accuulated fro tie 0, are not necessarily the sae. However, they are equal if a(t τ) = a(t) a(τ) (.35) for t > τ > 0. Figure. illustrates the evaluation of the future values at tie t of these two investents. Note that copound-interest accuulation satisfies the condition (.35). The principal at tie τ accuulates interest at the rate of i per year, whether the principal is invested at tie τ or is accuulated fro the past. Specifically, we have a(t) ( + i)t = a(τ) ( + i) τ =(+i)t τ = a(t τ). In contrast, siple-interest accuulation does not satisfy condition (.35). The future value at tie t of a unit payent at tie τ is a(t τ) =+r(t τ). However, we have a(t) a(τ) = +rt +rτ = +rτ + r(t τ) +rτ =+ r(t τ) +rτ < +r(t τ) =a(t τ). The discrepancy is due to the fact that only the principal of a(τ) invested at tie 0 is paid interest in the period τ to t, although the principal has accuulated to unit at tie τ. On the other hand, if unit of payent is invested at tie τ, thisisthe aount that generates interest in the period τ to t. Figure.: Present and future values of unit payent at tie τ future value a(t) a(τ) Dollar a(τ) present value future value a(t τ) Tie 0 τ t

24 24 CHAPTER Exaple.9: Let a(t) =0.02t 2 +. Calculate the future value of an investent at t =5consisting of a payent of now and a payent of 2 at t =3.Youay assue that future payents earn the current accuulation function. Solution: The future value at tie 5 is a(5) + 2 a(2) = [0.02(5) 2 +]+2[0.02(2) 2 +]=3.66. Exaple.20: Let δ(t) =0.0t. Calculate the future value of an investent at t =5consisting of a payent of now and a payent of 2 at t =2. You ay assue that future payents earn the current accuulation function. Solution: which is We first derive the accuulation function fro the force of interest, ( t ) a(t) =exp 0.0sds =exp(0.005t 2 ). 0 Thus, a(5) = exp(0.25) =.33 and a(3) = exp(0.045) =.0460, fro which we obtain the future value of the investent as.8 Equation of Value a(5) + 2 a(3) = (.0460) = Consider a strea of cash flows occurring at different ties. The present value of the cash flows is equal to the su of the present values of each payent. To fix ideas, assue the payents are of values C j occurring at tie j =0,,,n,and copound-interest accuulation is adopted. If the annual effective rate of interest is i with corresponding discount factor v, the present value P of the cash flows is given by n P = C j v j (.36) j=0 and the future value F of the cash flows at tie n is n n F =(+i) n P =(+i) n C j v j = C j ( + i) n j. (.37) j=0 j=0

25 Interest Accuulation and Tie Value of Money 25 Thus, equations (.36) and (.37) involve the quantities P, F, C j, i and n. 3 We call these the equations of value. GivenC j, i and n, P and F can be calculated straightforwardly. Alternatively, if C j and i are given, we can calculate the tie n given the present value P or future value F. The exaples below illustrate this point. Exaple.2: At the annual effective rate of interest i, when will an initial principal be doubled? Solution: This requires us to solve for n fro the equation (note that C j =0for j>0, andweletc 0 =and F =2) ( + i) n =2, fro which n = ln(2) ln( + i). Thus, n is generally not an integer but can be solved exactly fro the above equation. To obtain an approxiate solution for n, we note that ln(2) = so that n = i i ln( + i). We approxiate the last fraction in the above equation by taking i = 0.08 to obtain n i.0395 = i Thus, n can be calculated approxiately by dividing 0.72 by the effective rate of interest. This is called the rule of 72. It provides a surprisingly accurate approxiation to n for a wide range of values of i. For exaple, when i =2%,the approxiation gives n = 36 while the exact value is 35. When i = 4%, the approxiate value is 5.4 while the exact value is Exaple.22: How long will it take for $00 to accuulate to $300 if interest is copounded quarterly at the noinal rate of 6% per year? 3 Note that as we are assuing copound interest, (.35) holds. Thus, we can copute the future value of future payents as the future value of their corresponding present values, as in equation (.37).

26 26 CHAPTER Solution: Over one quarter, the interest rate is.5%. With C 0 = $00, C j =0 for j>0, andf = $300, fro (.37) the equation of value is (n is in quarters) 300 = 00(.05) n, so that i.e., 8.45 years. n = ln(3) ln(.05) =73.79 quarters, Another proble is the solution of the rate of interest that will give rise to a targeted present value or future value with given cash flows. The exaple below illustrates this point. Exaple.23: A student takes out a tuition loan of $5,000, and is required to pay back with a step-up payent $7,000 in year and $8,500 in year 2. What is the effective rate of interest she is charged? Solution: The equation of value is, fro (.36), 5,000 = 7,000v +8,500v 2. Solving for the quadratic equation (see Appendix A.3), we obtain, after dropping thenegativeroot, which iplies v = =0.979, i = =2.4%. Note that the solution of the above proble can be obtained analytically by solving a quadratic equation, as there are only two payents. When there are ore payents, the solution will require the use of nuerical ethods, the details of which can be found in Chapter 4.

27 Interest Accuulation and Tie Value of Money 27 Exaple.24: A savings fund requires the investor to pay an equal aount of installent each year for 3 years, with the first installent to be paid iediately. At the end of the 3 years, a lup su will be paid back to the investor. If the effective interest rate is 5%, what is the aount of the installent such that the investor can receive a lup su of $0,000? Solution: Let k be the installent. Fro (.37), the equation of value is 0,000 = k[(.05) 3 +(.05) ] = 3.3k, so that the installent is k = 0, =$3, It can be observed that if the horizon of the savings fund is long, the coputation of the above can be quite tedious. In the next chapter, we will see how the coputation can be ade easier for installents over a long horizon..9 Suary. The aount function and accuulation function trace the accuulation of an investent over a period of tie. The interest incurred in each period can be calculated fro the aount function. 2. The siple-interest and copound-interest ethods are two coonly used ethods for calculating interest incurred. For the copound-interest ethod, the frequency of copounding is an iportant deterinant of how interest accrues. To copare schees with different copounding frequencies, the effective rate of interest should be used. 3. Discount instruents are popular in raising short-ter loans. The effective rate of interest is always higher than the discount rate as the effective principal is less than the noinal principal or face value. 4. The force of interest is the instantaneous rate of interest as a percentage of the accuulated aount. It is a useful easure of how investent accuulates, especially when the rate of interest is tie varying. If the force of interest is constant, it is equal to the continuously copounded rate of interest. 5. The present value of a strea of payents is equal to the su of the present values of each payent. It aggregates the payents by properly weighting each of the by a discount factor that takes account of the differences in the tie the payents are ade.

28 28 CHAPTER 6. The equation of value links the present value or the future value with the installents, the period of the payents and the interest rate. It can be used to solve for one variable given the others. 7. Table.3 suarizes the accuulated value and present value for various accuulation ethods. Exercises. An investor invests $20,000 into a fund for 4 years. The annual noinal interest rate reains at 8% in each year although it is convertible seiannually in the first year, quarterly in the second year, bi-onthly in the third year, and onthly in the fourth year. Find the accuulated value of the fund at the end of the fourth year..2 For the copound-interest ethod over fraction of a copounding period, show that the second ethod described in Section.3 provides a larger accuulation aount over the first ethod..3 If A(4) =,200 and i(t) =0.0t 2,findI(5) and A(6)..4 It is given that A(0) = 300,I() = 5,I(2) = 7,I(3) = 9 and I(4) = 4. Find A(3) and i(4)..5 Find the accuulated value of $,000 at the end of the fourth year (a) if the siple discount rate is 6% per annu, (b) if the siple interest rate is 6% per annu, (c) if the effective rate of interest is 6% per annu, (d) if the annual noinal interest rate is 6% payable quarterly, (e) if the annual noinal discount rate is 6% copounded onthly, (f) if the constant force of interest is 6% per annu..6 Assue that a(t) =ln(0.5t 2 + e)+0.05t 0.3,find (a) i(2) and i(3), (b) the aount of interest earned in the third period if the principal is $,200.

29 Table.3: Suary of accuulated value and present value forulas (all rates quoted per year) Freq of Rate of Future value Present value of Accuulation conversion interest or of at tie t due at tie t Equations ethod per year discount (in years): a(t) (in years): /a(t) in book Rearks Copound r (= i) ( + r) t ( + r) t (.4) also applies to interest non-integer t>0 [ Copound r () interest + r() ] t [ + r() ] t (.6) > if copounding is ore frequent than annually Copound r e rt e rt (.9) r is the constant interest force of interest δ Copound d ( d) t ( d) t (.4), (.7) discount d() d() Copound d () [ ] t [ ] t (.9) >for loan period discount shorter than year Siple r +rt ( + rt) (.2) interest Siple d ( dt) dt (.23) discount Interest Accuulation and Tie Value of Money 29

30 30 CHAPTER.7 It is given that i(t) = t. (a) Find a(t) for t beinganinteger. (b) If the principal is $00, find the total aount of interest earned in year 3, 4 and 5..8 Which of the following is a valid accuulation function? (a) a(t) =t 2 +2t +2, (b) a(t) =t 2 2t +, (c) a(t) =t 2 +2t +..9 If d (4) =0.0985, find (a) i, (b) d (2), (c) r (4), (d) δ, (e) a(t)..0 A security is sold at a discount of 7%, copounded seiannually. Find the effective rate of interest.. Prove that i d = id by using forulas (.30) and (.3). What does this relation tell you?.2 $,500 is deposited into a savings account that pays 3.5% interest with onthly copounding. What is the accuulated aount after four and a quarter years? What is the aount of interest earned over this period?.3 $0,000 is deposited into a savings account that earns 3% per annu copounded quarterly. (a) How uch interest will be credited in the second onth? (b) How uch interest will be credited in the second year?.4 Using the rule of 72, calculate the approxiate nuber of years for an initial principal to double if the annual effective rate of interest is 4%. Copare your answer with the true value.

31 Interest Accuulation and Tie Value of Money 3.5 Find the present value of $500 to be paid at the end of 28 onths. (a) if the noinal interest rate is 6% convertible onthly, (b) if the noinal discount rate is 6% copounded every 4 onths, (c) if the noinal interest rate is 4% copounded seiannually, (d) if the noinal discount rate is 5% payable annually. Apply the siple-interest ethod for the reaining fraction of the conversion period if necessary..6 When does the rule of 72 hold exactly?.7 You wish to borrow $0,000. The following is the inforation you gathered fro two lenders: (a) Lender A charges 7% copounded quarterly, (b) Lender B charges an annual effective rate of interest of 7.25%. Which lender would you choose?.8 A Treasury Bill for $00 is purchased for $95 four onths before due. Find the noinal rate of discount convertible three ties a year earned by the purchaser..9 You are given the following strea of cash flows: t C t Find the present value and the future value (after 5 years) of the cash flows evaluated at i =2%..20 (a) Extend forula (.36) to the case when the interest rate is not constant over tie.

32 32 CHAPTER (b) It is given that t v(t) Find the present value of the following strea of payents:.2 Let v(t) = t. (a) Find i(),i(2) and i(3). (b) How is the interest credited? t C t ($) 0, Beda wishes to accuulate $3,000 in a fund at the end of 4 years. He akes deposits of $K, $2K, $3K at the beginning of year, 2 and 3, respectively. Find the value of K if the fund earns (a) siple interest of 5% per year, (b) copound interest of 5% per year..23 Eddy deposits $2,000 into a savings account. The bank credits interest at arateofi convertible yearly for the first 4 years and a noinal rate of 4i convertible quarterly thereafter. If the accuulated value of the account is $2,98.70 after 6 years, find the accuulated value of the account after 8.5 years..24 Ada pays $,000 today and $,200 after 2 years to Andrew in exchange for a payent of $2,98 one year fro today. (a) Set up an equation of value with i as unknown by equating the future values of the two cash-flow streas. (b) Solve the equation of value obtained in (a). (c) What can you observe?

33 Interest Accuulation and Tie Value of Money $2,000 is put into an account which credits interest at 4% per annu and $,500 is put into another account which credits interest at 6% per annu. How long does it take for the balances of the two accounts to be equal? Apply the copound-interest ethod for the reaining fraction of the conversion period..26 $2,000 is put into an account which credits interest at 4% per annu and $,500 is put into another account which credits interest at 6% per annu. How long does it take for the balances of the two accounts to be equal? Apply the siple-interest ethod for the reaining fraction of the conversion period. Copare your answer with that in Exercise Which of the following are valid graphs for the aount function A(t)? A(t) A(t) t t.28 Irene invests $5,000 in an account that credits interest during the first two years at a noinal interest rate of r (2) convertible seiannually. During the third year, the account earns interest at a noinal discount rate of d (4) convertible quarterly. At the end of the third year, the fund has accuulated to $5, If r (2) = d (4) = x,findx. A(t) A(t) t t.29 Betty deposits $200 at the beginning of each year for 0 years into an account which credits siple interest at an annual interest rate of i payable every year. Find i if the accuulated value is $2,440 after 0 years..30 You are given the following inforation:

34 34 CHAPTER t v(t) (a) You want to accuulate $0,000 five years fro today. What aount should you invest today? (b) You want to accuulate $0,000 five years fro today by two installents of equal aount, to be paid at the beginning of the first and the third year. How uch are the two installents? You ay follow the assuption in Section.7 for the coputation of the future value of future payents..3 The present value of two payents, the first payent of $500 paid at the end of n years, and the second payent of $,000 at the end of 2n years, is $,58. If i =4%, find the value of n..32 Albert takes out a loan of $30,000, and is required to pay back with a payent of $20,000 at the end of the second year, and $,000 at the end the fourth year. Interest is payable quarterly. What is the noinal rate of interest charged on the loan?.33 Let a(t) = for 0 t< t (a) Draw the graph of v(t) for 0 t<00. (b) How is the interest credited?.34 Let A(t) =t 2 +2t +4.Findδ(5)..35 Which of the following are properties of the force of interest δ(t)? (a) It is non-negative. (b) It is increasing with t. (c) It is continuous..36 Assue that δ(t) = and A(0) = 00. Find the aount of interest 0( + t) 3 earned in the fifth year.

35 Interest Accuulation and Tie Value of Money Calculate the future value of an investent at t =5consisting of a payent of at t =2and a payent of 3 at t =4,if (a) a(t) =+0.05t, (b) a(t) =(.05) t, (c) δ(t) =0.05t..38 In Fund P, oney accuulates at a force of interest δ(t) whose graph is given below (t) (a) Find a(5). (b) Fund Q accuulates at a constant force of interest δ Q.Ifthevalueof the two funds are equal at t =0and t =5,findδ Q..39 You are given the following strea of cash flows: t C t t Find the present value of the cash flows given that the noinal rate of interest is 8% copounded half-yearly..40 Certificates of deposit (CDs) are short- to ediu-ter investent instruents issued by banks which provides a fixed rate of interest for a period of tie. CDs offer stability in interest earned but have penalty for early withdrawal. A -year CD pays a noinal rate of interest of 8% copounded quarterly. You are offered two options of penalty for early withdrawal:

36 36 CHAPTER (a) loss of 3-onth interest; (b) a reduction in the noinal rate of interest to 6%. If you wish to withdraw after 9 onths, which option would you choose? Advanced Probles.4 (a) For c>0, letf(x) =+cx and g(x) =(+c) x,where0 x. () Show that f(0) = g(0), andf() = g(). (2) By using the fact that g( ) is increasing in x, show that f(x) g(x) for 0 x. (3) Letting c = r in (a)(2), what conclusion can you draw? (b) Let [x] be the integer part of x. For exaples, [4.2] = 4, [9.99] = 9 and [] =. By using the result in (a)(2), show that for t 0, ( ) t [t] ( ) + r() t [t] +r (). (c) Prove that when the loan period is not an integer ultiple of the copounding period, applying the siple-interest ethod for the reaining fraction of the conversion part always gives a larger accuulation function when copared with extending (.6) to any t The following graphs plot the force of interest for Fund X and Fund Y for 0 t 0. X (t) Y (t) t 2 0 t Find the points in tie at which the values of the accuulation functions of the two funds are equal.

37 Interest Accuulation and Tie Value of Money Assue that the force of interest is constant over tie. Let the constant force of interest be δ and the equivalent effective rate of interest and discount be i and d, respectively. (a) Express δ in ters of i. (b) Express δ in ters of d. (c) By using the Taylor s expansion ln( + x) = ( ) k x k k= k = x x2 2 + x3 3 x4 4 +, and the expressions in (a) and (b), show that δ = [ i + d i2 d 2 + i3 + d 3 + i4 d ] +. The forula above shows that δ is very close to the average of i and d.