2.6.3 Interest Rate 68 ESTOLA: PRINCIPLES OF QUANTITATIVE MICROECONOMICS

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1 68 ESTOLA: PRINCIPLES OF QUANTITATIVE MICROECONOMICS where price inflation p t/pt is subtracted from the growth rate of the value flow of production This is a general method for estimating the growth rate of flow of production from the growth rate of value flow of production, that is, growth rate of real flow from the nominal one 263 Interest Rate Let x measure the amount of money deposited on a bank account on which the bank pays interest The next example shows how the amount of deposited capital affects the interest revenues Example Let the annual interest rate the bank pays on its deposits be 4 %/y The annual interest revenues for capitals 00 e and 000 e are then 4 e/y and 40 e/y, respectively Due to this dependence of interest revenues on the amount of deposited capital, it is practical to measure the strength of change in the capital so that the amount of capital does not affect this measuring This can be done by relating the flow of money to the invested capital, which corresponds to the growth rate of the capital Example 2 The interest earnings in Example were 4 e/y and 40 e/y Dividing these by the corresponding capitals we get the growth rates of the capitals as 4 e/y/00 e = 004 /y and 40 e/y/000 e = 004 /y The growth rates are thus equal: 004 /y or 4 %/y Definition: Interest rate is the growth rate of a monetary quantity measured in units / t By interest rate we measure the strength of the growth of deposited, borrowed, or invested money It is common in economics to compare interest rates and the rates of return of financial and real investments, inflation, the growth rate of Gross Domestic Product GDP of an economy, etc This is legitimate because the rate of return of an investment is a quantity where the revenues from the project a flow of money in a time unit are divided by the invested capital If interest revenues are added in the invested capital, the obtained rate of return corresponds to the growth rate of the capital The growth rates of all quantities measured from time intervals of equal length can be directly compared to see which of the quantities has had the fastest growth Example 3 Let W e/h be nominal wage and p e/kg average price level in an economy Real wage W t/pt kg/h then measures the exchange rate between one hour of work and the amount of goods in the economy The

2 ESTOLA: MEASURING IN ECONOMICS 69 growth rate of the real wage during time unit t t 0 is: W t W t 0 pt pt 0 /t t 0 W t 0 = W t pt 0 t t 0 W t 0 pt = t t 0 pt 0 W t pt 0 W t 0 pt W t 0 pt 20 If, on the other hand, we calculate the growth rate of the nominal wage during the time unit, and subtract from this the growth rate of the average price level, we get almost equal result: [W t W t 0 ]/t t 0 [pt pt 0 ]/t t 0 W t 0 pt 0 = W t pt 0 W t 0 pt 2 t t 0 W t 0 pt 0 The difference in these two ways of measuring is in the denominator The smaller the change in the average price level, the more accurately 2 approximates that in 20 If the growth rate of the average price level and that of the nominal wage are equal, the real wage stays constant In continuous time, this approximation becomes exact: d dt W t pt W t pt = W t pt W tp t p 2 t W t pt = W t W t p t pt This example shows that the development in real wage can be approximated by subtracting inflation from the growth rate of nominal wage, when these are measured from the same time unit This result holds also for other real quantities obtained from nominal quantities by dividing them by an average price level; especially this result holds for interest rates Definition: When interest rate is measured as the growth rate of a monetary quantity, its is called nominal interest rate Real interest rate, on the other hand, measures the growth rate of purchasing power of the corresponding currency Let us denote the amount of money on a bank account at time moments t 0, t by xt 0 e, xt e, t > t 0, respectively, when only interest revenues are assumed to be added in the capital The interest revenues from period t t 0 are then xt xt 0 e The nominal interest rate r measured in units / t during time unit t t 0 is then: r = xt xt 0 t t 0 xt 0 = t t 0 xt xt 0 xt 0

3 70 ESTOLA: PRINCIPLES OF QUANTITATIVE MICROECONOMICS Let average price level in the economy be pt e/kg at moment t The growth rate of the average price level, ie the inflation i with unit / t at time unit t t 0, is then: i = pt pt 0 t t 0 pt 0 = t t 0 pt pt 0 pt 0 Real interest rate r R with unit / t measures the growth rate of the purchasing power xt kg of money xt e: pt r R = xt pt xt 0 pt 0 t t 0 xt 0 pt 0 = t t 0 pt0 xt pt xt 0 pt xt 0 22 Real interest rate can be approximated by subtracting inflation from the nominal interest rate: r i = pt0 xt pt xt 0 23 t t 0 pt 0 xt 0 The deviation between the approximation r i and the accurate measure r R is in the denominator; the smaller the difference pt pt 0, the more accurate the approximation The reader can check that similarly as in the case of real wage, in continuous time this approximation becomes exact Nominal lending and saving interest rates are positive because the lender saver requires a compensation for his lending saving In an inflationary economy, the purchasing power of the domestic currency decreases with time For the purchasing power of the lent saved money not to decrease with time, the nominal interest rate must be at least as great as inflation If this does not hold, lenders savers are not interested in lending saving money 27 Interest Calculation and Discounting In economic analysis, interest and discount 3 calculation have been defined in discrete and continuous time We start with the discrete one 27 Discounting in Discrete Time Suppose that at time moment t 0, the amount of xt 0 e is deposited on a bank account with fixed interest rate In order to separate the discrete 3 Name discount comes from the reduction in the future debt payments a debtor receives if he repays his debt before the terminal date

4 ESTOLA: MEASURING IN ECONOMICS 7 and continuous time calculations, in the following we denote discrete time interest rate by r d and continuous time interest rate by r Interest rate r d with unit / t is the following growth rate of deposited capital: [xt 0 + t xt 0 ]/ t = r d xt 0 + t xt 0 = r d txt 0 xt 0 xt 0 + t = xt 0 + r d txt 0 = + r d txt 0 The last form of the equation shows that the money on the bank account at moment t 0 + t, xt 0 + t, can be expressed by the interest rate r d Assuming r d to be fixed, we can calculate how a monetary capital increases during n time intervals when interest revenues are added in the capital at the last moment of every time unit We call this compound interest calculation Definition: By compound interest calculation we understand the calculation of interest returns in the way that interest revenues from one time unit are added into the invested capital after every time unit Table 24 Discrete time interest calculation time moment money on a bank account t 0 xt 0 t 0 + t xt 0 + t = + r d txt 0 t t xt t = + r d t 2 xt 0 t 0 + n t xt 0 + n t = + r d t n xt 0 The money on the bank account at the last moment of every time interval is shown in Table 24 The deposit at moment t 0, xt 0, grows during time unit t so that at the end of the time unit, the money on the bank account is xt 0 + t = xt 0 + r d txt 0 = + r d txt 0, where xt 0 is the invested capital and r d txt 0 the interest revenues from the time unit The dimensionless term + r d t, where the measurement units of r d and t cancel each other, is called the interest factor for one time unit At time moment t t the capital is, similarly, xt t = + r d txt 0 + r d t + r d txt 0 = + r d t 2 xt 0, where + r d txt 0 is the capital in the beginning of the time unit and r d t + r d txt 0 the interest revenues from the time unit Next we define the amount of money at time moment t 0, xt 0, which corresponds to xt 0 + t at moment t 0 + t, as + r d txt 0 = xt 0 + t xt 0 = + r d t xt 0 + t

5 72 ESTOLA: PRINCIPLES OF QUANTITATIVE MICROECONOMICS We call xt 0 the present value of xt 0 + t and / + r d t the discount factor between time moments t 0 and t 0 + t Discount factor is a dimensionless quantity like the interest factor In general, the discount factor between moments t 0 and t 0 + n t is / + r d t n If the interest rates of time units differ, r d r d2, where r dj is the interest rate of the j th time unit, the interest and discount factors between t 0 and t 0 + n t are: + r d t + r dn t and + r d t + r dn t 24 The reader can check this by constructing a table similar to Table 24 in the case the interest rates of the time units vary Example Let interest rate be 0 %/y = 0 /y and t = y Then, setting xt 0 + t = xt t = xt t = e we get the corresponding present values as: / + 0 = 09 e, / = 083 e and / = 075 e If interest rate is 5 %/y and t = y, the present values are: 095 e, 09 e and 086 e, respectively An increase in interest rate thus decreases present values Dimensionally, monetary quantities at every time unit belong in the dimension of money and are thus additive Positive interest rate means, however, that the value of one euro at a future time moment is smaller than that at current moment The reason for this is the interest revenues to be obtained for deposited money Current euro corresponds in the future to one euro plus interest revenues for one euro Definition: Interest and discount factors are transformation rules between monetary quantities at different time moments They work identically as the transformation equations between measurement units belonging in one dimension For example, the transformation equation between euros at moments t t and t 0 + t is xt t xt 0 + t = + r d t 4 xt 0 + r d t xt 0 xt t = + r d t 3 xt 0 + t etc xt t xt 0 + t = + r d t 3 Interest and discount calculation can be defined for real values too Suppose x e is deflated by average price level p of the economy The growth rate of the real quantity xt/pt then corresponds to the real interest rate r R : xt0 + t xt 0 pt 0 + t pt 0 xt 0 pt 0 / t = r R xt 0 + t pt 0 + t = xt 0 pt 0 + r R t 25

6 ESTOLA: MEASURING IN ECONOMICS 73 This can be seen by comparing equations 25 and 22 and remembering that t = t 0 + t or t = t t 0 The latter form of 25 defines the transformation equation between the values of real quantity x/p at different time units Quantity xt 0 + t/pt 0 + t measures the purchasing power of xt 0 + t at moment t 0 + t, that is, the amount of goods in the economy that can be obtained by x e at moment t 0 + t It is a matter of taste whether discounting is made in nominal or in real terms Essential is that if we operate with nominal real terms, the interest rate to be used in discounting is the nominal real one 272 Interest Calculation in Continuous Time* Analogously with discrete time, continuous time interest rate rt is defined as the instantaneous growth rate of a monetary quantity at time moment t, rt = x t/xt The solution of this differential equation is x t t t = rtxt xt = Ae rsds 0, 26 where A e is the integration constant, by s is denoted running time during the time interval t 0, t, and e is the base of the natural logarithm The reader can check the above result by taking the time derivative of the latter equation, substituting in this the solved function for xt, and verifying that the latter equation equals the former Setting t = t 0 in 26 we get: t t xt 0 = A xt = xt 0 e rsds 0 27 Eq 27 shows the money on the bank account at time moment t when xt 0 has been deposited at moment t 0 with continuous time interest rate r that may vary during t 0, t In continuous time compound interest calculation, interest revenues are added in the capital after every instant of time; this creates the exponential growth When t = t 0, xt = xt 0 in 27 Using the definition rs = x s/xs we get: t t 0 rsds = t We can thus express 27 as t 0 x s xs ds = t lnxs = ln t 0 xt = xt 0 e ln xt xt 0, xt xt 0 which is trivially true because the inverse operations exp and ln cancel each other The present value of xt at moment t 0 is obtained by solving

7 74 ESTOLA: PRINCIPLES OF QUANTITATIVE MICROECONOMICS equation 27 with respect to xt 0 : xt 0 = xte t t 0 rsds If the interest rate is constant during time interval t 0, t, we get: xt 0 = xte r t t 0 xt = xt 0 e r t t 0 28 The first form of 28 expresses the present value of xt at moment t 0, and the latter is the corresponding compound interest formula The continuous time correspondent for the discrete time discount factor + r d t n with constant interest rate is thus e r t t 0, and the corresponding interest factor is e r t t 0 These interest and discount factors are dimensionless because the measurement unit of r is / t and time unit t t 0 is measured in units t Remark! One essential difference exists between the two discount factors In the continuous time discount factor, the length of time interval t t 0 is measured in time units In the discrete time factor, however, exponent n is a pure number that represents the order of the time interval If time is measured in years in discrete analysis, then n = 3 implies that the discount factor is that between the initial and the third year etc 273 Parities Between Interest Rates What is the correct way to calculate interest revenues? There is no a unique answer to this question because banks calculate interest on their deposits in different ways Common is to take the end-of-the-day or the smallest amount of money on a bank account in a month as the capital for which the interest is calculated Discrete time interest returns are calculated for the deposited capital either on daily or monthly basis, and interest revenues are added in the capital usually once a year Some banks apply also continuous time interest calculation Thus the interest calculation principles vary between banks and between different accounts in a bank For this reason we present next conversation rules that make interest rates comparable when different kind of compound interest calculation methods are applied Discrete and Continuous Time Interest Rate* According to the previous sections, with compound interest calculation the discrete and continuous time calculated values of capitals at time moment t n, deposited on a bank account at moment t 0, are: xt n = xt 0 + r d t n tn and xt n = xt 0 e rtdt t0

8 ESTOLA: MEASURING IN ECONOMICS 75 Solving xt n /xt 0 from both equations and setting them equal, gives: + r d t n tn = e rtdt t0 29 Both methods give thus identical results if the two interest and discount factors are equal Suppose then that the two interest rates with unit / t are constant The amount of money on the bank account after time interval t, calculated by both methods, is then: xt = xt 0 + r d t and xt = xt 0 e r t t 0 Setting these equal and dividing by xt 0, gives: ln + r d t = r t t 0 or r d t = e r t 220 By using transformation 220 the continuous and discrete time compound interest calculation give identical results notice that t t 0 = t and r d t is a dimensionless quantity the numerical value of which equals with r d We can thus define the continuous time interest rate r c for time unit t, conformal with the corresponding discrete time rate, as: r c = t t 0 ln xt xt 0 = ln + r d t 22 t Notice that in 22, xt is calculated by using the corresponding discrete time interest rate and discrete time method of calculation Example We calculate the capital on a bank account by discrete and continuous time methods when 00 e is deposited at moment t 0 = 0, t n t 0 = 20 y, and r d = r = 0 /y The time paths of the capitals are graphed in Figure 22 The continuous time capital the curve somewhat overestimates the discrete time one the dots However, using r c = ln + r d t/ t = /y in continuous time the two time paths coincide, see Figure 23 Hull 2000 p 5-2 shows that even without the adjustment r = r c the continuous time interest calculation gives almost identical results as the daily discrete time analysis with normal levels of interest rates Discrete Time Interest Rates of Varying Time Units Another parity can be made between discrete time interest rates of varying time units Let us denote the discrete time interest rate for time unit k by r dk, and that for time unit z by r dz, where s = k/z k = sz is the transformation equation between the time units For example, y =

9 76 ESTOLA: PRINCIPLES OF QUANTITATIVE MICROECONOMICS 2 mn, where k = y, z = mn, and s = 2 In both cases, interest returns are added in the deposited capital at the ending moment of the corresponding time unit Let s > which implies k > z; r dz thus represents a more dense splitting of time, that is, the interest revenues are added in the capital more frequently In monthly annual compound interest calculation, interest revenues are added in the capital after every month year The deposited capital xt 0 e increases in both cases during time unit k as: xt k = xt 0 + r dk k and xt sz = xt 0 + r dz z s Setting these two capitals equal and dividing by xt 0 gives: r dc,k k = + r dz z s r dc,z z = + r dk k /s 222 These transformation rules make any two discrete time interest rates comparable with each other, because relation k = sz holds for every two time units k, z Eq 222 thus defines two discrete time interest rates conformal with each other in compound interest calculation Example Suppose r dy = 0 /y, r dm = 00 /m and y = 2 m Using the rules 222, the corresponding conformal monthly and annual interest rates are r dc,m = /m and r dc,y = 0268 /y, respectively These somewhat differ from the approximate ones we get by transforming simply by using the time units: r dm = /2 y/m 0 /y = /m and r dy = 2 m/y 00 /m = 02 /y Example 2 Figure 24 shows the differences in the capitals during 20 years when 00 e is deposited at time moment 0 and interest rates r dy = 0 /y, r dm = /m are applied the curve refers to monthly and dots to annual analysis This difference disappears when conformal monthly interest rate r dc,m = /m is applied, see Figure Present Values of Money Flows 28 Present Values in Discrete Time In calculating the present values of money flows, we transform the monetary quantities at various time moments comparable with current money, see Section 26 We assume money flow Nt 0 + j t e/ t, j =, 2,, where the money is assumed to be received at the ending moment of every time unit t, and a fixed interest rate r d / t In discrete time, the present

10 ESTOLA: MEASURING IN ECONOMICS 77 value of an n period flow at time moment t 0 is P t 0 = Nt 0 + t t + Nt t t + + Nt 0 + n t t + r d t + r d t 2 + r d t n n j = Nt 0 + j t t + r j= d t n = Nt 0 + j t t + r d t j, 223 j= where Nt 0 + j t t has unit e j means for all To compare the discrete and continuous time present values, we assume that the money flow is fixed, that is, Nt 0 + j t = N j We can then take the common factor N t of every term in front of the sum 223 and study the obtained sum of positive terms a j, a j = + r d t j, j =, 2,, n, 0 < a < because r d > 0 This sum is called a geometric series with positive terms Let us denote the sum of the series of n terms as: n S n = a + a 2 + a a n = a j Multiplying this sum every term of the sum by a, we get as n = a 2 + a 3 + a a n+ Subtracting these two series from each other, we can solve the sum of the geometric series of n terms as: S n as n = a a n+ as n = a a n S n = a an a 224 A geometric series with positive terms converges when 0 a < The sum of a convergent geometric series with an infinite number of terms is: lim n S n = a/ a because lim n a n = 0 when 0 a < We can then express the present value P t 0 with n time units as where a a n a P t 0 = N t a an a = +r d t +r d t n +r d t = + r d t n r d t j= = N r d [ + r d t n ], 225 = [ + r +r d t d t n ] +r d t +r d t

11 78 ESTOLA: PRINCIPLES OF QUANTITATIVE MICROECONOMICS A quick test for the correctness of the last form of 225 is made by dimensional analysis The measurement unit of the left hand side is e, that of N/r d is e, and the term in parenthesis is dimensionless At least the measurement units match; hopefully also the form of the equation Formula 225 is a complicated connection of four quantities: P t 0, N, r d and n There exists computer programs that solve r d from 225 in a numerical way to see what is the yield the internal rate of return per annum of an asset with initial price P t 0 = 0000 e that pays N t = 300 e at the end of every year in n = 0 years This yield can be compared with the prevailing risk-free interest rate in the economy to see whether this asset is competitive in creating revenues Another way to use Eq 225 is to calculate the necessary annual payment N related to an n period fixed flow with initial price P t 0 and discount rate r d as: N = r d P t r d t n One clear result can still be obtained from 225: the higher the interest rate r d, the smaller the present value of a fixed payment instrument Although quantities N, n and P t 0 are known at the moment of the investment decision, interest rate r d during the term to maturity of the asset is not known, and so there exists an interest rate risk in this investment decision Assuming an infinite time horizon lim n in 225, we get P t 0 = N r d 227 Notice that the measurement unit of the interest rate transforms the measurement unit of the present value as: e/ t// t = e Remark! In the calculation of present values of money flows, the interest rate applied in the discount factor represents the alternative rate of return that could be obtained for the money when invested elsewhere Many times the interest rates offered by government bonds see Section 042 are considered as a proper measure for the risk-free interest rate that can be used in discounting money flows of other financial instruments We calculate the present value of a money flow when 4 e is received after four weeks The flow is thus 4 e/4week even though the money is not received until the interval ends The interest rate is assumed to be 0 %/y = 4/520 /4week where the interest rate is transformed simply by using time units The present value of this flow is then 4 e/ t t + r t 4 e/4week 4week = + 4 = 4 e 520 4week 4week + 4 = 3969 e 520

12 ESTOLA: MEASURING IN ECONOMICS 79 Next we calculate the present value of the money flow where one dollar is received at the end of every week in a time unit of four weeks Notice that the total of the flow is the same as above: 4 e/4week Again, transforming the interest rate simply by time units, the weekly interest rate is 0 %/y = /520 /week and the present value is: e/ t t + e/ t t + r t + r t + e/ t t 2 + r t + e/ t t 3 + r t 4 e/week week e/week week = week week week week + e/week week 3 + week week e/week week 4 = 398 e week week Thus the more frequently the payments take place in a fixed flow, the greater the present value of a positive money flow The four week interest rate r d /4week, that gives the same present value as above when calculated on the four week basis, can be obtained as follows: 398 e = 4 e/4week 4week + r d /4week 4week = week r d = This is clearly smaller than the four week interest rate we used without making the compounding correction: 4/520 = /4week The weekly interest rate r d /week, that gives the same present value for the four week money flow when calculated on the weekly basis, is obtained by solving with numerical methods the interest rate r d from the equation: e/week week 3969 e = + r d /week week + e/week week + r d /week week 2 e/week week + + r d /week week + e/week week 3 + r d /week week Present Values in Continuous Time* Here we analyze the present values of money flows in continuous time Discrete time is transformed to continuous by letting t 0 This is done by dividing the time unit t into k equal subintervals, and letting k At the time interval t 0 + j t/k, the interest rate is r d t 0 + j t/k / t/k

13 80 ESTOLA: PRINCIPLES OF QUANTITATIVE MICROECONOMICS and the money flow is Nt 0 + j t/k e/ t/k, j =, 2,, nk With fixed k, the present value of this flow during nk time periods is: nk P t 0 = N j= t 0 + j t t + r d t 0 + j t k k k j t k When k, t/k dt, r d t 0 + j t/k rt 0 + jdt and Nt 0 + j t/k approaches the instantaneous flow at moment t 0 + jdt The above given discount factor can be modified as + r d t 0 + j t j t = + k k lim z k/ t r dt 0 +j t k see Chiang 974 p 290 Now we know that + z = e, z k/ t r dt 0 +j t k jr d t 0 +j t k k/ t and because lim k k/r d t 0 + j t/k t =, we can simplify the above formula by the definition of the number e The limiting process transforms the exponent of e into rt t t 0 This occurs because with k, j t/k jdt and r d t 0 + j t/kj t/k rt 0 + jdtjdt The following definition t = t 0 + jdt, j =, 2, for continuous time completes the proof Taking the limit k transforms the sum to an integral with integration limits t 0 and t 0 + n t = t n We can thus write N nk lim k j= tn = t 0 t 0 + j t + r d t 0 + j t k k Nte rt t t 0 dt, j t t k k where the instantaneous flow at moment t = t 0 + jdt is denoted by Nt Assuming r to be constant, the present value of an infinite fixed flow in continuous time becomes the following: Ne r t t0 dt = N t 0 t 0 r e r t t 0 = N e e 0 = N r r Thus if r d = r, we get equal present values for the same infinite fixed flow with discrete and continuous time discounting,

14 ESTOLA: MEASURING IN ECONOMICS 8 Next we calculate the present value of flow 4 e/4week in continuous time Time is first measured in units 4week and then in units week The present value of flow 4 e/4week is in the first case: t=t0 + t=t 0 + 4e r t t0 dt = 4 t=t 0 t=t 0 r e r t t 0 = 4 e r e 0 r = 520 e = 3985 e Notice that the marginal change in time, dt, is measured above in units 4week The complete form of the integrated factor is then 4 e/4week dt 4week e r t t0 = 4 dt e r t t0 e where e r t t0 is dimensionless Next time is measured in units week Flow 4 e/4week corresponds to 4/4 e/week = e/week The present value of this flow is t=t0 +4 t=t 0 +4 e r t t0 dt = t=t 0 t=t 0 r e r t t 0 = e 4r e 0 r = 520 e = 3985 e, where dt is measured in units week Thus, independent of the time unit applied we get equal present value for the same money flow in continuous time discounting Essential is that the limits of integration and the money flow are measured in the same time unit Present values of money flows in continuous time are a bit higher than the corresponding ones in discrete time The more dense the splitting of time in discrete analysis, the more close the present value is to that in continuous time 29 Economic Kinematics Kinematics bases on the innovations of Isaac Newton in describing the motion of a particle in mathematical terms Newton developed differential and integral calculus for this purpose Without knowing this Gottfried Wilhelm Leibniz developed the same calculus in mathematics for measuring changes in function values Later it was realized that the two techniques are identical This way a connection between mathematics and theoretical physics was found which is still active Various concepts defined in mathematics have correspondents in physics For example, vector the velocity vector of a particle, the algebra of vector calculation the calculation of resultant force vector, the potential of a function the potential energy of a particle etc The earlier presented connections between economics and physics mean that there exists a link between these sciences too Later we will see that also