Chapter 1 Interest Rates

Chapter 1 Interest Rates principal X = original amount of investment. accumulated value amount of interest S = terminal value of the investment I = S X rate of interest S X X = terminal initial initial accumulation function a(t) is any function with these properties: 1. a(0) = 1 2. a(t) is an increasing function 3. a(t) is a continuous function amount function A(t) : A(t) = Pa(t) So, A(t) enjoys these properties: 1. A(0) > 0 2. A(t) is an increasing function 3. A(t) is a continuous function Example. Suppose a(t) = αt 3 + β(t 1) + γ. We are given that a(1) = 1.6 and a(2) = 2.5. Find the coefficients. What will be the value at t = 3.1 of $5 invested at t = 0?. We have 1

a(0) = 1 a(1) = 1.6 a(2) = 2.5 β + γ = 1 (1) α + γ = 1.6 (2) 8α + β + γ = 2.5 (3) solve = α = 0.05 β = 0.55 γ = 1.55 a(t) = 0.05 t 3 + 0.55(t 1) + 1.55 a(3.1) = 4.1946 future value = 5 a(3.1) = (5)(4.1946) = 20.973 Definition. The effective rate of interest for a period [t 1, t 2 ] is defined to be the value A(t 2 ) A(t 1 ) A(t 1 ) = a(t 2) a(t 1 ) a(t 1 ) Example. For the accumulation function a(t) in the previous example fund the effective rate of interest for the period [0,3.1]. a(3.1) a(0) a(0) = 4.1946 1 1 = 3.1946 = 319.46% 2

1.2 Simple Interest Simple interest is used for short term transactions. In the case of simple interest the interest is paid on the initial investment only and not on the interest already earned. As an example, suppose that an account earns i% each year on your initial investment X = A(0). Then at the end of the first year the investment is worth X(1.4) and the end of second year it is worth X + ix + ix = X + 2iX,..., and generally at the end of the t-th year it is worth X + tix = X(1 + ti). So, if i is the simple interest rate, then for all t we have: simple interest AV(t) = X(1 + it) In this case of simple interest, interest in paid on the principle only and not on the interest that has previously accrued. Note. When the time t is not an integer, then the accumulated value AV t is calculated by pro-rating. Example. Maryam invested $100 on Jan. 1 2016 in an account that earns simple interest of 10%. (i) Calculate the value of the investment on September 1, 2020. (ii) Calculate the effective rate of interest. Solution to part (i). The period of study is 4 8 12 = 14 3 ( ) 14 AV 13 = 100(1 + 0.1) = 146.66 12 3 Solution to part (ii). years. So, AV terminal AV initial 146.66 100 = = 0.4666 = 46.66% AV initial 100 Note. Note that for a simple interest we have: 3

A(t) = A(0)(1 + it) A(t 1 ) = A(0)(1 + it 1 ) A(t 2 ) = A(0)(1 + it 2 ) A(t 2 ) A(t 1 ) = A(0)i(t 2 t 1 ) i = 1 A(t 2 ) A(t 1 ) average gain = A(0) t 2 t 1 initial investment noting that the difference t 2 t 1 must be expressed in the unit of one year. Example. I paid a friend $1010 for the $1000 he loaned me 50 days ago. What rate of simple interest did I pay? average gain = 10 50 365 i = = 365 5 = 73 average gain initial investment = 73 = 0.073 = 7.3% 1000 4

1.3 Compound Interest Suppose an account has earned the interest Xi in the first period on which the interest rate of i applies, then at the end of the period the invest is worth X(1 + i) = X + Xi. If for the second period the interest i is applied on both the principle X and the interest Xi, then at the end of the second period the account is worth X(1 + i)(1 + i) = X(1 + i) 2. Generally, at the end of the t-th period the account is worth AV t = X(1 + i) t. Note. When the time t is not an integer, then the accumulated value AV t is calculated by pro-rating. Her is a consistency argument for it: Consider a fractional time n 12 of a year, where n is the number of months elapsed. Suppose that a $1 invested at the initial time has grown to Y in the period [0, n 12 ]. To maintain consistency, at the end of period [ n 12, 2n 12 ] this value Y must grow to Y2, and at the end of [ 2n 12, 3n 12 ] it must grow to (Y2 )Y = Y 3, and so on, and finally at the end of period [ 11n 12, 12n 12 ] = [ 11n 12, n] it must grow to Y12. On the other hand, at the end of [ 11n 12, n] the initial value of $1 must be worth (1 + i)n with the yearly rate of i compounded. Then we must have the following equality: Y 12 = (1 + i) n Y = (1 + i) n 12 What this means is that the initial investment of $1 will be worth (1 + i) n 12 at the end of period [0, n 12 ]. So, as an example, a X invested at time t = 0 is worth X(1 + i) 5 12 after 5 months. And, it is worth X(1 + i) 273 365 after 273 days. Example. Maryam invest $100 on Jan. 1 2016 in an account that earns compound interest of 10%. (i) Calculate the value of the investment on September 1, 2020. (ii) Calculate the effective rate of interest. The period of study is 4 8 12 = 14 3 years. So, 5

AV 13 = 100(1 + 0.1) 14 3 = 156.01 12 AV terminal AV initial 156.01 100 = = 0.5601 = 56.01% AV initial 100 Note. Investments of one day term are called overnight money. Note. Consider the function f (t) = (1 + i) t (1 + it) f (t) = (1 + i) t ln(1 + i) i f (t) = 0 6

Discuss the graphs AV t = (1 + i) t AV t = 1 + it 1 Example. Consider the value function AV t = X(1 + i) t associate with compound interest. Calculate the effective interest rate for one period. AV t AV t 1 AV t 1 = X(1 + i)t X(1 + i) t 1 X(1 + i) t 1 = getting rid of a factor of (1 + i) t 1 (1 + i) 1 = = i 1 So, the effective interest rate is the same as i. This has been the reason behind the name for the annual effective interest rate given on page 9 of the textbook. Example. $1500 amounts to $2400 in 64 months. Calculate the annual effective interest rate. From the rule of pro-rating, we must have: 1500(1 + i) 64 12 = 2400 (1 + i) 64 12 = 2400 1500 i = ( ) 12 24 64 1 = 0.0921 = 9.21% 15 7

Example. With 6% annual effective rate, how long does it take for $100 to become $150. 100(1.06) t = 150 (1.06) t = 150 = 1.5 t ln(1.06) = ln(1.5) 100 t = ln(1.5) = 6.96 years ln(1.06) 8

1.5 Present Value The present value of an amount $X payable at time t assuming an annual compound interest rate of i is PV t = X (1+i) t Notations. The one year present value factor: v = 1 = (1 + i) 1 1 + i Compound interest present value factor: it is the present value of $1 to be made in t years: PV F t = v t = (1 + i) t simple interest present value factor: PV F t = 1 = (1 + it) 1 1 + it Note. According to the law of consistency, these formulas must hold for the non-integral values of time t too. Example. How much should we invest now to receive $10,000 in 15 years and 3 months time at 5% interest being compounded annually?. Also calculate its present value at time t = 2.5 (15 + 3 12 ) = 61 4 PV 61 4 = 10000(1.05) 61 4 = 4751.85 9

The time 2.5 is equivalent to 2 + 1 2 = 5 2. The time interval between this time and the time 61 4 is: 61 4 5 2 = 51 4 So, the answer to the second part of the question is: PV 51 4 = 10000(1.05) 51 4 = 5368.30 Example. Answer the previous question if the simple interest rate is used. PV 61 1 = 10000 4 1 + ( ) = 5673.76 61 4 (0.05) So, the answer to the second part of the question is: PV 51 1 = 10000 4 1 + ( ) = 6106.87 51 4 (0.05) 10

1.6 Rate of Discount Suppose you have already bought a movie ticket for $50 to watch a movie one week from now but you now have a new schedule and cannot go, therefore you decide to sell your ticket at a discount. You sell it to a friend for $40. The discount rate you are offering to you friend is 10 discount 50 = 20% = principle Here is another example: Consider the situation in which a bank lends you $10000 dollars for one period with at a rate of 10% but the bank charges the interest up front and give you $9000. The value taken from the value 10,000 is $1,000 ; the ratio 1000 10000 = 10000 9000 10000 is called the discount rate. You are supposed to pay back the amount of 10,000. Therefore, this quotient can be written as AV 1 AV 0 AV 1. In general, the effective discount rate for a period [t, t + 1] is defined to be d = AV t+1 AV t AV t+1 With the compound interest rate i, this ratio reduces to In fact: d = i 1+i d = AV t+1 AV t AV t+1 = X(1 + i)t+1 X(1 + i) t (1 + i) 1 X(1 + i) t+1 = = i 1 + i 1 + i Relationships between d and v: d + v = 1 iv = d Proof: d + v = i 1+i + 1 1+i = 1+i 1+i = 1 The relationships between i and d are: 11

d = i = i 1+i d 1 d We can accumulate and discount payments using the rate of discount: Present value factor for the compound rate of discount: PVF t = (1 d) t = present value of $1 to be made t years, under the compound rate of discount d Accumulated value factor for the compound rate of discount: AVF t = (1 d) t = accumulated value after t years of $1 invested now, under the compound rate of discount d Present value factor in the case of simple rate of discount: PVF t = (1 td) = present value of $1 to be made t years, under the simple rate of discount d Accumulated value factor in the case of simple rate of discount: AVF t = (1 td) 1 = accumulated value after t years of $1 invested now, under the simple rate of discount d Example. A bill for $785 due on 14th June is sold on the 28th March, the simple discount rate being 7% p.a. What are the proceeds? The number of days to run: March 3 April 30 May 31 June 14 discount = 785 0.07 78 365 = 11.74 proceeds = 785 11.74 = 773.26 12

Example. An investor pays $147 for a note maturing for $150 in 4 months. What is the (simple) discount rate? What simple interest rate does he earn on his investment? His rates in 4 months must be multiplied by 3 to get his yearly rates: d = 3 150 3 = 6% i = 3 147 3 = 6.12% Example. I wish to obtain $1000 from a bank as a 30-day loan. How much should I ask for if the bank charges 6% p.a. interest in advance? Let X be the amount asked for. Then X minus the interest in advance equals $1000 : X X 0.06 30 = 1000 X = 1004.96 365 Example. Smith lends Jones $900 for one year. After six months Jones and Smith agree to settle the debt at that time for $960. Which of the following correctly expresses the annual effective rate of discount for this transaction? (year 1979). 900(1 d) 0.5 = 960 (1 d) 0.5 = 960 900 0.12109 1 d = ( ) 900 2 d = 31 960 (16) 2 = 13

1.7 Constant force of interest Definition. A continuously compounded interest rate is called force of interest. If the annual effective interest rate is constant, then the force of interest is constant too. Mathematical Definition. Force of interest: δ t = d(av t ) dt AV t AV t+ t AV t t AV t = AV t+ t AV t AV t t = interest rate in the interval [t, t + t] t Under the compound interest rate of i: AV t = X(1+i) t d dt AV t = X(1+i) t ln(1+i) δ = AV t AV t = X(1 + i)t ln(1 + i) X(1 + i) t = ln(1+i) Constant force of interest rate : δ = ln(1 + i) Then : AV t = X(1 + i) t = Xe t ln(1+i) = Xe tδ Accumulated value factor in the case of constant force of interest: AVF t = e tδ Present value factor in the case of constant force of interest: PVF t = e tδ Example. What is the present value of $ 1000 to be made in 8 years if a constant force of interest of 4.2% is applied? 14

1000e 8δ = 1000e (8)(0.042) = $714.62 Example. A deposit of $ 500 is invested at time 5 years. The constant force of interest of 6 % per year is applied. Determine the accumulated value of the investment at the end of year 10. The investment has been made for 10-5=5 years. accumulated amount = 500e 5δ = 500e (5)(0.06) = $674.93 15

1.8 Varying force of interest When the force of interest is not constant, we may write: δ t = AV t AV t = d dt ln(av t) ( AVt2 ln AV t2 AV t1 AV t1 ) t2 = ln(av t2 ) ln(av t1 ) = δ t dt t 1 ( t2 ) = exp δ t dt t 1 Accumulated value factor in the case of varying force of interest: [ t2 ] AVF t = exp δ t dt t 1 Present value factor in the case of varying force of interest: [ t2 ] PVF t = exp δ t dt t 1 Example. If δ t = 0.01t, interval 0 t 2. 0 t 2, find the equivalent annual effective rate of interest over the ( 2 ) ( [0.01 (1 + i) 2 = exp 0.01tdt = exp 0 2 t2 ] 2 0 ) = exp(0.02) i = e 0.01 1 Example. Find the accumulated value of 1 at the end of 19 years if δ t =.04(1 + t) 2. ( 19 ) ( [0.04(1 A 19 = exp 0.04(1 + t) 2 dt = exp + t) 1 ] ) 19 = e 0.038 0 0 16

Example. In Fund X money accumulates at a force of interest δ t = 0.01t + 0.1 0 t 20 In Fund Y money accumulates at an annual effective interest rate i. An amount of 1 is invested in each fund for 20 years. The value of Fund X at the end of 20 years is equal to the value of Fund Y at the end of 20 years. Calculate the value of Fund Y at the end of 1.5 years. [ 20 ] A X 20 = exp 0 (0.01t + 0.1)dt = e 4 A Y 20 = (1 + i) 20 A Y 1.5 = (1 + i) 1.5 = [(1 + i) 20] 0.075 ( = e 4) 0.075 = e 0.3 = 1.35 17

1.9 Discrete changes in interest rates Here are some examples of when the interest rate changes at specific times. Example. A deposit of $ 2500 is invested at time t = 0. The annual effective rate of interest is 2.5% from t = 0 to t = 6. The annual effective rate of discount from t = 6 to t = 10 is 2.5%, and thereafter the annual force of interest is 2.5%. Find the accumulated value of investment at time t = 13. AVF [0,6] = (1.025) 6 AVF [6,10] = (1 0.025) 4 = (0.975) 4 AVF [10,13] = e 3 0.025 AV [0,13] = 2500AVF [0,6] AVF [6,10] AVF [10,13] = 2500(1.025) 6 (0.975) 4 e 3 0.025 = $3458.09 Example. If the effective rate of discount in year k is equal to 0.01k + 0.06 for k = 1,2,3, find the equivalent rate of simple interest over the three-year period. The effective rates of discount: d 1 = 0.07 d 2 = 0.08 d 3 = 0.09 ( )( )( ) ( )( )( ) 1 1 1 1 1 1 accumulated value = = 1 d 1 1 d 2 1 d 3 0.93 0.92 0.91 This must be equal to 1 + 3i ( )( )( ) 1 1 1 1 + 3i = 0.93 0.92 0.91 i = 0.948 = 9.48% 18