Math 360 Theory of Mathematical Interest Fall 2016

Transcription

1 Math 360 Fall 2016 Instructor: K. Dyke Math 360 Theory of Mathematical Interest Fall 2016 Instructor: Kevin Dyke, FCAS, MAAA 1

2 Math 360 Fall 2016 Instructor: K. Dyke LECTURE 1 AUG 31,

3 Time Value of Money Any long term investment or loan utilizes the concept of time value of money Fundamental component of all transactions utilized by actuaries Actuarial applications Calculating reserves for a portfolio of term life insurance policies issued at age 40 and currently aged 55 Determining the price for a 10 year annuity starting at age 65 Establishing the price for a professional liability policy covering an OB/Gyn physician for claims filed over the next 18 years Determine the current plan year contribution (normal cost) for a pension plan covering employees aged

4 Math 360 Fall 2016 Instructor: K. Dyke TVM - Real life applications Purchasing a car. Often dealers provide a choice between a cash rebate and a lower interest rate. You can use TVM to determine which is a better deal. $500 rebate with 6% interest rate vs. $0 rebate with 0% interest Mortgage loans. If you are buying a home, your monthly payment can be determined using TVM Credit cards. Paying off your credit cards monthly avoids interest charges. Otherwise you will be charged interest on your outstanding balance. You can estimate your interest using TVM. Investment returns. You want to calculate the return on an investment in a year. Auto insurance. You have been offered the option of monthly payments or a one-time annual payment on your auto insurance policy, where the one-time payment is 5% less than the sum of the monthly paymnts. Microbrewery. Your friend wants to start a microbrewery and offers you an investment share. You can earn 30% per year but there is a 50% chance it will fail within 5 years where you ll lose your entire principal. 4

5 Accumulated Amount Function Let s start with the generalized form of the growth of an investment A(t) = Value of the investment at time t A(0) = Initial value of investment (t=0) A(1) = Value of investment at end of first year (t=1) A(t) is known as the accumulated amount function Example: Carrie invests $1,000 in a mutual fund. At the end of the year, the fund value has grown to $1,100. A(0) = 1,000 A(1) = 1,100 What might you want to measure? 5

6 Accumulated Amount Function What is your return on investment (i)? i = A 1 A(0) A(0) i = 1,100 1,000 1,000 = 10% For any time = t, the Effective Annual Rate of Interest is defined as: A t + 1 A(t) Effective Annual Rate of Interest = i t+1 = A(t) Other rate of interests can be determined in the same manner, such as the 3 and 6 month returns. Partial years measure nominal returns instead of annual returns (more on that later). 6

7 Accumulation Functions A(t) represents the value of a fund at time t. If we were to assume the initial investment of 1 at time 0, the value of 1 at time t is the accumulation function represented as a(t). Accumulation function at time 0: a 0 1 If we represent the initial investment as k, such than A(0) = k, the relationship between A(t) and a(t) is defined as follows: A(t) = k a(t) Equivalence of Effective Rate of Interest A t + 1 A(t) k a t + 1 k a(t) i = = A(t) k a(t) = a t + 1 a(t) a(t) 7

8 Accumulation Functions Let s return to our previous example with Carrie. A(1) = 1,100 A(0) = 1,000 Determine the constant k and the value of the accumulation function at time 1. Answer: A(0) = 1000 = k a(0). Since a(0) = 1, then k = A(1) = k a(1) a 1 = A(1) = 1,100 = 1. 1 k 1,000 This makes sense because the investment grew by i=10% in the year. If we assumed the investment started with $1 instead of $1,000, then the value at the end of the year would be $

9 Accumulation Functions and Interest Accumulation functions take on many forms, but generally have properties of being a t = t a t = t 2 + 2t + 25 a t = 1 + it, where i = 10% a t = (1 + i) t, where i = 10% The latter two functions have special meaning in interest theory. The first is the definition of simple interest. The accumulation function is linear. In the example, the accumulation function a t = t, meaning each year the fund earns $0.10. After 20 years, the fund value is $3.00. $3.00 $2.50 $2.00 $1.50 $ Time (t) 9

10 Accumulation Functions and Interest The second is the definition of compound interest. The accumulation function is exponential. A t = (1 + i) t, where i = 10% Time (t) After 20 years, the fund is valued at almost $

11 Accumulation Functions and Interest Comparing the two functions illustrates the magic of compound interest as compared to simple interest (i = 10%) Simple Interest Compound Interest Time (t) 11

12 Problems in Interest Theory Joe borrows 10,000 from Lisa for 5 5%. Assuming he makes no interim payments, calculate Joe s settlement at year 5. a(t) = (1+i) t a(5) = (1.05) 5 = A(5) = 10000( ) = $12, Same arrangement as above, except Lisa simply tells Joe to pay him $13,000 at the end of year 5. Calculate the effective annual interest rate charged to Joe. A(5) = A(0) * a(5) 13,000 = 10,000 * (1+i) 5 13,000 / 10,000 = (1+i) 5 = ln(1+i) = ln 1.3 i = 5.384% 12

13 Problems in Interest Theory Joe borrows the same $10,000 from Lisa at a 5% rate but this time Lisa tells Joe not to pay her until the balance reaches $15,000. Determine when Joe needs to pay Lisa. A(t) = A(0) * a(t) 15,000 = 10,000 * (1.05) t 15,000 / 10,000 = (1.05) t = 1.5 t ln(1.05) = ln 1.5 t = 8.31 years 13

14 Equivalent Rates of Interest Rats are said to be equivalent if the produce the same accumulated values a(t) at each point in time. Another way to say it is the accumulation functions produce the same graph. Compounding at shorter intervals can produce the same interest A rate that is compounded monthly means that each month interest is credited to the account. Let s look at an example of equivalence between rates compounded monthly and annually. 14

15 Equivalent Rates of Interest $1 compounded monthly at 12% (1% each month) $1 compounded annually at 12.7% Month Monthly Annual (1.01)= (1.01)= (1.01)= (1.01) = (1.01)= (1.01)= (1.01)= (1.01)= (1.01)= (1.01)= (1.01)= (1.01)= (1.127)=1.127 Thus, an account with a 12% annual rate compounded monthly is equivalent to an account with a 12.7% rate compounded annually. We will discuss compounding at different intervals later in Chapter 1. 15

16 Annualized Return Given different annual returns over a period, an annualized return can be calculated. Example: An account earns 5% in year 1, 10% in year 2, 7% in year 3. a(1) = 1.05 a(2) / a(1) = 1.10 a(3) / a(2) = 1.07 a(3) = a(1) * a(2)/a(1) * a(3)/a(2) a(3) = 1.05 * 1.10 * 1.07 = Annualized return over the 3 year period = ( ) 1/3 = 1.073, or 7.3%. Generalized form a(t) = 1 + i i i 3 (1 + i t ) = (1 + i a ) 1/t Where i 1.. i t are the individual annual returns and i a is the annualized return 16

17 Lecture 1 Homework Read chapter 1 sections 1.0 and 1.1 Complete the following exercises: As a reminder, homework will not be collected. However as with most math courses, the material will become increasingly difficult. Delaying homework until close to the first exam on September 19 will reduce your likelihood of success. Be sure to take advantage of recitations and office hours if you need help with homework or want further explanation. 17