Mathematics of Finance PDF

Mathematics of Finance GE 112 Mathematics in the Modern World  Outline This module contains the following: 1. Introduction 2. Simple Interest 3. Future Value or Maturity Value 4. Compound Interest 5. Compound Amount 6. Present Value 7. Inflation 8. Module Exercises 9. R...

Mathematics of Finance GE 112 Mathematics in the Modern World  Outline This module contains the following: 1. Introduction 2. Simple Interest 3. Future Value or Maturity Value 4. Compound Interest 5. Compound Amount 6. Present Value 7. Inflation 8. Module Exercises 9. References Page 2  Simple Interest When you deposit money in a bank—for example, in a savings account—you are permitting the bank to use your money. The bank may lend the deposited money to customers to buy cars or make renovations on their homes. The bank pays you for the privilege of using your money. The amount paid to you is called interest. If you are the one borrowing money from a bank, the amount you pay for the privilege of using that money is also called interest. Mathematics of Finance Page 3  Simple Interest The amount deposited in a bank or borrowed from a bank is called the principal. The amount of interest paid is usually given as a percent of the principal. The percent used to determine the amount of interest is called the interest rate. Mathematics of Finance Page 4  Simple Interest ▼ Simple Interest Formula The simple interest formula is I = Prt where, I is the interest, P is the principal, r is the interest rate, and t is the time period. Note: the time t is expressed in the same period as the rate Mathematics of Finance Page 5  Simple Interest Example 1 Calculate the simple interest earned in 1 year on a deposit of ₱50,000 if the interest rate is 5%. Solution Use the simple interest formula. Substitute the following values into the formula: P = ₱50,000, r = 5% = 0.05, and t = 1. I = Prt I = 50,000(0.05)(1) I = 2,500 The simple interest earned is ₱2,500. Mathematics of Finance Page 6  Simple Interest Example 2 Calculate the simple interest due on a 4-month loan of ₱75,000 if the interest rate is 8.25%. Solution 4 P = 75,000, r = 8.25% = 0.0825, and t = 12 I = Prt 4 I = 75,000(0.0825) 12 I = 2,062.50 The simple interest due is ₱2,062.50. Mathematics of Finance Page 7  Simple Interest Example 3 Calculate the simple interest due on a 5-month loan of ₱55,000 if the interest rate is 1.25% per month. Solution P = 55,000, r = 1.25% = 0.0125 and t = 5. I = Prt I = 55,000(0.0125)(5) I = 3,437.50 The simple interest due is ₱3,437.50. Mathematics of Finance Page 8  Simple Interest Two Methods for Converting Time from Days to Years: Note: the ordinary method is used by most businesses Mathematics of Finance Page 9  Simple Interest Example 4 Calculate the simple interest due on a 120-day loan of ₱150,000 if the annual interest rate is 5.25%. Solution 120 P = 150000, r = 5.25% = 0.0525, and t =. 360 I = Prt 120 I = 150000(.0525) 360 I = 2,625.00 The simple interest due is ₱2,625.00. Mathematics of Finance Page 10  Simple Interest Example 5 How long will it take for ₱80,000 to earn ₱20,000, if it is invested at 7½% simple interest? Solution P = 80,000 , I = 20,000 , r = 7 ½% = 0.075, t = ? 𝐼 = 𝑃𝑟𝑡 𝐼 𝑡= 𝑃𝑥𝑟 20,000 = 80,000 𝑥 0.075 𝑡 = 3.33 𝑦𝑒𝑎𝑟𝑠 Mathematics of Finance Page 11  Simple Interest Example 6 A principal earns interest of ₱36,550 in 3 years and 7 months at a simple interest rate of 8½%. Find the principal invested. Solution 7 𝐼 = 36,550, 𝑡 = 3 , r = 8 1/2 % , P = ? 12 𝐼 = 𝑃𝑟𝑡 𝐼 𝑃= 𝑟𝑥𝑡 36,550 = 7.085 𝑥 312 𝑃 = 120,000 Mathematics of Finance Page 12  Simple Interest Future Value or Maturity Value Formula for Simple Interest The future or maturity value formula for simple interest is 𝑨 = 𝑷 + 𝑰 where, A is the amount after the interest, I, has been added to the principal, P. Mathematics of Finance Page 13  Simple Interest Future Value – the sum of the principal and the interest on an investment Mature Value – the sum of the principal and the interest on a loan Mathematics of Finance Page 14  Simple Interest Example 8 Calculate the maturity value of a simple interest, 8-month loan of ₱8000 if the interest rate is 9.75%. Solution Step 1: Find the interest. 8 P = 8000, r = 9.75% = 0.0975, and t = 12 I = Prt 8 I = 8000(0.0975) 12 I = 520 Mathematics of Finance Page 15  Simple Interest Step 2: Find the maturity value. A= P+I A = 8000 + 520 A = 8520 The maturity value of the loan is ₱8,520. Mathematics of Finance Page 16  Simple Interest Example 9 Find the future value after 1 year of ₱6,800 in an account earning 6.4% simple interest. Solution P = 6800, t = 1, r = 0.064. A=P+I A = P(1 + rt) A = 6800(1.064) A = 7235.20 The future value of the account after 1 year is ₱7235.20. Mathematics of Finance Page 17  Compound Interest Simple interest is generally used for loans of 1 year or less. For loans of more than 1 year, the interest paid on the money borrowed is called compound interest. Compound interest is interest calculated not only on the original principal, but also on any interest that has already been earned. Compounding period is the frequency with which the interest is compounded. Mathematics of Finance Page 18  Compound Interest Example 10 You deposit ₱5,000 in an account earning 6% interest, compounded semiannually. How much is in the account at the end of 1 year? Solution The interest is compounded every 6 months. Calculate 6 the amount in the account after the first 6 months, t =. 12 A = P(1 + rt) 6 A = 5000 1 + 0.06 12 A = 5,150 Mathematics of Finance Page 19  Compound Interest Calculate the amount in the account after the second 6 months. A = P(1 + rt) 6 A = 5150 1 + 0.06 12 A = 5,304.50 The total amount in the account at the end of 1 year is ₱5,304.50. Mathematics of Finance Page 20  Compound Interest In calculations that involve compound interest, the sum of the principal and the interest that has been added to it is called the compound amount. Mathematics of Finance Page 21  Compound Interest Compound Amount Formula The compound amount formula is 𝒓 𝒏𝒕 𝑨=𝑷 𝟏+ 𝒏 where, A is the compound amount, P is the amount of money deposited, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years. Mathematics of Finance Page 22  Compound Interest Mathematics of Finance Page 23  Compound Interest Example 11 Calculate the compound amount when ₱20,000 is deposited in an account earning 8% interest, compounded semiannually, for 4 years. Mathematics of Finance Page 24  Compound Interest Solution P = 20,000, r = 8% = 0.08, n = 2, t = 4 𝑟 𝑛𝑡 A=P 1+ 𝑛 0.08 2∗4 A = 20,000 1 + 2 8 A = 20,000(1 + 0.04) A = 20,000 (1.36856905) A = 27371.38 The compound amount after 4 years is approximately ₱27,371.38. Mathematics of Finance Page 25  Compound Interest Example 12 How much interest is earned in 5 years on ₱18,000 deposited in an account paying 9% interest, compounded monthly? Solution Calculate the compound amount. P =18000, r = 9%, n = 12, t = 5 𝑟 𝑛𝑡 A=P 1+ 𝑛 0.09 12∗5 A = 18000 1 + 12 Mathematics of Finance Page 26  Compound Interest A = 18000(1.0075)60 A = 18000(1.565681027) A = 28,182.26 Mathematics of Finance Page 27  Compound Interest Calculate the interest earned. I= A-P I = 28,182.26 - 18000 I = 10,182.26 The amount of interest earned is approximately ₱10,182.26. Mathematics of Finance Page 28  Compound Interest Present Value The present value of an investment is the original principal invested, or the value of the investment before it earns any interest. Therefore, it is the principal, P, in the compound amount formula. Present value is used to determine how much money must be invested today in order for an investment to have a specific value at a future date. Mathematics of Finance Page 29  Compound Interest Present Value Formula The present value formula is 𝑨 𝑷= 𝒏𝒕 𝒓 𝟏+ 𝒏 where, P is the original principal invested, A is the compound amount, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years. Mathematics of Finance Page 30  Compound Interest Example 13 How much money should be invested in an account that earns 8% interest, compounded quarterly, in order to have ₱50,000 in 4 years? Solution A = 50,000, r = 8%, n = 4, t = 4 𝐴 P= 𝑟 𝑛𝑡 1+𝑛 50,000 P= 0.08 4∗4 1+ 4 Mathematics of Finance Page 31  Compound Interest Solution A = 50,000, r = 8%, n = 4, t = 4 𝐴 P= 𝑟 𝑛𝑡 1+𝑛 50,000 P= 0.08 4∗4 1+ 4 50,000 P= (1.02)16 50,000 P= ≈ 36,422.29 1.372785705 ₱36,422.29 should be invested in the account in order to have ₱50,000 in 4 years. Mathematics of Finance Page 32  Inflation Suppose the price of a large-screen TV is ₱75,000. You have enough money to purchase the TV, but decide to invest the ₱75,000 in an account paying 6% interest, compounded monthly. After 1 year, the compound amount is ₱79,625.84. But during that same year, the rate of inflation was 7%. The large-screen TV now costs ₱75,000 + (7% of ₱75000) = ₱75000 + 5250 = ₱80,250 Because ₱79,625.84 < ₱80,250, you have actually lost purchasing power. At the beginning of the year, you had enough money to buy the large-screen TV; at the end of the year, the compound amount is not enough to pay for that same TV. Your money has actually lost value because it can buy less now than it could 1 year ago. Mathematics of Finance Page 33  Inflation Inflation is an economic condition during which there are increases in the costs of goods and services. Inflation is expressed as a percent; for example, we speak of an annual inflation rate of 7%. Mathematics of Finance Page 34  Inflation Calculate the Effect of Inflation on Salary Suppose your annual salary today is ₱35,000. You want to know what an equivalent salary will be in 20 years— that is, a salary that will have the same purchasing power. Assume a 6% inflation rate. Mathematics of Finance Page 35  Inflation Solution: Use the compound amount formula, with P = 35,000, r = 6% , t = 20. The inflation rate is an annual rate, so n = 1. 𝑟 𝑛𝑡 A=P 1+ 𝑛 0.06 1∗20 A = 35,000 1 + 1 A = 35,000(3.20713547) ≈ 112,249.74 Twenty years from now, you need to earn an annual salary of approximately ₱112,249.74 in order to have the same purchasing power. Mathematics of Finance Page 36  Inflation Calculate the Effect of Inflation on Future Purchasing Power Suppose you purchase an insurance policy in 2018 that will provide you with ₱500,000 when you retire in 2050. Assuming an annual inflation rate of 8%, what will be the purchasing power of the ₱500,000 in 2050? Solution Use the present value formula. A = 500,000, r = 8% , t = 32 , n = 1 Mathematics of Finance Page 37  Inflation Solution 𝐴 P= 𝑟 𝑛𝑡 1+ 𝑛 500,000 P= 0.08 1∗32 1+ 1 500,000 P= 1.0832 500,000 P = 11.737083 P = 42,600.02 Assuming an annual inflation rate of 8%, the purchasing power of ₱500,000 will be about ₱ 42,600.02 in 2050. Mathematics of Finance Page 38  Inflation Mathematics of Finance Page 39  Reference Aufmann, R., Lockwood, J., Nation, R. and Clegg, D. (2013). Mathematical Excursions (3rd Edition). Brooks/Coole: Cengage Learning 1 Topic 1 Page 40