Time Value of Money Concepts PDF

BMSH2201 TIME VALUE OF MONEY Concepts in Time Value Opportunity Cost People are constantly making choices among various financial decisions. People must give up something to gain something else in making those choices. For example, if y...

BMSH2201 TIME VALUE OF MONEY Concepts in Time Value Opportunity Cost People are constantly making choices among various financial decisions. People must give up something to gain something else in making those choices. For example, if you want to save up for a new phone, you may need to give up eating out on weekends. The opportunity cost of a new phone is eating out on weekends. Therefore, the opportunity cost is something you give up to achieve something. The time value of money states that all things being equal, a peso today is worth more than a peso in the future. The peso you have today is something that you can use now and earn interest, while you cannot use a future peso. Because money is a limited resource, you cannot save and spend at the same time. If an individual decides to save his/her money, then he/she gives up the opportunity to spend it. Good money management is needed in personal life and business. Interest The formula for interest is as follows: 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 × 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 × 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 This formula can be summarized as 𝐼𝐼 = 𝑃𝑃𝑃𝑃𝑃𝑃. Three (3) other variations of this formula can be used to find 𝑃𝑃, 𝑟𝑟, and 𝑇𝑇: 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝑃𝑃 = ; 𝑟𝑟 = ; 𝑇𝑇 = 𝑟𝑟𝑟𝑟 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃 To compute for the maturity amount (interest plus principal), this formula will be used: 𝐴𝐴 = 𝑃𝑃 (1 + 𝑟𝑟𝑟𝑟) Note that when using formulas, the 𝑟𝑟 and 𝑇𝑇 should be in the same time units. For example, adjustments must be made if the rate is monthly and the time is given yearly. Compound interest – The interest in the first compounding period is added on the principal, which will be the basis for the interest computed for the next period. So, in our earlier example, the interest earned in the first year is equal to 500,000 𝑥𝑥.08 = 40,000. The 40,000 interest will be added to the 500,000 principal, which will be the basis for interest computation for the second year; 540,000 𝑥𝑥.08 = 43,200, and so on. The formula below summarizes the effects of adding on the interest, where 𝑚𝑚 is the compounding frequency. 𝑟𝑟 𝑇𝑇𝑇𝑇 𝐼𝐼 = 𝑃𝑃 × 1 + − 𝑃𝑃 𝑚𝑚 In the story above: 𝑃𝑃 = 500,00; 𝑟𝑟 = 8%; 𝑇𝑇 = 5 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦; 𝑚𝑚 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 Thus, 0.08 5(1) 𝐼𝐼 = 500,000 × 1 + − 500,000 = 𝑃𝑃234,664.04 1 Compounding Frequency – The number of times interest is computed on a certain principal in one (1) year. 08 Handout 1 *Property of STI  [email protected] Page 1 of 6  BMSH2201 If the investment pays annually, the interest is the same as computed above since 𝑚𝑚 = 1. If the investment pays semi-annually, the total interest will be equal to: 0.08 5(2) 𝐼𝐼 = 500,000 × 1 + − 500,000 = 𝑃𝑃240,122.14 2 Effective annual rate – It is the actual interest paid or earned. It should be distinguished from the nominal rate or the stated contractual rate, which is the interest charged by a lender or promised by a borrower. It does not reflect the effect of compounding frequency. 𝑟𝑟 𝑚𝑚 𝐸𝐸𝐸𝐸𝐸𝐸 = [ 1 + ]−1 𝑚𝑚 Future Value and Present Value Future Value There are three (3) reasons why a peso today is worth more than a peso in the future: Preference for present consumption - Individuals prefer present consumption to future consumption. There needs to be a strong incentive to convince people to give up present consumption for future consumption. For example, a businessman would be willing to deposit money if he is assured that it will bring a higher interest. Inflation – When there is inflation, the value of the currency decreases over time. The greater the inflation, the greater the difference in value between a peso today and a peso tomorrow. Risk – If there is any uncertainty or risk associated with the cash flow in the future, people will think it is better to spend their money now. The formula for the future value is: 𝐹𝐹𝑉𝑉𝑡𝑡 = 𝑃𝑃𝑃𝑃 (1 + 𝑖𝑖 )𝑡𝑡 Where: 𝐹𝐹𝐹𝐹 = Future value in 𝑡𝑡 period 𝑃𝑃𝑃𝑃 = Present value or the money invested today 𝑖𝑖 = Interest rate 𝑡𝑡 = Number of periods As an example, let us continue from the previous interest problem. How much will it earn if you deposit P10,000 at 3% compounded annual interest in the second year? Using the formula, we compute the future value of the deposit: 𝐹𝐹𝑉𝑉𝑡𝑡 = 𝑃𝑃10,000 (1 + 3%)2 𝐹𝐹𝑉𝑉𝑡𝑡 = 𝑃𝑃10,609 The future value of the original deposit at the end of the second year is P10, 609. Complete the table: Rate 3% Year Principal Interest Total 08 Handout 1 *Property of STI  [email protected] Page 2 of 6  BMSH2201 1 10,000.00 300.00 10,300.00 2 10,300.00 309.00 10,609.00 3 10,609.00 318.27 10,927.27 4 10,927.27 327.82 11,255.09 5 11,255.09 337.65 11,592.74 6 11,592.74 347.78 11,940.52 7 11,940.52 358.22 12,298.74 8 12,298.74 368.96 12,667.70 9 12,667.70 380.03 13,047.73 10 13,047.73 391.43 13,439.16 Present Value The present value formula is: 𝐹𝐹𝐹𝐹 𝑃𝑃𝑃𝑃 = (1 + 𝑖𝑖 )𝑡𝑡 Where: 𝐹𝐹𝐹𝐹 = Future value in 𝑡𝑡 period 𝑃𝑃𝑃𝑃 = Present value or the money invested today 𝑖𝑖 = Interest rate 𝑡𝑡 = Number of periods You need P25,000.00 to buy a laptop when you enter college two (2) years from now. How much must you invest now if the interest rate is 6% per annum? 𝐹𝐹𝐹𝐹 25,000 𝑃𝑃𝑃𝑃 = = = 𝑃𝑃22,249.91 (1 + 𝑖𝑖 )𝑡𝑡 (1.06)2 You need to invest PHP22,249.91 to have PHP25,000.00 by the end of 2 years. Loan Amortization A loan is a money lent at an interest rate for a certain period of time. Loans are normally secured from different financial institutions, and the most common are banks. A bond is also a form of a loan but can be traded through Philippine Dealing and Exchange (PDEX) System. How Amortization Works Amortization happens when a person pays off a debt with regular, equal payments over time. With each monthly payment, a portion of the money goes towards the interest costs and reduces the loan balance. At the beginning of the loan, the interest costs are high. For long-term loans, the majority of each monthly payment is an interest expense, and only a small amount is deducted from the loan. As time goes on, more payment goes towards the principal and less on interest. Amortized loans are designed so that the last loan payment will completely pay off the loan balance after a certain amount of time. 08 Handout 1 *Property of STI  [email protected] Page 3 of 6  BMSH2201 The term loan amortization refers to the computation of equal periodic loan payments. These payments provide the lender with a specified interest return and loan principal repayment over a specified period. The loan amortization process involves finding out the future payments over the term of the loan, which will pay the loan plus the interest. Lenders use the loan amortization schedule to determine these payment amounts and allocate each payment to interest and principal. Auto/car loans, home loans, and personal loans are usually amortized loans. Credit card loans, however, are not amortized. The relationship of the time value of money in loan/bond pricing a. Present value To get the present value of a bond maturing a year from now, let us illustrate what we have learned in the previous module regarding present value. Let’s say that you are willing to invest a sum of money that will yield PHP100,000 at the end of year 1, what amount should you invest today? If the investment earns 10%, then the amount to be invested or the present value should be equal 1 to 𝑃𝑃𝑃𝑃𝑃𝑃100,000 𝑥𝑥 (1.10)1 = 𝑃𝑃𝑃𝑃𝑃𝑃90,909.09 If this amount will be received in two (2) years, then the present value is equal to 1 𝑃𝑃100,000 𝑥𝑥 (1.10)2 = 𝑃𝑃82,644.63. We may also use the present value table. b. Present value of interest payments (annuities) In addition to receiving the face value at maturity, the investor/lender also receives periodic interest payments over the bond's life. As mentioned in the previous module, these periodic payments are called annuities. To illustrate, assume that you will receive P10,000 annually for three (3) years, and the interest rate is 10%. c. Present value of a bond To calculate the present value or the price of a bond, we need to combine both the present values of the face value and the annuity payments. In our previous example, suppose that a bond with a face value of PHP100,000 pays interest of 10% annually and matures in 3 years. What is the price of the bond? First, we find the present value of the face value of the bond. That is equal to 1 𝑃𝑃𝑃𝑃𝑃𝑃100,000 𝑥𝑥 ( ) = 𝑃𝑃𝑃𝑃𝑃𝑃75,131.40. (1.10)3 Next, we compute for the present value of annuity payments of P10,000 annual interest (100,000 𝑥𝑥 0.1). This was computed earlier in our example, PHP24,868.6. Therefore, the price of the bond is: P100,000 face value at 10% for 3 years P75,131.40 08 Handout 1 *Property of STI  [email protected] Page 4 of 6  BMSH2201 P10,000 interest for 3 years 24,868.60 Price of the bond P100,000.00 Take note that interest payments may be made semi-annually or even monthly. In this case, we need to adjust the interest rates and time periods accordingly. d. Bonds issued at a discount When bonds are issued below the face or par value, they are issued at a discount. A discount occurs when the required rate of return is greater than the nominal rate of return. For example, let’s say we have a PHP100,000 bond with a stated rate of 10% and an effective rate (required rate of return) of 12% that pays interest semi-annually and has a maturity of 3 years. At what price should the bond be issued? First, we need to compute the amount of interest payment per semi-annual period equal to 100,000 x 10% x 6/12 = 5,000. The total period is 6 (3 years x 2), and the discount rate to be used is 6% (12%/2). The price of the bond is as follows: P100,000 face value at 6% for 6 P70,496.05 periods P5,000 interest for 6 periods 24,568.62 Price of the bond P95,082.67 e. Bonds issued at a price premium When bonds are issued above par, they are issued at a premium. It occurs when the required rate (effective rate) is below the stated or nominal rate. Let us recall our previous example but use 8% effective rate instead of 12%. It will result in the following bond price: P100,000 face value at 4% for 6 P79,031.45 periods P5,000 interest for 6 periods 26,210.68 Price of the bond P105,242.13 08 Handout 1 *Property of STI  [email protected] Page 5 of 6  BMSH2201 Constructing an Amortization Table In an amortization schedule, the payments would be the same amount, but there would be a different principal, interest, and balance for each payment. Amortization payments can be computed using this formula: 𝑖𝑖 × 𝑃𝑃 × (1 + 𝑖𝑖 )𝑛𝑛 𝐴𝐴 = (1 + 𝑖𝑖 )𝑛𝑛 − 1 Where: 𝐴𝐴 = Payment per period 𝑃𝑃 = Principal 𝑖𝑖 = Interest per payment period 𝑛𝑛 = Total number of payments Example: Period 6 semi-annual periods (3 years) Effective interest 12% Stated rate 10% Face value 100,000 Issue price 95,082.68 Solution: Interest to be paid Interest expense Amortization of discount Carrying Period 100,000 × 5% = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 × 6% = 𝑖𝑖𝑖𝑖𝑖𝑖. 𝑒𝑒𝑒𝑒𝑒𝑒 − 𝑖𝑖𝑖𝑖𝑖𝑖. 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 balance 95,082.68 1 5,000.00 5,704.96 704.96 95,787.64 2 5,000.00 5,747.26 747.26 96,534.90 3 5,000.00 5,792.09 792.09 97,326.99 4 5,000.00 5,839.62 839.62 98,166.61 5 5,000.00 5,890.00 890.00 99,056.61 6 5,000.00 5,943.40 943.40 100,000.00 References Mayo, H. B. (2017). Business Finance: Theory and Practice. Quezon City: Abiva Publishing House, Inc. Titman, S., Keown, A. J., & Martin, J. D. (2018). Financial Management: Principles and Applications (13th ed.). New York: Pearson Education, Inc. 08 Handout 1 *Property of STI  [email protected] Page 6 of 6