Simple Interest GenMath PDF

SH1902 Interest I. Simple Interest Simple interest is a fixed percentage of the total amount invested paid to an investor each year. Let 𝐼𝐼 be the simple interest, 𝑃𝑃 be the amount invested called the principal, 𝑟𝑟 be the fixed...

SH1902 Interest I. Simple Interest Simple interest is a fixed percentage of the total amount invested paid to an investor each year. Let 𝐼𝐼 be the simple interest, 𝑃𝑃 be the amount invested called the principal, 𝑟𝑟 be the fixed percentage or the rate, and 𝑡𝑡 be the number of years the principal is invested. Then, 𝐼𝐼 = 𝑃𝑃𝑃𝑃𝑃𝑃. The formula can be algebraically manipulated to obtain a formula when 𝑃𝑃, 𝑟𝑟, or 𝑡𝑡 is unknown. When 𝑃𝑃 is unknown When 𝑟𝑟 is unknown When 𝑡𝑡 is unknown 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝑡𝑡 = 𝑃𝑃 = 𝑟𝑟 = 𝑃𝑃𝑃𝑃 𝑟𝑟𝑟𝑟 𝑃𝑃𝑃𝑃 (in years) Table 1.1 Example: Suppose you deposited ₱10,000 in a bank that gives 10% simple interest. How much interest will you receive after three (3) years? Solution: Given: 𝑃𝑃 = ₱10,000 𝑟𝑟 = 10% = 0.1 𝑡𝑡 = 3 Solving for the interest 𝐼𝐼, we use the formula 𝐼𝐼 = 𝑃𝑃𝑃𝑃𝑃𝑃. 𝐼𝐼 = 𝑃𝑃𝑃𝑃𝑃𝑃 = ₱10,000(0.1)(3) = ₱3,000 Example: Cyndi borrowed ₱30,000 from a bank. If the interest she paid was ₱3,375 for nine (9) months, at what rate would she pay? Solution: Given: 𝐼𝐼 = ₱3,375 𝑃𝑃 = ₱30,000 Since time is in months, we first have to convert it into years. 1 year 9 3 9 months ⋅ = year = year 12 months 12 4 3 Hence, 𝑡𝑡 = 4. 06 Handout 1 *Property of STI  [email protected] Page 1 of 6  SH1902 𝐼𝐼 𝑟𝑟 = 𝑃𝑃𝑃𝑃 ₱3,375 = 3 ₱30,000 4 = 0.15 As we see, time can be expressed not only in months. But before continuing with the formulas, we must always convert time into years first. Time can also be expressed in days. In such cases, we consider two (2) ratios. Exact interest (𝐼𝐼𝑒𝑒 ) is simple interest computed based on the ratio 1 year : 365 days. Ordinary interest (𝐼𝐼𝑜𝑜 ), on the other hand, is simple interest computed based on the ratio 1 year : 360 days. Example: Miss Uy borrowed ₱80,000 from a bank. The borrowed amount earned ₱5,400 at 9% ordinary interest. How many days did it take Miss Uy to repay the loan? Solution: Given: 𝐼𝐼𝑜𝑜 = ₱5,400 𝑃𝑃 = ₱80,000 𝑟𝑟 = 9% = 0.09 𝐼𝐼 Solving for the interest 𝑡𝑡, we use the formula 𝑡𝑡 = 𝑃𝑃𝑃𝑃𝑜𝑜. 𝐼𝐼𝑜𝑜 𝑡𝑡 = 𝑃𝑃𝑃𝑃 ₱5,400 = ₱80,000(0.09) = 0.75 But note from Table 1.1 that 𝑡𝑡 is expressed in years. But the question requires time to be expressed in days. Hence, we convert 0.75 years into days using the ratio for ordinary interest. 360 days 0.75 years ⋅ = 0.75(360) days = 270 days 1 year Hence, it took 270 days before Miss Uy repaid the loan. Example: Mr. Sandoval paid the bank ₱1,200 exact interest at 8% for 146 days. How much did he borrow? Solution: Given: 𝐼𝐼𝑒𝑒 = ₱1,200 06 Handout 1 *Property of STI  [email protected] Page 2 of 6  SH1902 𝑟𝑟 = 8% = 0.08 1 2 𝑡𝑡 = 146 ⋅ = 365 5 𝐼𝐼 Solving for the principal 𝑃𝑃, we use the formula 𝑃𝑃 = 𝑟𝑟𝑟𝑟𝑒𝑒. 𝐼𝐼𝑒𝑒 𝑃𝑃 = 𝑟𝑟𝑟𝑟 ₱1,200 = 2 0.08 5 = ₱37,500 An investment grows depending on the amount of interest it accumulates over a period of time. The final amount of an investment (or debt) after interest is added is called maturity value (𝑀𝑀𝑀𝑀). By definition, 𝑀𝑀𝑀𝑀 = 𝑃𝑃 + 𝐼𝐼. Since 𝐼𝐼 = 𝑃𝑃𝑃𝑃𝑃𝑃, then 𝑀𝑀𝑀𝑀 = 𝑃𝑃 + 𝑃𝑃𝑃𝑃𝑃𝑃. Factoring out 𝑃𝑃, we arrive with an equivalent formula for the maturity value, 𝑀𝑀𝑀𝑀 = 𝑃𝑃(1 + 𝑟𝑟𝑟𝑟). 𝐼𝐼 Also, since 𝑃𝑃 = 𝑟𝑟𝑟𝑟, then 𝐼𝐼(1 + 𝑟𝑟𝑟𝑟) 𝑀𝑀𝑀𝑀 =. 𝑟𝑟𝑟𝑟 Example: Find the maturity value for each item. a. 𝑃𝑃 = ₱40,000, and 𝐼𝐼 = ₱9,600 b. 𝑃𝑃 = ₱32,000 , 𝑟𝑟 = 8%, and 𝑡𝑡 = 180 days (ordinary interest) c. 𝐼𝐼𝑒𝑒 = ₱594, 𝑟𝑟 = 9%, and 𝑡𝑡 = 73 days (ordinary interest) Solution: a. 𝑀𝑀𝑀𝑀 = 𝑃𝑃 + 𝐼𝐼 = ₱40,000 + ₱9,600 = ₱49,600 b. 𝑀𝑀𝑀𝑀 = 𝑃𝑃(1 + 𝑟𝑟𝑟𝑟) 1 = ₱32,000 1 + 0.08 180 ⋅ 360 = ₱33,280 𝐼𝐼𝑒𝑒 (1+𝑟𝑟𝑟𝑟) c. 𝑀𝑀𝑀𝑀 = 𝑟𝑟𝑟𝑟 1 ₱594 1 + 0.09 73 ⋅ = 365 1 0.09 73 ⋅ 365 = ₱33,594 06 Handout 1 *Property of STI  [email protected] Page 3 of 6  SH1902 II. Compound Interest Suppose no deposits or withdrawals were made in a savings account. After a period, the interest due is added to the previous balance and the sum then becomes the new principal for the next period. The interest that is added to the old principal to make a new principal on which interest is again calculated for the next period is called the compound interest, denoted by 𝐼𝐼𝐶𝐶. The total amount at the end of the last period is called the compound amount, denoted by 𝐴𝐴. The compound amount is computed using the formula 𝑟𝑟 𝑛𝑛 𝐴𝐴 = 𝑃𝑃 1 +. 𝑚𝑚 𝑃𝑃 is the principal or the present value 𝑟𝑟 is the annual rate 𝑚𝑚 is the number of compounding periods in a year Some common values for 𝑚𝑚 o 𝑚𝑚 = 1 if compounded annually o 𝑚𝑚 = 2 if compounded semi-annually o 𝑚𝑚 = 4 if compounded quarterly o 𝑚𝑚 = 12 if compounded monthly o 𝑚𝑚 = 365 if compounded daily 𝑛𝑛 is the total number of compounding periods The value of 𝑛𝑛 usually is the product of 𝑚𝑚 and the total number of years. The compound interest therefore is the difference between the compound amount and the principal, that is, 𝐼𝐼𝐶𝐶 = 𝐴𝐴 − 𝑃𝑃. Example: Esther deposited ₱8,000 in a savings account for four (4) years at 0.5% compounded quarterly. How much will she have in her account at the end of four (4) years? How much will be the interest of her investment? Solution: Let 𝑃𝑃 = ₱8,000, 𝑟𝑟 = 0.005, 𝑚𝑚 = 4, and 𝑛𝑛 = 4(4) = 16. Then 𝑟𝑟 𝑛𝑛 𝐴𝐴 = 𝑃𝑃 1 + 𝑚𝑚 0.005 16 = ₱8,000 1 + 4 = ₱8,161.51. Thus, Esther will have ₱8,161.51 in her account at the end of four (4) years. The interest in her investment therefore is 06 Handout 1 *Property of STI  [email protected] Page 4 of 6  SH1902 𝐼𝐼𝐶𝐶 = 𝐴𝐴 − 𝑃𝑃 = ₱8,161.51 − ₱8,000 = ₱161.51. Example: How much interest is earned after 20 years by ₱20,000 at 4% interest compounded monthly? Solution: Let 𝑃𝑃 = ₱20,000, 𝑟𝑟 = 0.04, 𝑚𝑚 = 12, 𝑛𝑛 = 12(20) = 240. Then the compound amount is 𝑟𝑟 𝑛𝑛 𝐴𝐴 = 𝑃𝑃 1 + 𝑚𝑚 0.04 240 = ₱20,000 1 + 12 = ₱44,451.64, and the compound interest is 𝐼𝐼𝐶𝐶 = 𝐴𝐴 − 𝑃𝑃 = ₱44,451.64 − ₱20,000 = ₱24,451.64. Since the maturity value of an amount invested or borrowed is the amount it is worth in the future at a given interest rate, it is also known as future value. The future value (𝐹𝐹𝐹𝐹) is the amount that matured from the principal, that is, the present value (𝑃𝑃𝑃𝑃). The relationship between the future value and the present value is similar to the relationship of 𝑟𝑟 𝑛𝑛 compound amount and principal 𝐴𝐴 = 𝑃𝑃 1 + 𝑚𝑚. We just replace 𝐴𝐴 with 𝐹𝐹𝐹𝐹 and replace 𝑃𝑃 with 𝑃𝑃𝑃𝑃 so that 𝑟𝑟 𝑛𝑛 𝐹𝐹𝐹𝐹 = 𝑃𝑃𝑃𝑃 1 +. 𝑚𝑚 Sometimes, we may be interested to know the present value in order to figure out how much money must be invested today to achieve a target amount in the future. Manipulating the equation above, we arrive with the formula for 𝑃𝑃𝑃𝑃 𝐹𝐹𝐹𝐹 𝑃𝑃𝑃𝑃 =. 𝑟𝑟 𝑛𝑛 1 + 𝑚𝑚 Example: How much will ₱15,000 compounded semi-annually for five (5) years grow at 6% interest? Solution: Let 𝑃𝑃𝑃𝑃 = ₱15,000, 𝑟𝑟 = 0.06, 𝑚𝑚 = 2, and 𝑛𝑛 = 2(5) = 10. Solving for the future value, 𝑟𝑟 𝑛𝑛 𝐹𝐹𝐹𝐹 = 𝑃𝑃𝑃𝑃 1 + 𝑚𝑚 0.06 10 = ₱15,000 1 + 2 = ₱20,158.75. 06 Handout 1 *Property of STI  [email protected] Page 5 of 6  SH1902 Example: If you target to accumulate ₱2,000,000 for retirement at 3% interest compounded quarterly, how much would you need to invest today to have this amount in 45 years? Solution: Let 𝐹𝐹𝐹𝐹 = ₱2,000,000, 𝑟𝑟 = 0.03, 𝑚𝑚 = 4, 𝑛𝑛 = 4(45) = 180. Then the present value should be 𝐹𝐹𝐹𝐹 𝑃𝑃𝑃𝑃 = 𝑟𝑟 𝑛𝑛 1 + 𝑚𝑚 ₱2,000,000 = 0.03 180 1 + 4 = ₱521,098.87. Thus, you need to invest ₱521,098.87 today to have ₱2,000,000 in 45 years. The formula for future value can also be algebraically manipulated to obtain a formula when interest rate (𝑟𝑟), number of compounding periods in a year (𝑚𝑚), or total number of compounding periods (𝑛𝑛) is unknown. When 𝑟𝑟 is unknown When 𝑚𝑚 is unknown When 𝑛𝑛 is unknown 1 𝑟𝑟 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝑛𝑛 𝑚𝑚 = 1 ln 𝑃𝑃𝑃𝑃 𝑟𝑟 = 𝑚𝑚 − 1 𝐹𝐹𝐹𝐹 𝑛𝑛 𝑛𝑛 = 𝑟𝑟 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃 − 1 ln 1 + 𝑚𝑚 Table 1.2 Note that when time is unknown, we simply divide 𝑛𝑛 by 𝑚𝑚, that is, 𝑛𝑛 𝑡𝑡 =. 𝑚𝑚 References: Domain and range of exponential and logarithmic functions. (n.d.). In Varsity Tutors. Retrieved from https://www.varsitytutors.com/hotmath/hotmath_help/topics/domain-and-range-of- exponential-and-logarithmic-functions Chua, R., Ubarro, A., & Wu, Z. (2016). Soaring 21st century mathematics (general mathematics). Quezon City: Phoenix Publishing House. Fernando, O. (2016) Next century mathematics (general mathematics). Quezon City: Phoenix Publishing House. Lim, Y., Nocon E., Nocon, R., & Ruivivar L. (2016). Math for engaged learning (general mathematics). Quezon City: Sibs Publishing House. Melosantos, L. (2016). Math connections in the digital age (general mathematics). Quezon City: Sibs Publishing House. Zorilla, R. (2016). General mathematics for senior high school. Malabon City: Mutya Publishing House. 06 Handout 1 *Property of STI  [email protected] Page 6 of 6  SH1902 Annuities An annuity is an amount to be paid, usually in equal amounts at equal time intervals (Regacho, Benjamin, & Oryan, 2017, p. 165). Annuities are usually applied in payments for large purchases. It may also be considered as an investment with the promise that it will be paid over a certain number of periods. Common Applications: Houses Cars Condominiums Insurance plans Annuities are classified according to payment intervals and interest conversion periods. Payment interval - The time between successive payments of an annuity Term of an annuity - The number of periods from the first payment interval to the last payment interval Simple annuity – Classification of annuity wherein payment intervals and interest conversion periods are the same General annuity – Classification of annuity wherein payment intervals and interest conversion periods are unequal Simple and general annuities are further classified into three (3) classifications. Ordinary annuity (annuity-immediate) – Periodic payments are made at the end of the payment intervals. Annuity due – Periodic payments are made at the beginning of payment intervals. Deferred annuity – Periodic payment is due at some later date. Accumulation phase – The time money is put into the annuity until it is released Distribution phase – The time after the release of annuity If each phase consists of a single payment, then there is no annuity. If each phase consists of more than one payment, then an annuity exists. The present value or future value of a sequence of payments can be computed like the present value or future value of a single amount. But, recursive computation for every payment and adding them together would be tedious. Thus, you use a formula to calculate the present value (𝑃𝑃𝑃𝑃) and the future value (𝐹𝐹𝐹𝐹) of an ordinary annuity given periodic payment (𝑅𝑅), interest rate per period (𝑖𝑖), and number of payment periods (𝑛𝑛). (1 + 𝑖𝑖)𝑛𝑛 − 1 (1 + 𝑖𝑖)𝑛𝑛 − 1 𝑃𝑃𝑃𝑃 = 𝑅𝑅 𝐹𝐹𝐹𝐹 = 𝑅𝑅 𝑖𝑖(1 + 𝑖𝑖)𝑛𝑛 𝑖𝑖 07 Handout 1 *Property of STI  [email protected] Page 1 of 5  SH1902 Example: Money is worth 12% compounded quarterly. What are the present and future values of an ordinary annuity paying ₱1500 quarterly for 12 years? Solution: 0.12 With 𝑖𝑖 = 4 = 0.03, 𝑅𝑅 = ₱1,500, and 𝑛𝑛 = 12(4) = 48, then, (1 + 𝑖𝑖)𝑛𝑛 − 1 (1 + 𝑖𝑖)𝑛𝑛 − 1 𝑃𝑃𝑃𝑃 = 𝑅𝑅 𝐹𝐹𝐹𝐹 = 𝑅𝑅 𝑖𝑖(1 + 𝑖𝑖)𝑛𝑛 𝑖𝑖 (1 + 0.03)48 − 1 (1 + 0.03)48 − 1 = ₱1500 = ₱1500 0.03(1 + 0.03)48 0.03 (1.03)48 − 1 (1.03)48 − 1 = ₱1500 = ₱1500 0.03(1.03)48 0.03 = ₱37,900.06 = ₱156,612.59 Therefore, the present value of the annuity is ₱37,900.06, and its future value is ₱156,612.59. Example: Benicio paid ₱150,000 as down payment for a vacant lot and agreed to pay ₱21,000 at the end of every six (6) months for five (5) and a half years to complete the purchase of the lot. If money is worth 10% compounded semiannually, what is the equivalent cash price of the lot? Solution: The cash price of the lot is equal to the sum of the down payment and the annuity of 11 payments. 0.1 Note that 𝑅𝑅 = ₱21,000, 𝑖𝑖 = 2 = 0.05, and 𝑛𝑛 = 5.5(2) = 11. Hence, (1 + 𝑖𝑖)𝑛𝑛 − 1 𝑃𝑃𝑃𝑃 = 𝑅𝑅 𝑖𝑖(1 + 𝑖𝑖)𝑛𝑛 (1 + 0.05)11 − 1 = ₱21,000 0.05(1 + 0.05)11 (1.05)11 − 1 = ₱21,000 0.05(1.05)11 = ₱174,434.70. Therefore, the cash price of the lot is ₱150,000 + ₱174,434.70 = ₱324,434.70. The present value of an annuity due (𝑃𝑃𝑉𝑉due ) is the sum of the present values of the periodic payments. The future value of an annuity due (𝐹𝐹𝑉𝑉due ) is the sum of the accumulated values of the payments at the end of the term. That is, given the periodic payment 𝑅𝑅, the interest rate 𝑖𝑖, and the number of payment periods 𝑛𝑛, 𝑃𝑃𝑉𝑉due and 𝐹𝐹𝑉𝑉due are solved as follows: 1 − (1 + 𝑖𝑖)1−𝑛𝑛 (1 + 𝑖𝑖)𝑛𝑛+1 − 1 𝑃𝑃𝑉𝑉due = 𝑅𝑅 + 1 𝐹𝐹𝑉𝑉due = 𝑅𝑅 − 1 𝑖𝑖 𝑖𝑖 07 Handout 1 *Property of STI  [email protected] Page 2 of 5  SH1902 Example: If money is worth 8% compounded quarterly, what is the present value and the future value of an annuity due paying ₱2,500 quarterly for a term of two (2) years? Solution: The problem could be presented using the time diagram below. 0.08 With 𝑖𝑖 = 4 = 0.02, 𝑅𝑅 = ₱2,500, and 𝑛𝑛 = 2(4) = 8, then, 1 − (1 + 𝑖𝑖)1−𝑛𝑛 (1 + 𝑖𝑖)𝑛𝑛+1 − 1 𝑃𝑃𝑉𝑉due = 𝑅𝑅 + 1 𝐹𝐹𝑉𝑉due = 𝑅𝑅 − 1 𝑖𝑖 𝑖𝑖 1 − (1 + 0.02)1−8 (1 + 0.02)8+1 − 1 = ₱2,500 + 1 = ₱2,500 − 1 0.02 0.02 1 − (1.02)−7 (1.02)9 − 1 = ₱2,500 + 1 = ₱2,500 − 1 0.02 0.02 = ₱18,679.98 = ₱21,886.57 07 Handout 1 *Property of STI  [email protected] Page 3 of 5  SH1902 An annuity whose term does not begin until the expiration of specified time is called a deferred annuity. To say that an annuity is deferred for a certain time means that the term of the annuity starts at the end of this time. The present value of a deferred annuity (𝑃𝑃𝑉𝑉def) whose term is 𝑛𝑛 interest periods and is deferred 𝑑𝑑 periods, is equal to the present value of all payments for 𝑛𝑛 + 𝑑𝑑 periods minus the present value of the 𝑑𝑑 periods. Present Value of 𝑛𝑛 + 𝑑𝑑 periods (1 + 𝑖𝑖)𝑛𝑛+𝑑𝑑 − 1 𝑅𝑅 𝑖𝑖(1 + 𝑖𝑖)𝑛𝑛+𝑑𝑑 𝑛𝑛 + 𝑑𝑑 periods 𝑑𝑑 periods deferral 𝑛𝑛 interest periods Present Value of 𝑑𝑑 periods (1 + 𝑖𝑖)𝑑𝑑 − 1 𝑅𝑅 𝑖𝑖(1 + 𝑖𝑖)𝑑𝑑 Given the periodic payment 𝑅𝑅, the interest rate per period 𝑖𝑖, and the number of deferred periods 𝑑𝑑, then (1 + 𝑖𝑖)𝑛𝑛+𝑑𝑑 − 1 (1 + 𝑖𝑖)𝑑𝑑 − 1 𝑃𝑃𝑉𝑉def = 𝑅𝑅 − 𝑅𝑅 𝑖𝑖(1 + 𝑖𝑖)𝑛𝑛+𝑑𝑑 𝑖𝑖(1 + 𝑖𝑖)𝑑𝑑 (1 + 𝑖𝑖)𝑛𝑛+𝑑𝑑 − 1 (1 + 𝑖𝑖)𝑑𝑑 − 1 = 𝑅𝑅 − 𝑖𝑖(1 + 𝑖𝑖)𝑛𝑛+𝑑𝑑 𝑖𝑖(1 + 𝑖𝑖)𝑑𝑑 (1 + 𝑖𝑖)𝑛𝑛+𝑑𝑑 − 1 (1 + 𝑖𝑖)𝑛𝑛+𝑑𝑑 − (1 + 𝑖𝑖)𝑛𝑛 = 𝑅𝑅 − 𝑖𝑖(1 + 𝑖𝑖)𝑛𝑛+𝑑𝑑 𝑖𝑖(1 + 𝑖𝑖)𝑛𝑛+𝑑𝑑 (1 + 𝑖𝑖)𝑛𝑛+𝑑𝑑 − 1 − [(1 + 𝑖𝑖)𝑛𝑛+𝑑𝑑 − (1 + 𝑖𝑖)𝑛𝑛 ] = 𝑅𝑅. 𝑖𝑖(1 + 𝑖𝑖)𝑛𝑛+𝑑𝑑 Simplifying further, we arrive at the formula below. (1 + 𝑖𝑖)𝑛𝑛 − 1 𝑃𝑃𝑉𝑉def = 𝑅𝑅 𝑖𝑖(1 + 𝑖𝑖)𝑛𝑛+𝑑𝑑 07 Handout 1 *Property of STI  [email protected] Page 4 of 5  SH1902 Example: What is the present value of an annuity whose sequence of 12 annual payments of ₱3,000 at a rate of 5%, with the first one due at the end of five (5) years? Solution: The time diagram for the problem can be made as follows. 16 periods 4 periods of Term: 12 annual payments deferral Let 𝑅𝑅 = ₱3,000, 𝑛𝑛 = 12, 𝑑𝑑 = 4, and 𝑖𝑖 = 0.05, then (1 + 0.05)12 − 1 𝑃𝑃𝑉𝑉def = ₱3,000 0.05(1 + 0.05)12+4 (1.05)12 − 1 = ₱3,000 0.05(1.05)16 = ₱21,875.46 References: Chua, R., Ubarro, A., & Wu, Z. (2016). Soaring 21st century mathematics (general mathematics). Quezon City: Phoenix Publishing House. Fernando, O. (2016) Next century mathematics (general mathematics). Quezon City: Phoenix Publishing House. Lim, Y., Nocon E., Nocon, R., & Ruivivar L. (2016). Math for engaged learning (general mathematics). Quezon City: Sibs Publishing House. Melosantos, L. (2016). Math connections in the digital age (general mathematics). Quezon City: Sibs Publishing House. Regacho, C., Benjamin, M., & Oryan, S. (2017). Mathematics skills for life. Quezon City: Abiva Publishing House, Inc. Zorilla, R. (2016). General mathematics for senior high school. Malabon City: Mutya Publishing House. 07 Handout 1 *Property of STI  [email protected] Page 5 of 5  SH1902 Basic Concepts of Stocks and Bonds Stocks and Bonds Stocks and bonds are financial assets but they differ from each other. A stock is a share in the ownership of a company, representing a claim on the company's assets and earnings. A stockholder is a person who holds a company's stock, and this means that person is one of the many owners or shareholders of the company. A stockholder's return on his/her investment is called dividend. S/he may have voting rights to some important company decisions, but only depending on the privileges the company attaches to stock s/he holds. A stockholder’s ownership stake in the company becomes greater when s/he buys more stocks. A stock certificate is a proof of ownership, which represents a stock. Stocks are bought and sold in a stock exchange, also called the stock market. The Philippine Stock Exchange (PSE) regulates the stock market in the country. The original price set by a company for stocks when they are first issued is called par value. The market value is the highest estimated price a buyer would pay, and a seller would accept for a stock in an open and competitive market. The true value of a share of stock is how much an investor is willing to pay for it. Many stocks are issued with little or no par value. These are some important features of stocks: 1. Issuing stocks does not require the company to pay interest. 2. Liability is limited. A stockholder is not personally liable if the company is not able to pay its debts. 3. The more shares a stockholder owns, the larger the portion of the profits s/he gets. A bond is bought to grant credit to a company. When a person buys bonds from a company, s/he becomes a lender to that company instead of a part-owner. The company would now be obliged to pay him/her back its "loan." Bonds may be bought and sold at any time. The term "bond" also refers to the written contract between the issuer of the bond (borrower) and the investor (lender, or the buyer of the bond) that specifies the following: 1. the face value or denomination of the bond on the front of the bond; 2. the redemption date or maturity date on which the loan will be repaid; 3. the bond rate or coupon rate which the bond pays on its face value at regular time intervals until the maturity date; and 4. the redemption value which is the amount promised to be paid on the redemption date. Stocks and bonds are traded through brokers in exchanges either online or in traditional transaction. Brokerage firms can set their own commission charge for bond transactions. 08 Handout 1 *Property of STI  [email protected] Page 1 of 7  SH1902 Kinds of Stocks and Their Dividends Common stock gives the owner the right to share in the profits and to vote on company policy. The share of the profits will vary from year to year, depending on how much profit the company makes and how much is distributed to the stockholders. Preferred stock pays the owner a fixed percentage of the stock's par value each year. Dividends for preferred stocks must be paid before any dividend can be distributed to owners of common stock. However, preferred stock owners generally do not have voting rights in the company. A preferred stock can be cumulative (cumulative preferred stock), that is, a company has the option not to distribute it for a period of time, and the dividends accumulate until they are finally distributed by the company. Example: Consider a stock report of a certain company below. Type of Stock Number of Shares Par Value Stock Dividend Value Cumulative Preferred 10,000 ₱100 9% of par value Common 145,000 ₱10 ₱3.50 How much would stockholders of each kind receive? Solution: Stockholders with cumulative preferred stocks would receive 9% of par value, that is ₱9. 9% × ₱100 = ₱9 Stockholders with common stocks would receive dividends of ₱3.50 per share. The total amount for each stock is as follows. For preferred stock: ₱9 × 10,000 shares = ₱90,000 For common stock: ₱3.50 × 145,000 shares = ₱507,500 Retained earnings is the remaining amount of profit of the company after stock dividends are subtracted. Example: Suppose the company in the previous example had net income of ₱825,000 in one year. How much is the retained earnings that year? Solution: Total dividends distributed is ₱90,000 + ₱507,500 = ₱597,500. Thus, the retained earnings is ₱825,000 − ₱597,500 = ₱227,500. Market indices are indicators of the value of a certain stock used to quantify and compare relative values of stocks and bonds. 08 Handout 1 *Property of STI  [email protected] Page 2 of 7  SH1902 Market Indices for Stocks Stock Yields (percent yield) – Ratio of annual dividend to price per share Annual dividend Percent yield = Price per share Current Yield – Ratio of annual dividend to current price Annual dividend Current yield = Price per share Earnings per Share (EPS) – Ratio of net income to number of outstanding shares Net income EPS = Number of outstanding shares Price-Earnings Ratio (PE ratio) – Ratio of price per share to annual earnings per share Price per share PE ratio = Annual earnings per share The PE ratio measures how expensive a stock is. Rate of Return on a Stock Investment – Ratio of total gains to total cost Total gains Rate of return = Total cost Consider these examples of illustrating ways of quantifying relative values of stocks. Example: Ms. Bautista bought shares from a company at ₱90. If she receives ₱3 annual dividend, what percent yield did she have on her investment? Solution: Annual dividend Percent yield = Price per share ₱3 = ₱90 = 3.33% Example: A company sells stocks at ₱85 and paying ₱2.50 annual dividend. What would be the current yield? 08 Handout 1 *Property of STI  [email protected] Page 3 of 7  SH1902 Solution: Annual dividend Current yield = Current price ₱2.50 = ₱85 = 2.94% Example: A company reported net income of ₱870,000 and had 40,000 outstanding shares of stocks. If the current price is ₱75, what are the EPS and the PE ratio of the stock? Solution: Net income Price per share EPS = PE ratio = Number of outstanding shares Annual earnings per share ₱870,000 ₱75 = = 40,000 ₱21.75 = ₱21.75 = 3.45 Example: With the help of stockbroker Phebe, Timothy purchased 525 shares at ₱18 and received his dividends for that year. He sold his shares the next year at ₱21 per share, again with Phebe’s help. If Phebe charges 3% broker commissions and the current dividend is ₱1 per share, what is the company’s rate of return? Solution: Consider the table that summarizes the situation above. Expenses Gains 525 × ₱18 = ₱9,450 525 × ₱21 = ₱11,025 Broker’s Commission Broker’s Commission 3% × ₱9,450 = ₱283.50 3% × ₱11,025 = ₱330.75 Total Cost Total Received ₱9,450 + ₱283.50 = ₱9,733.50 ₱11,025 + ₱330.75 = ₱11,355.75 Net gain = Total received − Total cost = ₱11,355.75 − ₱9,733.50 = ₱1,622.25 Total gain = Net gain + Dividends (₱1 per share) = ₱1,622.25 + ₱525 = ₱2,147.25 Thus, Total gains Rate of return = Total cost ₱2,147.25 = ₱9733.50 = 22.06%. 08 Handout 1 *Property of STI  [email protected] Page 4 of 7  SH1902 Market Indices for Bonds Bond Yields (percent yield) – Ratio of annual dividend to price per share Annual dividend Percent yield = Price per share Current Yield – Ratio of annual dividend to current price Annual dividend Current yield = Current price Approximate Yield to Maturity 2(𝑛𝑛𝑛𝑛 + 𝑃𝑃 − 𝐶𝐶) Approximate Yield to Maturity = 𝑛𝑛(𝑃𝑃 + 𝐶𝐶) 𝐼𝐼 is the amount of annual interest 𝑃𝑃 is the par value of bond 𝐶𝐶 is the cost or current price 𝑛𝑛 is the number of years to maturity Example: A ₱7,000 bond bears 6% interest. Find the percent yield under the conditions below. a. The bond is bought at 95. b. The bond is bought at 105. c. The face value (₱7,000) is paid. Solution: Annual dividend is 6% × ₱7,000 = ₱420. Price per share varies for each condition. a. Since the bond is bought at 95, then Price per share = 95% × ₱7,000 = ₱6,650. Therefore, Annual dividend Percent yield = Price per share ₱420 = ₱6,650 = 6.32%. b. Since the bond is bought at 105, then Price per share = 105% × ₱7,000 = ₱7,350. Therefore, 08 Handout 1 *Property of STI  [email protected] Page 5 of 7  SH1902 Annual dividend Percent yield = Price per share ₱420 = ₱7,350 = 5.71%. c. Since face value is paid, then the price per share is 100% of ₱7,000, that is, Price per share = ₱7,000. Therefore, Annual dividend Percent yield = Price per share ₱420 = ₱7,000 = 6% Example: Consider the same bond in the previous example, that is, ₱7,000 bond bearing 6% interest. Find the current yield under the conditions below. a. The bond is priced at 92. b. The bond is priced at 104. Solution: a. Since bond is priced at 92, then Current price = 92% × ₱7,000 = ₱6,440. Therefore, Annual dividend Current yield = Current price ₱420 = ₱6,440 = 6.52%. b. Since bond is priced at 104, then Current price = 104% × ₱7,000 = ₱7,280. Therefore, Annual dividend Current yield = Current price ₱420 = ₱7,280 = 5.77%. Example: What is the approximate yield to maturity of a ₱6,000 bond bearing 5% interest, bought at 108, with 10 years left to maturity? 08 Handout 1 *Property of STI  [email protected] Page 6 of 7  SH1902 Solution: Given 𝐼𝐼 = 5% × ₱6,000 = ₱300 𝑃𝑃 = ₱6,000 𝐶𝐶 = 108% × ₱6,000 = ₱6,480 𝑛𝑛 = 10 Therefore, 2(𝑛𝑛𝑛𝑛 + 𝑃𝑃 − 𝐶𝐶) Approximate Yield to Maturity = 𝑛𝑛(𝑃𝑃 + 𝐶𝐶) 2(10(₱300) + ₱6,000 − ₱6,480) = 10(₱6,000 + ₱6,480) = 4.04%. Theory of Efficient Markets The Theory of Efficient Markets holds that stocks are already accurately priced and they reflect all available information about companies. Hence, there is no way to predict future stock prices and the only way to earn higher returns is by purchasing higher risk investments. Given how broad the original theory was, Eugene Fama divided the theory into three (3) sub-theories: The weak form assumes that current stock prices fully reflect all historical information, including past returns. Thus investors would gain little from technical analysis, or the practice of studying a stock's price chart in an attempt to determine where the stock price is going to go in the future. The semi-strong form assumes that stock prices fully reflect all historical information and all current publicly available information. Thus, investors gain little from fundamental analysis, or the practice of examining a company's financial statements and recent developments. The strong-form states that prices reflect not just historical and current publicly available information, but insider information, too. Investors therefore can't benefit from technical analysis, fundamental analysis, or insider information References: Chua, R., Ubarro, A., & Wu, Z. (2016). Soaring 21st century mathematics (general mathematics). Quezon City: Phoenix Publishing House. Fernando, O. (2016) Next century mathematics (general mathematics). Quezon City: Phoenix Publishing House. Lim, Y., Nocon E., Nocon, R., & Ruivivar L. (2016). Math for engaged learning (general mathematics). Quezon City: Sibs Publishing House. Melosantos, L. (2016). Math connections in the digital age (general mathematics). Quezon City: Sibs Publishing House. Regacho, C., Benjamin, M., & Oryan, S. (2017). Mathematics skills for life. Quezon City: Abiva Publishing House, Inc. Zorilla, R. (2016). General mathematics for senior high school. Malabon City: Mutya Publishing House. 08 Handout 1 *Property of STI  [email protected] Page 7 of 7  SH1902 Logic Propositions A proposition is a declarative statement that can be evaluated either true or false but cannot be both. A simple proposition consists of one (1) declarative sentence or statement. A compound proposition consists of two (2) or more simple propositions joined together by logical operators. Logical Operators Logical operators are words that either: combine two (2) or more simple propositions to form a new compound proposition; or modify the meaning of a proposition. Propositions can be denoted by variables (usually uppercase). Consider two (2) simple propositions below denoted 𝑃𝑃 and 𝑄𝑄. 𝑃𝑃: The animal barks. 𝑄𝑄: The animal is a dog. A. Conjunction: Uses the word “and” to join together two (2) propositions Symbol: ∧ Example: 𝑃𝑃 ∧ 𝑄𝑄 Read as: 𝑃𝑃 AND 𝑄𝑄 (The animal barks AND it is a dog.) TRUE: When ALL its components are TRUE FALSE: When AT LEAST ONE (1) of its components is FALSE B. Disjunction: Uses the word “or” to join together two (2) propositions Symbol: ∨ Example: 𝑃𝑃 ∨ 𝑄𝑄 Read as: 𝑃𝑃 OR 𝑄𝑄 (The animal barks or it is a dog.) TRUE: When AT LEAST ONE (1) of its components is TRUE FALSE: When ALL of its components are FALSE C. Implication: Uses “if-then” to construct a new proposition from two (2) propositions In an implication, the first proposition is called the premise while the second is called the conclusion. Symbol: → Example: 𝑃𝑃 → 𝑄𝑄 Read as: IF 𝑃𝑃, THEN 𝑄𝑄 (IF the animal barks, THEN it is a dog.) 09 Handout 1 *Property of STI  [email protected] Page 1 of 4  SH1902 (Premise: The animal barks; Conclusion: The animal is a dog.) TRUE: When the premise is FALSE, or when the premise and the conclusion are BOTH TRUE FALSE: When the premise is TRUE but the conclusion is FALSE D. Biconditional: Uses “if and only if” or “is equivalent to” to construct a new proposition from two (2) propositions A biconditional statement simply states that two propositions are equivalent, that is, if the first one is true, then the second must also true, and if the second is true, then the first must also be true. Symbol: ↔ Example: 𝑃𝑃 ↔ 𝑄𝑄 Read as: 𝑃𝑃 IF AND ONLY IF 𝑄𝑄 (The animal barks IF AND ONLY IF it is a dog.) TRUE: When BOTH propositions have the same truth value, that is, when BOTH are TRUE, or when BOTH are FALSE FALSE: When the propositions have opposite truth value, that is, one is TRUE but the other is FALSE E. Negation: Precedes a proposition with the word “not” Symbol: ∼ Example: ∼ 𝑃𝑃 Read as: The animal does NOT bark. TRUE: When the proposition is FALSE FALSE: When the proposition is TRUE TRUTH TABLES 𝑃𝑃 𝑄𝑄 𝑃𝑃 ∧ 𝑄𝑄 𝑃𝑃 ∨ 𝑄𝑄 𝑃𝑃 → 𝑄𝑄 𝑃𝑃 ↔ 𝑄𝑄 ∼ 𝑃𝑃 ∼ 𝑄𝑄 T T T T T T F F T F F T F F F T F T F T T F T F F F F F T T T T Tautologies and Contradictions A tautology is a proposition that is always true under any circumstance. On the other hand, a contradiction is always false under any circumstance. In determining whether a proposition is a tautology, a contradiction, or neither, truth tables are used. Example: Let 𝑃𝑃 and 𝑄𝑄 be simple propositions. Determine which of these compound propositions are tautology/ies, contradiction/s, or neither. a. 𝑃𝑃 ∨ (∼ 𝑃𝑃) c. 𝑃𝑃 ∨ (∼ 𝑄𝑄) b. 𝑄𝑄 ∧ (∼ 𝑄𝑄) d. (𝑃𝑃 ∨ 𝑄𝑄) ∧ [(∼ 𝑃𝑃) ∧ (∼ 𝑄𝑄)] 09 Handout 1 *Property of STI  [email protected] Page 2 of 4  SH1902 Solutions: a. 𝑃𝑃 ∨ (∼ 𝑃𝑃) 𝑃𝑃 ∼ 𝑃𝑃 𝑃𝑃 ∨ (∼ 𝑃𝑃) T F T F T T Conclusion: It is a tautology. b. 𝑄𝑄 ∧ (∼ 𝑄𝑄) 𝑄𝑄 ∼ 𝑄𝑄 𝑄𝑄 ∧ (∼ 𝑄𝑄) T F F F T F Conclusion: It is a contradiction. c. 𝑃𝑃 ∨ (∼ 𝑄𝑄) 𝑃𝑃 𝑄𝑄 ∼ 𝑄𝑄 𝑃𝑃 ∨ (∼ 𝑄𝑄) T T F T T F T T F T F F F F T T Conclusion: It is NEITHER a tautology nor a contradiction. d. (𝑃𝑃 ∨ 𝑄𝑄) ∧ [(∼ 𝑃𝑃) ∧ (∼ 𝑄𝑄)] 𝑃𝑃 𝑄𝑄 ∼ 𝑃𝑃 ∼ 𝑄𝑄 𝑃𝑃 ∨ 𝑄𝑄 (∼ 𝑃𝑃) ∧ (∼ 𝑄𝑄) (𝑃𝑃 ∨ 𝑄𝑄) ∧ [(∼ 𝑃𝑃) ∧ (∼ 𝑄𝑄)] T T F F T F F T F F T T F F F T T F T F F F F T T F T F Conclusion: It is a contradiction. Rules of inference are rules that provide the way of drawing a correct conclusion from a given premise. With these rules are syllogisms that draw correct conclusion from two (2) or more premises. Examples: If 𝑄𝑄 is the consequence of 𝑃𝑃, and 𝑃𝑃 happens, then 𝑄𝑄 also happens. (This rule of inference is known as Modus Ponens.) 09 Handout 1 *Property of STI  [email protected] Page 3 of 4  SH1902 If 𝑄𝑄 is the consequence of 𝑃𝑃, and 𝑄𝑄 did not happen, then 𝑃𝑃 does not happen. (This rule of inference is known as Modus Tollens.) The classmate I have without a pen is Mike. The classmate who borrowed my pen is my only classmate without a pen. Therefore, the person who borrowed my pen is Mike. All dolphins are mammals. All mammals have kidneys. Therefore, all dolphins have kidneys. (The last two (2) arguments in which the conclusion absolutely follows from the given premises is called syllogism.) A fallacy is a kind of reasoning in which the conclusion does not necessarily or logically follow from the premise. Hence, it is also known as faulty, invalid, or erroneous reasoning. Examples: In a class of 50 students, 35 receive daily allowance worth above P100. Hence, all 50 students receive allowance above P100. (The error of attributing to the whole what is observed to some is known as the fallacy of composition.) If you do anything you want, then you will find joy in life. (The error of failing to give logical connection between the premise and the conclusion, but rather the arguments appeal to one’s emotion is known as the fallacy of relevance.) References: Chua, R., Ubarro, A., & Wu, Z. (2016). Soaring 21st century mathematics (general mathematics). Quezon City: Phoenix Publishing House. Fernando, O. (2016) Next century mathematics (general mathematics). Quezon City: Phoenix Publishing House. Lim, Y., Nocon E., Nocon, R., & Ruivivar L. (2016). Math for engaged learning (general mathematics). Quezon City: Sibs Publishing House. Melosantos, L. (2016). Math connections in the digital age (general mathematics). Quezon City: Sibs Publishing House. Regacho, C., Benjamin, M., & Oryan, S. (2017). Mathematics skills for life. Quezon City: Abiva Publishing House, Inc. Zorilla, R. (2016). General mathematics for senior high school. Malabon City: Mutya Publishing House. 09 Handout 1 *Property of STI  [email protected] Page 4 of 4