Mathematics of Finance PDF

GE1707 The Mathematics of Finance Simple Interest Simple Interest A fixed percentage of the principal (the total amount invested) paid to a depositor or an investor each year the principal is left on deposit or has been invested; usually denoted...

GE1707 The Mathematics of Finance Simple Interest Simple Interest A fixed percentage of the principal (the total amount invested) paid to a depositor or an investor each year the principal is left on deposit or has been invested; usually denoted by 𝐼𝐼 Fixed amount paid to a bank or any lender (one who gives money to a borrower) each year the principal has been borrowed Formula: 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑥𝑥 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑥𝑥 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 or simply 𝐼𝐼 = 𝑃𝑃𝑃𝑃𝑃𝑃 Where: 𝐼𝐼 - Simple Interest 𝑃𝑃 - Principal 𝑟𝑟 - Rate 𝑡𝑡 - Time Then the final amount or Maturity value (𝐹𝐹) at the end of 𝑡𝑡 years can be solved using 𝐹𝐹 = 𝑃𝑃 + 𝐼𝐼 Derived Formulas 𝐼𝐼 1. 𝑃𝑃 = 𝑟𝑟𝑟𝑟 𝐼𝐼 2. 𝑟𝑟 = 𝑃𝑃𝑃𝑃 𝐼𝐼 3. 𝑡𝑡 = 𝑃𝑃𝑃𝑃 4. 𝐹𝐹 = 𝑃𝑃 + 𝐼𝐼 = 𝑃𝑃 + 𝑃𝑃𝑃𝑃𝑃𝑃 𝐹𝐹 = 𝑃𝑃(1 + 𝑟𝑟𝑟𝑟) Example 1: If ₱1,500 was borrowed at 8% simple interest, how much will the interest be for 2 years? Given: 𝑃𝑃 = ₱1,500; 𝑟𝑟 = 8% 𝑜𝑜𝑜𝑜.08; 𝑡𝑡 = 2 years Find: 𝐼𝐼 Example 2: If ₱300 is the interest at 9% after 4 months, how much was borrowed? 4 1 Given: 𝐼𝐼 = ₱300; 𝑟𝑟 = 9% 𝑜𝑜𝑜𝑜.09; 𝑡𝑡 = 𝑜𝑜𝑜𝑜 year 12 3 Find: 𝑃𝑃 Example 3: If ₱1,912.50 is the interest for investing ₱9,000 for 2 years and 6 months, find the rate of interest. Given: 𝐼𝐼 = ₱1,912.50; 𝑃𝑃 = ₱9,000; 𝑡𝑡 = 2.5 years Find: 𝑟𝑟 Example 4: Accumulate ₱8,000 for 1 year and 6 months at 10% simple interest. Given: 𝑃𝑃 = ₱8,000; 𝑡𝑡 = 1.5 years;𝑟𝑟 = 10% 𝑜𝑜𝑜𝑜.1 Find: 𝐹𝐹 05 Handout 1 *Property of STI Page 1 of 13  GE1707 Simple Discount Discount 𝐷𝐷 is a deduction from maturity amount 𝐹𝐹 of an obligation allowed for paying it currently. The formula is 𝐷𝐷 = 𝐹𝐹𝐹𝐹𝐹𝐹, where: 𝐷𝐷 − discount 𝐹𝐹 − amount of maturity 𝑑𝑑 − discount rate 𝑡𝑡 − time or term of discount To find 𝑃𝑃, use 𝑃𝑃 = 𝐹𝐹 − 𝐷𝐷 or 𝑃𝑃 = 𝐹𝐹(1 − 𝑑𝑑𝑑𝑑) Derived Formulas are 𝐷𝐷 𝐷𝐷 𝐷𝐷 𝑑𝑑 = ; 𝑡𝑡 = ; and 𝐹𝐹 = 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝑑𝑑𝑑𝑑 Example: Find the present value of ₱2,000, which is due at the end of 90 days at 5% simple discount. 1 Given: 𝐹𝐹 = ₱2,000; 𝑡𝑡 = ; 𝑑𝑑 =.05 4 Actual Time This is the actual number of days between two (2) dates. Approximate Time This method considered that there were 30 days in each month or 360 days in one (1) year. Example 1: A note dated February 28 is due to be paid August 1. How many days will the note run? Example 2: Find the due date for a 130-day note dated July 7. Example 3: Determine the actual time and approximate time from March 3, 2015 to September 10, 2015. 05 Handout 1 *Property of STI Page 2 of 13  GE1707 Ordinary and Exact Interest Exact Interest denoted by 𝐼𝐼𝑒𝑒 , is a type of simple interest computed based on 365 days, that is, the exact number 𝟏𝟏 of days in a year. In other words, the time 𝑡𝑡 for 1 day is 𝑡𝑡 = year. 𝟑𝟑𝟑𝟑𝟑𝟑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝐼𝐼𝑒𝑒 = Pr 365 Ordinary Interest denoted by 𝐼𝐼𝑜𝑜 , is a type of interest computed based on 360 days, that is, assuming each month 1 in a year has 30 days. In other words, the time 𝑡𝑡 for 1 day is 𝑡𝑡 = year. 360 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝐼𝐼𝑜𝑜 = Pr 360 Example 1: Find the interest on ₱28, 700 at 7.3% from March 14, 2016 to August 16, 2016 using the following: a. ordinary interest using actual time c. exact interest using actual time b. ordinary interest using approximate time d. exact interest using approximate time Solution: Given: 𝑃𝑃 = ₱28,700 𝑟𝑟 = 7.3% = 0.073 The first step is to determine the approximate time and actual time of the term, then compute for the ordinary interest and exact interest. Month Actual Time Approximate Time March 14, 2016 31-14=17 30-14=16 April 30 30 May 31 30 June 30 30 July 31 30 August 16, 2016 16 16 Total 155 152 a. Ordinary Interest using Actual Time: 𝑡𝑡 = 155 days 𝐼𝐼 = 𝑃𝑃𝑃𝑃𝑃𝑃 155 𝐼𝐼 = 28,700(0.073) 360 𝐼𝐼 = ₱902.06 b. Ordinary Interest using Approximate Time: 𝑡𝑡 = 152 days 𝐼𝐼 = 𝑃𝑃𝑃𝑃𝑃𝑃 155 𝐼𝐼 = 28,700(0.073) 360 𝐼𝐼 = ₱884.60 c. Exact Interest using Actual Time: 𝑡𝑡 = 155 days 𝐼𝐼 = 𝑃𝑃𝑃𝑃𝑃𝑃 155 𝐼𝐼 = 28,700(0.073) 365 𝐼𝐼 = ₱889.70 d. Exact Interest using Approximate Time: 𝑡𝑡 = 155 days 𝐼𝐼 = 𝑃𝑃𝑃𝑃𝑃𝑃 155 𝐼𝐼 = 28,700(0.073) 365 𝐼𝐼 = ₱872.48 05 Handout 1 *Property of STI Page 3 of 13  GE1707 Compound Interest Compound Interest is interest computed on the sum of the original principal of a deposit or loan and the interest accumulated. It is denoted by 𝐼𝐼𝑐𝑐. Computing for the Total Number of Conversion Periods When the interest rate is compounded annually, interest is computed once a year. Thus, one (1) conversion period is equivalent to one (1) year. The frequency of conversion, denoted by 𝑚𝑚, is the number of times that the interest is computed in the span of one (1) year. The time or the number of years of the term compounded interest is denoted by 𝑡𝑡. Thus, the total number of conversion periods for the entire term, denoted by 𝑛𝑛, is the product of the number of years 𝑡𝑡 and the frequency of conversion 𝑚𝑚. 𝑛𝑛 = 𝑡𝑡𝑡𝑡 The following table shows the different values of 𝑛𝑛 given a particular frequency conversion 𝑚𝑚 and time 𝑡𝑡. Total Number of Conversions for the Entire Term at Various Conversion Periods Frequency of Description Conversion Period Time (𝑡𝑡) Value of 𝑛𝑛 Conversion (𝑚𝑚) Annually 1 year 1 year 1 1 Semiannually 6 months 3 years 2 6 Quarterly 3 months 2 years 4 8 Monthly 1 month 5 years 12 60 The value of 𝑛𝑛 (total number of conversions for the entire term) is dependent of 𝑚𝑚 (frequency of conversion). If the term 𝑡𝑡 is kept constant, 𝑛𝑛 increases as 𝑚𝑚 increases. Similarly, 𝑛𝑛 is also dependent on 𝑡𝑡. If 𝑚𝑚 is held constant, 𝑛𝑛 increases at 𝑡𝑡 increases. Computing Interest Rates per Period Suppose you want to know how much the interest rate per period is for compound interest charged on a particular loan or investment. How do you get this value? First, determine the nominal rate, or the rate charge that may be converted several times per year, say semiannually. This type of rate is denoted by 𝑗𝑗. For example, you are investing ₱36,000 for 5 years in a bank that pays 3% compounded semiannually. The nominal rate 𝑗𝑗 is 3% or 0.03. Now, to get the interest per period, denoted by 𝑖𝑖, divide the nominal rate by the frequency of conversion per year as follows: 𝑗𝑗 𝑖𝑖 = 𝑚𝑚 The following table shows the different interest rates per period given a particular nominal rate: Interest Rate per Period for Various Conversion Periods Frequency of Interest Rate per Description Interest Period Nominal Rate (𝑗𝑗) Conversion (𝑚𝑚) Period (𝑖𝑖) Annually 1 year 1 10% 10% or 0.10 Semiannually 6 months 2 12% 6% or 0.06 Quarterly 3 months 4 14% 3.5% or 0.035 Monthly 1 month 12 16% 1.33% or 0.0133 The accumulated value of the principal 𝑃𝑃 at the end of the term is called the compound amount, denoted by the variable 𝐹𝐹. The formula for compound amount at the end of 𝑛𝑛 periods is given by 𝐹𝐹 = 𝑃𝑃(1 + 𝑖𝑖)𝑛𝑛 Where: 𝐹𝐹 is the compound amount or accumulated value of the principal 𝑃𝑃 at the end of the term 05 Handout 1 *Property of STI Page 4 of 13  GE1707 𝑃𝑃 is the present value or original principal 𝑖𝑖 is the interest rate per period 𝑛𝑛 is the total number of conversions periods. Example 1 If Mrs. De Leon invested ₱12,900 for 4 years in a bank that pays 3% compounded semiannually, how much will she receive after 4 years? Given: 𝑃𝑃 = ₱12,900; 𝑡𝑡 = 4 years; 𝑚𝑚 = 2; 𝑗𝑗 = 0.03 Solution: First solve for 𝑖𝑖 and 𝑛𝑛. 𝑗𝑗 0.03 𝑖𝑖 = = = 0.015 and 𝑛𝑛 = 𝑡𝑡𝑡𝑡 = 4(2) = 8 𝑚𝑚 2 Since the total number of conversion periods in 4 years is 8 and the interest rate per period is 1.5% or 0.015, it follows that 𝐹𝐹 = 𝑃𝑃(1 + 𝑖𝑖)𝑛𝑛 = (₱12,900.00)(1 + 0.015)8 = ₱14,531.75. Mrs. De Leon will receive ₱14,531.75 after 4 years. Computing Compound Interest Compound interest is the total interest earned for the entire term. You can obtain it by getting the difference between the compound amount 𝐹𝐹 and the principal or present value 𝑃𝑃. 𝐼𝐼𝑐𝑐 = 𝐹𝐹 − 𝑃𝑃 Example 2 Refer to the previous example. How much interest will Mrs. De Leon’s investment earn? Given: 𝑃𝑃 = ₱12,900.00; 𝐹𝐹 = ₱14,531.75 Solution: The compounded interest is computed as 𝐼𝐼𝑐𝑐 = 𝐹𝐹 − 𝑃𝑃 = ₱14,531.75 − ₱12,900.00 = ₱1,631.75. Mrs. De Leon’s investment will earn total interest of ₱1,631.75 after 4 years. Example 3 Find the compound interest earned at the end of 20 months if ₱150,000 is invented in a fund that pays 20% compounded monthly. Given: 𝑃𝑃 = ₱150,000; 𝑗𝑗 = 20% 𝑜𝑜𝑜𝑜 0.2; 𝑚𝑚 = 12; 𝑡𝑡 = 20 months’ Solution: First solve for the interest rate per period. 𝑗𝑗 𝑖𝑖 = 𝑚𝑚 0.2 = 12 = 0.01666 20 Next, solve for the total number of conversion periods. Note that 𝑡𝑡 = 20 months means 𝑡𝑡 = years. 12 𝑛𝑛 = 𝑡𝑡𝑡𝑡 20 = (12) 12 = 20 Compute the compound amount. 𝐹𝐹 = 𝑃𝑃(1 + 𝑖𝑖)𝑛𝑛 = ₱150,000(1 + 0.01666)20 = ₱208,767.53 Lastly, compute for the compound interest. 𝐼𝐼𝑐𝑐 = 𝐹𝐹 − 𝑃𝑃 = ₱208,767.53 − ₱150,000 = ₱58,767.53 05 Handout 1 *Property of STI Page 5 of 13  GE1707 The compounded interest earned for ₱150,000.00 invested in a fund that pays 20% compounded monthly at the end of 20 months is ₱58,767.53. Example 4 Accumulate ₱30,000 for 3 years and 6 months at 16% compounded semiannually. 6 Given: 𝑃𝑃 = ₱30,000; 𝑡𝑡 = 3 years; 𝑗𝑗 = 16% (𝑚𝑚 = 2) 12 Solution: To accumulate means to find the compound amount. First, solve for the interest per period. 𝑗𝑗 0.16 𝑖𝑖 = = = 0.08 𝑚𝑚 2 Next, solve for the total number of conversion periods. 6 𝑛𝑛 = 𝑡𝑡𝑡𝑡 = 3 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 (2) = (3.5)(2) = 7 12 Compute the compound amount. 𝐹𝐹 = 𝑃𝑃(1 + 𝑖𝑖)𝑛𝑛 𝐹𝐹 = ₱30,000.00(1 + 0.08)7 = ₱51,414.73 1 The compound amount at the end of 3 years of ₱30,000.00 invested at 16% (m=2) is ₱51,414.73. 2 Example 5 3 If ₱1.5 million is invested in a fund that pays 23 % compounded monthly for 10 years and 8 months, how much 4 will be in the fund at the end of the term? 3 Given: 𝑃𝑃 = ₱1,500,000 𝑗𝑗 = 23 % 4 𝑚𝑚 = 12 𝑡𝑡 = 10 years and 8 months Solution: First, solve for the interest per period 𝑖𝑖. 3 𝑗𝑗 23 % 23.75% 0.2375 𝑖𝑖 = = 4 = = = 0.01979 𝑚𝑚 12 12 12 Next, solve for the total number of conversion periods. 8 𝑛𝑛 = 𝑡𝑡𝑡𝑡 = 10 (12) = 128 12 Compute the compound amount. 𝐹𝐹 = 𝑃𝑃(1 + 𝑖𝑖)𝑛𝑛 𝐹𝐹 = ₱1,500,000.00(1 + 0.01979)128 𝐹𝐹 = ₱18,431,385.42 3 If ₱1.5 million is invested for 10 years and 8 months in a fund that pays 23 % compounded monthly, the 4 compound amount at the end of the term is ₱18,431,385.42. Present Value and Compound Interest Present Value refers to the value of a certain sum of money at the present time. 𝐹𝐹 𝑃𝑃 = 𝑜𝑜𝑜𝑜 𝑃𝑃 = 𝐹𝐹(1 + 𝑖𝑖)−𝑛𝑛 (1 + 𝑖𝑖)𝑛𝑛 where 𝑃𝑃 is the present value 𝐹𝐹 is the future value or compound amount 𝑖𝑖 is the interest per period 𝑛𝑛 is the total number of conversion periods Example 6 What is the present value of ₱35,000 due in 7 years and 6 months if the rate is 12% compounded quarterly? Given: 𝐹𝐹 = ₱35,000; 𝑡𝑡 = 7 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝑎𝑎𝑎𝑎𝑎𝑎 6 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚ℎ𝑠𝑠 𝑜𝑜𝑜𝑜 7.5 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦; 𝑗𝑗 = 12% = 0.12; 𝑚𝑚 = 4 Solution: Solve for 𝑖𝑖 and 𝑛𝑛. 𝑗𝑗 0.12 𝑖𝑖 = = = 0.03 𝑚𝑚 4 𝑛𝑛 = 𝑡𝑡𝑡𝑡 = (7.5)(4) = 30 Substitute the values of 𝐹𝐹, 𝑖𝑖, and 𝑛𝑛 in the formula for 𝑃𝑃. 05 Handout 1 *Property of STI Page 6 of 13  GE1707 𝐹𝐹 ₱35,000 𝑃𝑃 = = = ₱14,419.54 (1 + 𝑖𝑖) 𝑛𝑛 (1 + 0.03)30 Another way to solve the problem is by using the equivalent formula for 𝑃𝑃, 𝑃𝑃 = 𝐹𝐹(1 + 𝑖𝑖)−𝑛𝑛 = ₱35,000(1 + 0.03)−30 = ₱35,000(1.03)−30 = ₱14,419.54 The present value of ₱35,000.00 that is due at the end of 7.5 years is ₱14,419.54. Example 7 A certain principal 𝑃𝑃 was invested at 6% compounded semiannually. If this principal amounted to ₱94,500 at the end of 3 years, how much was the principal? Find the compound interest earned. Given: 𝐹𝐹 = ₱94,500; 𝑡𝑡 = 3 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦; 𝑗𝑗 = 6% 𝑜𝑜𝑜𝑜 0.06; 𝑚𝑚 = 2 Solution: Solve first for 𝑖𝑖 and 𝑛𝑛. 𝑗𝑗 0.06 𝑖𝑖 = = = 0.03 𝑚𝑚 2 𝑛𝑛 = 𝑡𝑡𝑡𝑡 = (3)(2) = 6 Use the values of 𝐹𝐹, 𝑖𝑖 and 𝑛𝑛 to solve for 𝑃𝑃. 𝑃𝑃 = 𝐹𝐹(1 + 𝑖𝑖)−𝑛𝑛 = ₱94,500.00(1 + 0.03)−6 = ₱79,142.26 The compound interest 𝐼𝐼𝑐𝑐 is then computed as 𝐼𝐼𝑐𝑐 = 𝐹𝐹 − 𝑃𝑃 = ₱94,500.00 − ₱79,142.26 = ₱15,357.74 The present value of ₱94,500.00 that is due at the end of 3 years is ₱79, 142.26. The compounded interest that the investment earned is ₱15,357.74. Stocks and Bonds Stocks A stock is a type of security that signifies ownership in a corporation and represents a claim on part of the corporation’s assets and earnings. Stocks are classified into two (2) types: the common stock and the preferred stock. In the Philippines, stocks are traded regularly at various marketplaces such as Philippines Stock Exchange (PSE) and Market Stock Exchange (MSE). These are places where an investor can buy or sell stocks through a licensed broker authorized to transact business in the market stock places. Bonds A bond is a certificate or a written contract in which the debtor promises to pay its holder a specified amount of money, plus a certain rate of interest at a stated future date. A company that needs money can borrow from investors by selling bonds. A bond is a debt covering a long term such as 10, 20 or more years. The investor is the bondholder who is guaranteed to be repaid at a specified future date. Typical bonds which are issued in various denominators are payable at par value or face value at maturity date. The par value or face value is the principal borrowed as stated in the bond. The par value is usually the price the investor pays when buying the bond from the issuing company. An investor may sell the bond at any time to another investor. Bonds are traded among investors. Hence, the value of a bond fluctuates up and down during its lifetime, depending on how many investors are willing to pay for it. The amount the investor actually pays for the bond is called the market. 05 Handout 1 *Property of STI Page 7 of 13  GE1707 A bond is sold at par when it is sold at a price equal to the face or par value, and is sold at premium when it is sold at a price higher than its face or par value. The bond is bought at a discount if it is bought for less than its par value. Dividends on Preferred and Common Stocks Dividends are the distribution of a company’s profits to its shareholders. Shareholders are persons who own shares of stocks in a corporation. There are two (2) types of stocks: preferred and common stocks. Preferred stocks is a class of corporate stock in which the investor has preferential rights over the common shareholders to dividends and a company’s assets. Common stocks is a class of corporate stock in which the investor has voting rights and shares directly in the success or failure of the business. A par value is an arbitrary monetary figure specified in the corporate charter for each share of stock and printed on each stock certificate. The dividend for par value preferred stock is quoted as a percent of the par value. Alternatively, a no-par value stock is a stock that does not have a par value and the dividend is quoted as a peso amount per share. Cumulative preferred stock is a type of preferred stock that receives a dividend each year. The dividend in arrears is the amount of dividends that accumulate and are owned to cumulative preferred shareholders before for past years in which no dividends are paid. Preferred stocks are categorized as nonparticipating and participating. Nonparticipating stock means that shareholders receive only fixed dividend, while participating stock means that the shareholders may receive additional dividends if the company perform well. Convertible preferred means the stock may be exchanged for a specified number of common shares in the future. The following are the formulas used in stocks: 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 × 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 – 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 (𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝) = 𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 × 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 Example 1: The AUS Enterprises has 1,500,000 shares of common stock outstanding. If a dividend of ₱30,000,000 was declared by the company directors last year, what are the dividends per share of common stock? Solution: Because the company has no preferred stock, the common shareholders will receive the entire dividends. Given: 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = ₱30,000,000 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) = 1,500,000 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑟𝑟 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) 30,000,000 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) = 1,500,000 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) = ₱20 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 The dividend per share of stock is ₱20 Example 2: The board of directors of SSS, Inc. has declared a dividend of ₱18,000,000. The company has 40,000 shares preferred stock that pay ₱60 per share and 80,000 shares of common stock. Calculate the amount of dividends due the preferred shareholders and the dividend per share of common stock. Solution: 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺: 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 (𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝) = 40,000 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 (𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝) = ₱60 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) = 80,000 05 Handout 1 *Property of STI Page 8 of 13  GE1707 The total amount of dividends of a preferred stock is 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 × 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 40,000(60) 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = ₱2,400,000 Total amount of common dividend is 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 − 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 18,000,000 − 2,400,000 𝑇𝑇𝑇𝑇𝑡𝑡𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = ₱15,600,000 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) = ₱195 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑟𝑟𝑟𝑟 The dividend per share of common stock is ₱195. Example 3: Neri Corporation has 70,000 shares of ₱2,000 per value, 5% cumulative preferred stock and 250,000 shares of common stock. Although no dividend was declared last year, a ₱25,000,000 dividend was declared this year. Determine the amount of dividends due the preferred shareholders and the dividend per share of common stock. Solution: Take note that preferred stock is cumulative and the company did not pay a dividend in the previous year, the preferred shareholders are entitled to the dividends in arrears and the dividend for the current period. Given: 𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 = ₱2,000 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 = 5% = 0.05 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 (𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝) = 70,000 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑟𝑟𝑟𝑟𝑟𝑟 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) = 250,000 𝑇𝑇ℎ𝑒𝑒 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑜𝑜𝑜𝑜 𝑎𝑎 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 (𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝) = 𝑃𝑃𝑃𝑃𝑃𝑃 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 × 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 (𝑝𝑝𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟) = 2,000(.05) 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 (𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝) = ₱100 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 The total dividend on a preferred stock in arrears is 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑒𝑒𝑒𝑒 × 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 70,000(100) 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = ₱7,000,000 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 7,000,000 (𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎) + 7,000,000 (𝑐𝑐𝑐𝑐𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦) 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = ₱14,000,000 The amount of dividends due for preferred stock is ₱14,000,000 The amount of dividends due for common stock is 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 − 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 25,000,000 – 14,000,000 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = ₱11,000,000 Computing for the dividend per share 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) 11,000,000 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) = 250,000 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) = ₱44 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 The dividend per share of common stock is ₱44. 05 Handout 1 *Property of STI Page 9 of 13  GE1707 Stock Valuation Stock Valuation is the process of calculating the values of goods or materials owned by a company or available for sale in a store at a particular time. A. Current Yield for a Stock To measure how much you have earned on a stock as compared with other investments, compute for the current yield. The current yield is a way of determining the current value of a stock. The current yield shows how much dividend you can get as a percentage of the current price of the stock per share. If a stock pays no dividend, there is no current yield. The current yield is computed using the formula: 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 Example 4: If MMDN Corporation paid a dividend of ₱142.60 per share last year. If yesterday’s last price was ₱2,300, what is the current yield on the stock? Solution: Given: 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = ₱142.60 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = ₱2,300 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 142.60 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 = 2,300 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 = 0.062 𝑜𝑜𝑜𝑜 6.2% Example 5: Calculate the current yield for MMDN Corporation stock, which pays a dividend of ₱70 per year and is currently selling at ₱1,400 per share. Solution: Given: 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = ₱70 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = ₱1,400 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 70 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 = 1,400 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 = 0.05 𝑜𝑜𝑜𝑜 5% The current yield rate per share is 5%. B. Price-Earnings Ratio of Stock Another thing that some people use to help them decide which stock to buy is the price-earnings ratio. This ratio is found using the formula: 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑃𝑃𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 − 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 = 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 Example 6: WSS Inc. is currently selling for ₱2,685 per share. If the company had earnings per share of ₱89.50 in the past year, what is the price-earnings ratio for WSS? Solution: Given: 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = ₱2,685 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = ₱89.50 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 = 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑟𝑟𝑒𝑒 2,685 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 = 89.50 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 = 30 𝑜𝑜𝑜𝑜 30: 1 This means investors are currently willing to pay 30 times the earnings for one share of WSS stock. 05 Handout 1 *Property of STI Page 10 of 13  GE1707 Example 7: Sofia would like to own stocks in SSS and GSIS, but she does not know if either stock is a good buy. One thing she can do is to look at the price-earnings ratio for each. a. SSS, price share ₱2,464, annual net income per share ₱88, and b. GSIS, price share ₱1,900, annual net income per share ₱76. Solution: Use the formula for price-earnings ratio to get a. Price-earnings ratio for SSS Given: 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = ₱2,464 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 2,464 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 = = = 28 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑝𝑝𝑝𝑝𝑟𝑟 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 88 The price-earnings ratio of SSS stock is 28 times per share. b. Price-earnings ratio for GSIS Given: 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = ₱1,900 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑒𝑒 1,900 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 = = = 25 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 76 The price-earnings ratio of GSIS stock is 25 times per share. The price-earnings ratio is not a perfect guide to future market behavior of a stock. Occasionally a low price- earnings ratio implies that the stock is undervalued in the market-in order words a good buy. At times a low price-earnings ratio denotes that investors see a poor future for the company. C. Cost, Proceeds, and Gain (or loss) of a Stock Proceeds are the amount of money that an investor receives after selling a stock. It is computed as the value of shares less the broker’s commission. The stockbroker’s commission is the fee charges for assisting in the purchase or sale of shares of stocks; percent of the cost of the stock transaction. A stockbroker is a professional in stock market trading and investment who acts as an agent in the selling and buying of stocks or other securities. The gain (or loss) is the difference between the cost of purchasing the stock and the proceeds and received when selling the stock. One more factor affecting the commission is whether the amount of shares purchased is a round lot (multiple of 100 shares), or an odd lot (less than 100 shares). The commission rate on a round lot is generally a bit lower than an odd lot. The following formula will be used: 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑜𝑜𝑜𝑜 𝑆𝑆ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 × 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝑟𝑟 ′ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 × 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 + 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝑟𝑟 ′ 𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 × 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 − 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝑟𝑟 ′ 𝑠𝑠 𝑐𝑐𝑐𝑐𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 (𝑜𝑜𝑜𝑜 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙) 𝑜𝑜𝑜𝑜 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 Example 8: Shiela purchased 250 shares of AUS Inc. common stock at ₱3,500 per share. A few months later, you sell the shares at ₱4,000. Her stockbroker charges 3% commission on round lots and 4% on odd lots. Calculate the (a) total cost, (b) the proceeds, and (c) the gain or loss on the transaction. Solution: Given: Price per share = ₱3,500 Commission rate (round lots) =3%=0.03 Number of shares = 250 Commission rate (odd lots) =4%=0.04 a. Cost of purchasing stock 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 × 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 3,500(250) = ₱875,000 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝑟𝑟 ′ 𝑠𝑠 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 × 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑙𝑙𝑙𝑙𝑙𝑙 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 200 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 × 3,500 × 0.03 = ₱21,000 05 Handout 1 *Property of STI Page 11 of 13  GE1707 𝑂𝑂𝑂𝑂𝑂𝑂 𝑙𝑙𝑙𝑙𝑙𝑙 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 50 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 × 3,500 × 0.04 = ₱7,000 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝑟𝑟 ′ 𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 21,000 + 7,000 = 28,000 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 + 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝑟𝑟 ′ 𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 875,000 + 28,000 = 903,000 The total cost of 250 shares of common stock is ₱903,000. b. Proceeds from selling stock 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑟𝑟𝑟𝑟 × 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 4,000(250) = ₱1,000,000 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝑟𝑟 ′ 𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 × 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑙𝑙𝑙𝑙𝑙𝑙 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 200 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 × 4,000 × 0.03 = ₱24,000 𝑂𝑂𝑂𝑂𝑂𝑂 𝑙𝑙𝑙𝑙𝑙𝑙 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑜𝑜𝑜𝑜 = 50 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 × 4,000 × 0.04 = ₱8,000 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝑟𝑟 ′ 𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 24,000 + 8,000 = ₱32,000 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 − 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝑟𝑟 ′ 𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 1,000,000 − 32,000 = ₱968,000 The proceeds on common stock are ₱968,000. c. Gain (or loss) on the transaction 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 (𝑜𝑜𝑜𝑜 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙)𝑜𝑜𝑜𝑜 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 = 968,000 − 903,000 = ₱65,000 The transaction gain is ₱65,000. Bond Valuation Bond valuation is a technique for determining the fair value of a particular bond. A. Current Yield of Bond The current yield of a bond is computed by dividing the annual interest by the purchase price of a bond. The current yield is obtained using the formula: 𝐴𝐴𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑜𝑜𝑜𝑜 𝑎𝑎 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 Example 9: What is the current yield of a bond whose face value is ₱14,500 and pays a yearly interest of 12% if purchased at face value at ₱13,920? Solution: When the price of bond is ₱14,500 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖. 12(14,500) 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 = = = 12% 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑜𝑜𝑜𝑜 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 14,500 When the price of bond is ₱13,920 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼. 12(14,500) 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 = = = 12.5% 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑜𝑜𝑜𝑜 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 13,920 B. Price of a Bond using General Method The same with stocks, when bonds are sold or bought, charge is commonly added to the price of the bond. The following variables will be in our mathematical treatment of bonds: 𝑉𝑉 = 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑜𝑜𝑜𝑜 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝐹𝐹 = 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑜𝑜𝑜𝑜 𝑝𝑝𝑝𝑝𝑝𝑝 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑃𝑃 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑜𝑜𝑜𝑜 𝑎𝑎 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑟𝑟 = 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑘𝑘 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑗𝑗 = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑚𝑚 = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑝𝑝𝑝𝑝𝑝𝑝 𝑦𝑦𝑦𝑦𝑦𝑦𝑟𝑟 𝑡𝑡 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 (𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡) 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑜𝑜𝑜𝑜 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑏𝑏 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑖𝑖 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 The regular (periodic) interest payment from the bond will be: 𝐶𝐶𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 × 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 05 Handout 1 *Property of STI Page 12 of 13  GE1707 𝑘𝑘 = 𝐹𝐹𝐹𝐹 The price of a bond is computed using the formula below: 1 − (1 + 𝑖𝑖)−𝑛𝑛 𝑃𝑃 = 𝑉𝑉(1 + 𝑖𝑖)−𝑛𝑛 + 𝑘𝑘 𝑖𝑖 Example 10: A ₱3,200, at 9% bond pays coupons quarterly and will be redeemed on July 7, 2016. Find the price if the bond is bought on July 7, 2012 to yield 8% compounded quarterly if the bond is redeemed at par, (b) if the bond if redeemable at 110%. Given: 𝑉𝑉 = ₱3,200 𝑗𝑗 = 8% 𝑜𝑜𝑜𝑜.08 𝑡𝑡 = 4 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝑚𝑚 = 4 𝑟𝑟 = 9% 𝑜𝑜𝑜𝑜.09 𝑛𝑛 = 𝑡𝑡𝑡𝑡 = 4(4 = 16) 𝑟𝑟 0.09 𝑗𝑗. 08 𝑏𝑏 = = =.0225 𝑖𝑖 = = =.02 𝑚𝑚 4 𝑚𝑚 4 The redemption value is 𝑘𝑘 = 𝐹𝐹𝐹𝐹 = 3,200(. 0225) = ₱72 The coupon payments are ₱72. 1 − (1 + 𝑖𝑖)−𝑛𝑛 𝑃𝑃 = 𝑉𝑉(1 + 𝑖𝑖)−𝑛𝑛 + 𝑘𝑘 𝑖𝑖 1 − (1 +.02)−16 𝑃𝑃 = 3,200(1 +.02)−16 + 72. 02 1 − (1.02)−16 𝑃𝑃 = 3,200(1.02)−16 + 72. 02 𝑃𝑃 = 3,200(0.7284458137) + 72(13.57770931) 𝑃𝑃 = 2,331.026604 + 977.5950706 𝑃𝑃 = ₱3,308.62 The price of the bond to yield 8% is ₱3,308.62. At 110% redemption means 𝑉𝑉 = 3,200(1.10) = ₱3,520 1 − (1 + 𝑖𝑖)−𝑛𝑛 𝑃𝑃 = 𝑉𝑉(1 + 𝑖𝑖)−𝑛𝑛 + 𝑘𝑘 𝑖𝑖 1 − (1 + 0.02)−16 𝑃𝑃 = 3,520(1 + 0.02)−16 + 72 0.02 𝑃𝑃 = 3,520(0.7284458137) + 72(13.57770931) 𝑃𝑃 = 2,564.129264 + 977.5950706 𝑃𝑃 = ₱3,541.72 The price of the bond is ₱3,541.72. REFERENCES: Cordova, W., Gotauco, C., Ledesma, F., & Tabuloc, M.C. (2017). Mathematics of finance. Quezon City: Abiva Publishing House, Inc. Regacho, C., Benjamin, JB., & Oryan, S. (2017). Mathematics skills for life. Quezon City: Abiva Publishing House, Inc. Sirug, W. (2016). General mathematics. Manila: Mindshapers Co., Inc. 05 Handout 1 *Property of STI Page 13 of 13