Commercial Arithmetic. Grade 12 Mathematics:Commercial Arithmetic Subtopic Navigator Commercial Arithmetic Simple Interest Compound Interest Annuity Amortization Bonds Shares Cumulative Exercises Conclusion Lesson Objectives Solve complex problems involving simple and compound interest with varying time periods Analyze and calculate annuities with different payment frequencies and interest rates Apply amortization techniques to loan repayment schedules Evaluate bond prices, yields, and returns under different market conditions Calculate share valuations, dividends, and investment returns Apply commercial arithmetic concepts to real-world financial scenarios Commercial Arithmetic Commercial arithmetic forms the mathematical foundation of modern finance and investment. Understanding these concepts is essential for personal financial planning, business decision-making, and professional careers in finance, economics, and accounting. This lesson explores advanced applications of financial mathematics with emphasis on real-world problem solving. Simple Interest Simple interest is calculated only on the principal amount, without considering any accumulated interest from previous periods. Formula: [latex]I = P times r times t[/latex] [latex]A = P(1 + rt)[/latex] Where: I = Interest P = Principal amount r = Annual interest rate (as decimal) t = Time in years A = Total amount after interest Example 1: Intermediate Level A principal of $8,500 is invested at 6.5% simple interest. If the investment earns $2,210 in interest, how long was the money invested? Solution: Using [latex]I = P times r times t[/latex]: [latex]2,210 = 8,500 times 0.065 times t[/latex] [latex]t = frac{2,210}{8,500 times 0.065} = frac{2,210}{552.5} = 4[/latex] years Example 2: Difficult Level Maria invested different amounts in two simple interest accounts. In Account A, she invested $12,000 at 5% p.a., and in Account B, she invested $8,000 at an unknown rate. After 3 years, the total interest from both accounts was $3,420. What was the interest rate for Account B? Solution: Interest from Account A: [latex]12,000 times 0.05 times 3 = 1,800[/latex] Let r be the rate for Account B Interest from Account B: [latex]8,000 times r times 3 = 24,000r[/latex] Total interest: [latex]1,800 + 24,000r = 3,420[/latex] [latex]24,000r = 1,620[/latex] [latex]r = frac{1,620}{24,000} = 0.0675 = 6.75%[/latex] Simple Interest Problems At what simple interest rate will $15,000 triple itself in 12 years? A sum of money doubles itself in 8 years at simple interest. How long will it take to triple itself at the same rate? John invested $25,000 at 4.5% simple interest and another $35,000 at 5.2% simple interest. What is his total interest after 4.5 years? If the simple interest on $18,000 for 5 years exceeds the interest on $12,000 for 6 years by $1,260, find the interest rate. A sum of money amounts to $31,200 in 4 years and to $35,200 in 6 years at simple interest. Find the principal and rate of interest. Compound Interest Compound interest is calculated on the initial principal and also on the accumulated interest of previous periods. Formulas: Annual compounding: [latex]A = P(1 + r)^t[/latex] Multiple compounding periods per year: [latex]A = P(1 + frac{r}{n})^{nt}[/latex] Continuous compounding: [latex]A = Pe^{rt}[/latex] Where: A = Future value P = Principal r = Annual interest rate (decimal) t = Time in years n = Number of compounding periods per year Example 1: Intermediate Level Calculate the compound interest on $15,000 for 3 years at 6% per annum, compounded annually. Solution: [latex]A = 15,000(1 + 0.06)^3 = 15,000(1.06)^3 = 15,000 times 1.191016 = 17,865.24[/latex] Interest = [latex]17,865.24 - 15,000 = 2,865.24[/latex] Example 2: Difficult Level A principal of $20,000 is invested at 5% per annum, compounded semi-annually. How long will it take for the investment to reach $30,000? Solution: [latex]30,000 = 20,000(1 + frac{0.05}{2})^{2t}[/latex] [latex]1.5 = (1.025)^{2t}[/latex] Taking natural logarithms: [latex]ln(1.5) = 2t times ln(1.025)[/latex] [latex]0.405465 = 2t times 0.0246926[/latex] [latex]2t = frac{0.405465}{0.0246926} = 16.424[/latex] [latex]t = 8.212[/latex] years ≈ 8 years 2.5 months Compound Interest Problems Find the compound interest on $50,000 for 3 years at 6% per annum compounded quarterly. At what rate percent per annum compound interest will $10,000 amount to $13,310 in 3 years? The difference between compound interest and simple interest on a certain sum at 5% per annum for 2 years is $25. Find the sum. A sum of money placed at compound interest doubles itself in 5 years. In how many years will it become eight times? If $25,000 is invested at 4.5% compounded monthly, what will be the accumulated amount after 8 years? Annuity An annuity is a series of equal payments made at regular intervals. Common types include ordinary annuities (payments at period end) and annuities due (payments at period beginning). Formulas: Future Value of Ordinary Annuity: [latex]FV = PMT times frac{(1 + r)^n - 1}{r}[/latex] Present Value of Ordinary Annuity: [latex]PV = PMT times frac{1 - (1 + r)^{-n}}{r}[/latex] Where: PMT = Periodic payment r = Interest rate per period n = Number of periods Example 1: Intermediate Level What is the future value of an ordinary annuity of $2,000 per year for 15 years at 7% annual interest? Solution: [latex]FV = 2,000 times frac{(1 + 0.07)^{15} - 1}{0.07}[/latex] [latex]FV = 2,000 times frac{2.75903 - 1}{0.07}[/latex] [latex]FV = 2,000 times frac{1.75903}{0.07} = 2,000 times 25.129 = 50,258[/latex] Future Value = $50,258 Example 2: Difficult Level John wants to retire in 25 years and believes he'll need $1,500,000 at retirement. If he can earn 6.5% annual return, how much must he invest at the end of each year to reach his goal? Solution: Using the future value of annuity formula: [latex]1,500,000 = PMT times frac{(1 + 0.065)^{25} - 1}{0.065}[/latex] [latex](1.065)^{25} = 4.827[/latex] [latex]1,500,000 = PMT times frac{4.827 - 1}{0.065} = PMT times frac{3.827}{0.065} = PMT times 58.877[/latex] [latex]PMT = frac{1,500,000}{58.877} = 25,477.18[/latex] Annual payment needed = $25,477.18 Annuity Problems Find the present value of an annuity of $1,200 payable at the end of each quarter for 5 years at 8% p.a. compounded quarterly. A person invests $800 at the end of each month in an account paying 6% compounded monthly. What will be the amount after 10 years? What equal annual payment should be made to discharge a debt of $50,000 due in 6 years at 7% p.a. compound interest? The future value of an annuity is $75,000 after 12 years of annual payments of $4,000. Find the interest rate. A car costs $35,000. If paid in installments, $10,000 is paid immediately and the balance in 24 equal monthly installments at 12% p.a. compounded monthly. Find the monthly installment. Amortization Amortization is the process of paying off a debt (such as a loan) through regular payments over time. Each payment covers both interest and principal repayment. Formulas: Amortization Payment: [latex]PMT = PV times frac{r(1 + r)^n}{(1 + r)^n - 1}[/latex] Remaining Balance: [latex]B = PMT times frac{1 - (1 + r)^{-(n - k)}}{r}[/latex] Where: PMT = Periodic payment PV = Present value (loan amount) r = Interest rate per period n = Total number of payments k = Number of payments made Example 1: Intermediate Level A home loan of $300,000 is to be repaid over 25 years at 5.4% p.a. compounded monthly. Find the monthly payment. Solution: [latex]PV = 300,000[/latex] [latex]r = frac{0.054}{12} = 0.0045[/latex] [latex]n = 25 times 12 = 300[/latex] [latex]PMT = 300,000 times frac{0.0045(1 + 0.0045)^{300}}{(1 + 0.0045)^{300} - 1}[/latex] [latex](1.0045)^{300} = 3.847[/latex] [latex]PMT = 300,000 times frac{0.0045 times 3.847}{3.847 - 1} = 300,000 times frac{0.0173115}{2.847} = 300,000 times 0.006081 = 1,824.30[/latex] Monthly payment = $1,824.30 Example 2: Difficult Level A $250,000 business loan at 6% p.a. compounded monthly is amortized over 20 years with monthly payments. After 5 years, what is the remaining balance on the loan? Solution: First, calculate the monthly payment: [latex]r = frac{0.06}{12} = 0.005[/latex] [latex]n = 20 times 12 = 240[/latex] [latex]PMT = 250,000 times frac{0.005(1 + 0.005)^{240}}{(1 + 0.005)^{240} - 1}[/latex] [latex](1.005)^{240} = 3.3102[/latex] [latex]PMT = 250,000 times frac{0.005 times 3.3102}{3.3102 - 1} = 250,000 times frac{0.016551}{2.3102} = 250,000 times 0.007164 = 1,791[/latex] After 5 years, 60 payments have been made, with 180 payments remaining: [latex]B = 1,791 times frac{1 - (1 + 0.005)^{-180}}{0.005}[/latex] [latex](1.005)^{-180} = 0.4075[/latex] [latex]B = 1,791 times frac{1 - 0.4075}{0.005} = 1,791 times frac{0.5925}{0.005} = 1,791 times 118.5 = 212,233.50[/latex] Remaining balance = $212,233.50 Amortization Problems A loan of $180,000 is to be amortized over 20 years with monthly payments at 5.25% p.a. Find the monthly payment and total interest paid. A car loan of $35,000 at 7.5% p.a. is to be repaid in 5 years with monthly payments. After 2 years, what is the remaining balance? Construct an amortization schedule for the first three payments of a $100,000 loan at 6% p.a. for 15 years with monthly payments. A business takes a $500,000 loan at 5.8% p.a. for 25 years. After 10 years, what percentage of the loan has been paid off? If extra payments of $200 per month are made on a $300,000 mortgage at 4.5% for 30 years, how much sooner will the loan be paid off? Bonds Bonds are debt instruments where an investor loans money to an entity (corporate or governmental) that borrows the funds for a defined period at a fixed or variable interest rate. Formulas: Bond Price: [latex]P = C times frac{1 - (1 + r)^{-n}}{r} + F times (1 + r)^{-n}[/latex] Current Yield: [latex]CY = frac{Annual Coupon Payment}{Current Market Price}[/latex] Yield to Maturity: Solved numerically from the bond price formula Where: P = Bond price C = Coupon payment per period r = Yield per period n = Number of periods until maturity F = Face value Example 1: Intermediate Level A 5-year bond with face value $1,000 pays 6% annual coupons. If the required yield is 5%, what is the bond's price? Solution: Annual coupon = [latex]1,000 times 0.06 = 60[/latex] [latex]P = 60 times frac{1 - (1 + 0.05)^{-5}}{0.05} + 1,000 times (1 + 0.05)^{-5}[/latex] [latex]P = 60 times frac{1 - 0.78353}{0.05} + 1,000 times 0.78353[/latex] [latex]P = 60 times frac{0.21647}{0.05} + 783.53 = 60 times 4.3294 + 783.53 = 259.76 + 783.53 = 1,043.29[/latex] Bond price = $1,043.29 Example 2: Difficult Level A 10-year bond with face value $1,000 pays semi-annual coupons at 5% p.a. and is priced at $950. What is the yield to maturity? Solution: Semi-annual coupon = [latex]frac{1,000 times 0.05}{2} = 25[/latex] Number of periods = [latex]10 times 2 = 20[/latex] We need to solve for r in: [latex]950 = 25 times frac{1 - (1 + r)^{-20}}{r} + 1,000 times (1 + r)^{-20}[/latex] Using trial and error or financial calculator: Try r = 0.03 (6% annual): [latex]P = 25 times frac{1 - 1.03^{-20}}{0.03} + 1,000 times 1.03^{-20}[/latex] [latex]P = 25 times 14.8775 + 1,000 times 0.55368 = 371.94 + 553.68 = 925.62[/latex] (too low) Try r = 0.0275 (5.5% annual): [latex]P = 25 times frac{1 - 1.0275^{-20}}{0.0275} + 1,000 times 1.0275^{-20}[/latex] [latex]P = 25 times 15.3262 + 1,000 times 0.58201 = 383.16 + 582.01 = 965.17[/latex] (too high) Try r = 0.028 (5.6% annual): [latex]P = 25 times frac{1 - 1.028^{-20}}{0.028} + 1,000 times 1.028^{-20}[/latex] [latex]P = 25 times 15.0986 + 1,000 times 0.57437 = 377.47 + 574.37 = 951.84[/latex] (close) Try r = 0.0281 (5.62% annual): [latex]P = 25 times frac{1 - 1.0281^{-20}}{0.0281} + 1,000 times 1.0281^{-20}[/latex] [latex]P = 25 times 15.0808 + 1,000 times 0.57317 = 377.02 + 573.17 = 950.19[/latex] (very close) Yield to maturity ≈ 5.62% p.a. Bond Problems A 7-year bond with face value $1,000 pays 5.5% annual coupons. If the market requires a 6% yield, what is the bond's price? A zero-coupon bond with face value $10,000 matures in 8 years. If it's priced at $6,500, what is the yield to maturity? A bond with 12 years to maturity pays semi-annual coupons at 4.8% p.a. and is priced at $1,050. If the face value is $1,000, find the yield to maturity. Which bond has higher duration: a 5-year bond with 3% coupon or a 5-year bond with 6% coupon, both with the same yield? A bond portfolio has two bonds: Bond A (5 years, 4% coupon, $50,000 face value) and Bond B (10 years, 5% coupon, $30,000 face value). If both yield 5%, what is the portfolio's market value? Shares Shares represent ownership in a corporation. Share valuation involves determining the theoretical fair value of a company's stock. Formulas: Dividend Discount Model: [latex]P_0 = frac{D_1}{r - g}[/latex] (constant growth) Price-Earnings Ratio: [latex]P/E = frac{Market Price per Share}{Earnings per Share}[/latex] Dividend Yield: [latex]DY = frac{Annual Dividend per Share}{Price per Share}[/latex] Where: P₀ = Current stock price D₁ = Expected dividend next year r = Required rate of return g = Constant growth rate Example 1: Intermediate Level A company is expected to pay a $2.50 dividend next year, with dividends growing at 4% annually. If the required return is 10%, what is the stock's intrinsic value? Solution: Using the constant growth dividend discount model: [latex]P_0 = frac{2.50}{0.10 - 0.04} = frac{2.50}{0.06} = 41.67[/latex] Intrinsic value = $41.67 Example 2: Difficult Level ABC Corp is expected to have irregular dividend growth: $1.50 next year, $1.80 in year 2, $2.00 in year 3, and then constant growth of 5% thereafter. If the required return is 12%, what is the stock worth today? Solution: This is a multi-stage growth model. First, calculate present value of dividends during the abnormal growth period: PV(D₁) = [latex]frac{1.50}{1.12} = 1.339[/latex] PV(D₂) = [latex]frac{1.80}{(1.12)^2} = frac{1.80}{1.2544} = 1.435[/latex] PV(D₃) = [latex]frac{2.00}{(1.12)^3} = frac{2.00}{1.4049} = 1.424[/latex] At the end of year 3, we switch to the constant growth model: D₄ = [latex]2.00 times 1.05 = 2.10[/latex] P₃ = [latex]frac{2.10}{0.12 - 0.05} = frac{2.10}{0.07} = 30.00[/latex] PV(P₃) = [latex]frac{30.00}{(1.12)^3} = frac{30.00}{1.4049} = 21.355[/latex] Total value = [latex]1.339 + 1.435 + 1.424 + 21.355 = 25.553[/latex] Stock value = $25.55 Share Problems A stock pays a $1.20 dividend expected to grow at 6% annually. If the required return is 11%, what is the stock's value? XYZ Corp has earnings per share of $3.50 and a P/E ratio of 15. What is its stock price? A company's stock is priced at $45 with expected dividends of $1.80 next year growing at 5%. What is the implied required return? Calculate the dividend yield for a stock priced at $75 that pays annual dividends of $2.25. If a stock's beta is 1.2, the risk-free rate is 3%, and the market risk premium is 5%, what is the required return using CAPM? Cumulative Exercises An investment of $75,000 earns 5.8% compounded quarterly. What will it be worth after 7 years? What annual payment is required to accumulate $250,000 in 15 years at 6.5% interest? A $400,000 mortgage at 4.2% for 25 years has monthly payments. After 10 years, what is the remaining balance? A 15-year bond with $1,000 face value pays 4.5% semi-annual coupons. If the yield is 5.2%, what is the bond's price? A stock pays a $2.40 dividend expected to grow at 7%. If the required return is 12%, what is its value? How long will it take for money to double at 6% compounded monthly? What is the effective annual rate for 5.6% compounded quarterly? A perpetuity pays $5,000 annually. If the required return is 8%, what is its present value? Which investment is better: 6% compounded monthly or 6.1% compounded annually? A project requires $150,000 initial investment and returns $45,000 annually for 5 years. If the required return is 10%, what is the NPV? Show/Hide Answers Problem: An investment of $75,000 earns 5.8% compounded quarterly. What will it be worth after 7 years? Answer: [latex]A = 75,000(1 + frac{0.058}{4})^{4 times 7} = 75,000(1.0145)^{28} = 75,000 times 1.496 = 112,200[/latex] Future value = $112,200 Problem: What annual payment is required to accumulate $250,000 in 15 years at 6.5% interest? Answer: [latex]250,000 = PMT times frac{(1.065)^{15} - 1}{0.065}[/latex] [latex](1.065)^{15} = 2.5718[/latex] [latex]250,000 = PMT times frac{1.5718}{0.065} = PMT times 24.1815[/latex] [latex]PMT = frac{250,000}{24.1815} = 10,338.65[/latex] Annual payment = $10,338.65 Problem: A $400,000 mortgage at 4.2% for 25 years has monthly payments. After 10 years, what is the remaining balance? Answer: Monthly rate = 0.042/12 = 0.0035, Total payments = 300, Payments made = 120, Remaining = 180 Monthly payment = [latex]400,000 times frac{0.0035(1.0035)^{300}}{(1.0035)^{300} - 1} = 2,157.27[/latex] Remaining balance = [latex]2,157.27 times frac{1 - (1.0035)^{-180}}{0.0035} = 2,157.27 times 142.324 = 307,000[/latex] Remaining balance = $307,000 Problem: A 15-year bond with $1,000 face value pays 4.5% semi-annual coupons. If the yield is 5.2%, what is the bond's price? Answer: Semi-annual coupon = $22.50, Periods = 30, Semi-annual yield = 0.026 Price = [latex]22.50 times frac{1 - (1.026)^{-30}}{0.026} + 1,000 times (1.026)^{-30}[/latex] Price = [latex]22.50 times 19.046 + 1,000 times 0.4632 = 428.54 + 463.20 = 891.74[/latex] Bond price = $891.74 Problem: A stock pays a $2.40 dividend expected to grow at 7%. If the required return is 12%, what is its value? Answer: [latex]P_0 = frac{2.40}{0.12 - 0.07} = frac{2.40}{0.05} = 48.00[/latex] Stock value = $48.00 Problem: How long will it take for money to double at 6% compounded monthly? Answer: [latex]2 = (1 + frac{0.06}{12})^{12t}[/latex] [latex]ln 2 = 12t times ln(1.005)[/latex] [latex]0.6931 = 12t times 0.0049875[/latex] [latex]t = frac{0.6931}{12 times 0.0049875} = frac{0.6931}{0.05985} = 11.58[/latex] years Problem: What is the effective annual rate for 5.6% compounded quarterly? Answer: [latex]EAR = (1 + frac{0.056}{4})^4 - 1 = (1.014)^4 - 1 = 1.0572 - 1 = 0.0572 = 5.72%[/latex] Problem: A perpetuity pays $5,000 annually. If the required return is 8%, what is its present value? Answer: [latex]PV = frac{5,000}{0.08} = 62,500[/latex] Present value = $62,500 Problem: Which investment is better: 6% compounded monthly or 6.1% compounded annually? Answer: EAR for 6% monthly = [latex](1 + frac{0.06}{12})^{12} - 1 = 0.06168 = 6.168%[/latex] 6.168% > 6.1%, so 6% compounded monthly is better Problem: A project requires $150,000 initial investment and returns $45,000 annually for 5 years. If the required return is 10%, what is the NPV? Answer: NPV = [latex]-150,000 + 45,000 times frac{1 - (1.10)^{-5}}{0.10}[/latex] NPV = [latex]-150,000 + 45,000 times 3.7908 = -150,000 + 170,586 = 20,586[/latex] NPV = $20,586 Conclusion/Recap Commercial arithmetic provides the mathematical foundation for personal and business financial decision-making. Mastering these concepts enables analysis of investment opportunities, loan structures, and financial instruments. The ability to calculate interest, annuities, amortization schedules, bond prices, and stock valuations is essential for financial literacy and professional success in business, finance, and economics. Clip It! Share your ANSWER in the Chat. 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