# Financial Applications: Compound Interest and Annual Depreciation [78bab9ca474d] ## Compound Interest [379fc50b8ac1] 1. Simple interest is interest that provides a fixed amount each period, for example, 5% of the original amount you put in. 2. **Compound interest**, meanwhile, increases each period by a percentage of the **current** amount. 3. This doesn't sound like a big difference, but** exponential growth** is a powerful thing, and a bank account with compound interest can end up significantly richer than one with simple interest. [{"_key":"c631c8a313ad","_type":"block","asset":null,"children":[{"_key":"cdee04da95f2","_type":"span","marks":[],"text":"Let's deposit $\\tex{\\$}5000$ into two accounts, both with "},{"_key":"604e0874cffe","_type":"span","marks":["strong"],"text":"5% annual interest"},{"_key":"fd66ed5a42f4","_type":"span","marks":[],"text":". "}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"34655ca844d8","_type":"block","asset":null,"children":[{"_key":"58de90f5a2a6","_type":"span","marks":[],"text":"Account $A$ has "},{"_key":"33bbe3a843b5","_type":"span","marks":["strong"],"text":"simple interest,"},{"_key":"0413cb3dd0dd","_type":"span","marks":[],"text":" and account $B$ has "},{"_key":"18bffb490e14","_type":"span","marks":["strong"],"text":"compound interest"},{"_key":"3d70b764421a","_type":"span","marks":[],"text":"."}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"6056aab9adc7","_type":"block","asset":null,"children":[{"_key":"f6bcdd984f2d","_type":"span","marks":["strong"],"text":"Account A's balance"},{"_key":"eb0b264b9855","_type":"span","marks":[],"text":" "},{"_key":"fa0697a65761","_type":"span","marks":["strong"],"text":"increases"},{"_key":"37ff1d8c7e97","_type":"span","marks":[],"text":" every year by $\\text{\\$}(0.05\\times5000)=\\text{\\$}250$."}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"6f76eb4e39ae","_type":"block","asset":null,"children":[{"_key":"0d40d013b900","_type":"span","marks":[],"text":"Meanwhile, "},{"_key":"66001edc3b68","_type":"span","marks":["strong"],"text":"account B's balance"},{"_key":"fd17d9160e32","_type":"span","marks":[],"text":" "},{"_key":"dbbdd69f665c","_type":"span","marks":["strong"],"text":"gets"},{"_key":"8de5834b1802","_type":"span","marks":[],"text":" "},{"_key":"2ca4c922f201","_type":"span","marks":["strong"],"text":"multiplied"},{"_key":"85e55b73e589","_type":"span","marks":[],"text":" by $1.05$ every year."}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"35655a1a7dba","_type":"block","asset":null,"children":[{"_key":"078f0e91bacf","_type":"span","marks":[],"text":"After 50 years, account A will have a balance of $$\\text{\\$}(5000+50(0.05\\times5000))=\\text{\\$}17500$$"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"8d560088b438","_type":"block","asset":null,"children":[{"_key":"5d5202d01648","_type":"span","marks":[],"text":"After 50 years, account B will have a balance of $$\\text{\\$}5000(1.05)^{50}\\approx \\text{\\$}57337$$"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"36cc9feb1387","_type":"block","asset":null,"children":[{"_key":"27df3b79ea62","_type":"span","marks":[],"text":"You can see here how "},{"_key":"fa5eab8c8fd5","_type":"span","marks":["strong"],"text":"compound"},{"_key":"def8356c6866","_type":"span","marks":[],"text":" interest is a lot more "},{"_key":"766422c83d82","_type":"span","marks":["strong"],"text":"profitable"},{"_key":"495f0731461f","_type":"span","marks":[],"text":" than simple interest!"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"}] ### The Basic Formula [67838814ab23] The compound interest formula is: $$ A = P(1 + r)^n $$ where: 1. $A$ is final amount 2. $P$ is principal (initial investment) 3. $r$ is interest rate (as a decimal) 4. $n$ is number of compounding periods [{"_key":"Uk7kePxaUXt2OXbBhw24xK","_type":"block","asset":null,"children":[{"_key":"Uk7kePxaUXt2OXbBhw24y5","_type":"span","marks":[],"text":"When converting "},{"_key":"88428278adfa","_type":"span","marks":["strong"],"text":"interest rates to"},{"_key":"503f9a055555","_type":"span","marks":[],"text":" "},{"_key":"cc92bfa02c41","_type":"span","marks":["strong"],"text":"decimals"},{"_key":"d6be65487c16","_type":"span","marks":[],"text":", divide the percentage by 100. "}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"b4071381c55c","_type":"block","asset":null,"children":[{"_key":"914db9fdfb05","_type":"span","marks":[],"text":"For example, 5% becomes 0.05"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"}] 1. This might look familiar. 2. This is because this is really just a **geometric series** where the common ratio is $1 + r$. 3. For example, if the compound interest rate is $5\%$, the common ratio the balance increases by each year is $1.05$. ### Different Compounding Periods [a800c2a52573] 1. Interest can be compounded at **different frequencies**: 1. Annually (once per year) 2. Semi-annually (twice per year) 3. Quarterly (four times per year) 4. Monthly (twelve times per year) 5. Daily (365 times per year) 2. For **non-annual compounding**, we modify our formula to: $$ A = P(1 + \frac{r}{k})^{kn} $$ where $k$ is the number of times interest is compounded per year. [{"_key":"Uk7kePxaUXt2OXbBhw24yq","_type":"block","asset":null,"children":[{"_key":"Uk7kePxaUXt2OXbBhw24zb","_type":"span","marks":[],"text":"If you invest $1000 at 6% annual interest compounded monthly for 2 years:"}],"markDefs":[],"style":"normal"},{"_key":"Uk7kePxaUXt2OXbBhw2517","_type":"block","asset":null,"children":[{"_key":"Uk7kePxaUXt2OXbBhw251s","_type":"span","marks":[],"text":"$P = 1000$"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"Uk7kePxaUXt2OXbBhw253O","_type":"block","asset":null,"children":[{"_key":"Uk7kePxaUXt2OXbBhw2549","_type":"span","marks":[],"text":"$r = 0.06$"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"Uk7kePxaUXt2OXbBhw255f","_type":"block","asset":null,"children":[{"_key":"Uk7kePxaUXt2OXbBhw256Q","_type":"span","marks":[],"text":"$k = 12$ (monthly)"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"Uk7kePxaUXt2OXbBhw257w","_type":"block","asset":null,"children":[{"_key":"Uk7kePxaUXt2OXbBhw258h","_type":"span","marks":[],"text":"$n = 2$ (years)"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"Uk7kePxaUXt2OXbBhw25AD","_type":"block","asset":null,"children":[{"_key":"Uk7kePxaUXt2OXbBhw25Ay","_type":"span","marks":[],"text":"$$A = 1000(1 + \\frac{0.06}{12})^{24} = 1127.49$$"}],"markDefs":[],"style":"normal"}] ## Annual Depreciation [c8e30aa7dc71] 1. **Depreciation** is the opposite of compound interest, i.e. when the value of an asset decreases by a fixed percentage each year. 2. This usually happens to physical assets that degrade over time, such as cars or real estate. ### Declining Balance Depreciation [f30f67ec6198] 1. This method assumes the** asset loses a fixed percentage** of its value each year. 2. Formula: $$ V = P(1-r)^n $$ where: 1. $V$ is final value 2. $P$ is initial value 3. $r$ is rate of depreciation 4. $n$ is number of years [{"_key":"Uk7kePxaUXt2OXbBhw25Bj","_type":"block","asset":null,"children":[{"_key":"Uk7kePxaUXt2OXbBhw25CU","_type":"span","marks":[],"text":"Don't confuse the "},{"_key":"1f57b865581b","_type":"span","marks":["strong"],"text":"declining balance"},{"_key":"8ead14de0838","_type":"span","marks":[],"text":" formula with the "},{"_key":"773ad88686b4","_type":"span","marks":["strong"],"text":"compound interest"},{"_key":"02c0a57ce6f8","_type":"span","marks":[],"text":" formula! "}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"1fe9ea7e6219","_type":"block","asset":null,"children":[{"_key":"3e66404d89b0","_type":"span","marks":[],"text":"While they look similar, one shows growth (compound interest) and the other shows reduction (depreciation)."}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"}] [{"_key":"fa397bc286fd395f391695a94b805271","_type":"block","asset":null,"children":[{"_key":"6d319ad9431e","_type":"span","marks":[],"text":"A car worth $\\text{\\$}25,000$ depreciates at $15\\%$ per year. "}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"854e4b791056","_type":"block","asset":null,"children":[{"_key":"6ab7c6c7ee34","_type":"span","marks":[],"text":"After 3 years: $$ V = 25000(1-0.15)^3 = 15,438.28 $$"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"}] [{"_key":"Uk7kePxaUXt2OXbBhw25DF","_type":"block","asset":null,"children":[{"_key":"Uk7kePxaUXt2OXbBhw25E0","_type":"span","marks":[],"text":"In real-world applications, depreciation rates can vary based on:"}],"markDefs":[],"style":"normal"},{"_key":"Uk7kePxaUXt2OXbBhw25FW","_type":"block","asset":null,"children":[{"_key":"Uk7kePxaUXt2OXbBhw25GH","_type":"span","marks":[],"text":"Type of asset"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"Uk7kePxaUXt2OXbBhw25Hn","_type":"block","asset":null,"children":[{"_key":"Uk7kePxaUXt2OXbBhw25IY","_type":"span","marks":[],"text":"Industry standards"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"Uk7kePxaUXt2OXbBhw25K4","_type":"block","asset":null,"children":[{"_key":"Uk7kePxaUXt2OXbBhw25Kp","_type":"span","marks":[],"text":"Tax regulations"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"Uk7kePxaUXt2OXbBhw25ML","_type":"block","asset":null,"children":[{"_key":"Uk7kePxaUXt2OXbBhw25N6","_type":"span","marks":[],"text":"Company policies"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"}] [{"_key":"fe09199cdca0","_type":"block","asset":null,"children":[{"_key":"179e7ca050b3","_type":"span","marks":[],"text":"Problem-Solving Strategy"}],"markDefs":[],"style":"h4"},{"_key":"040a2929e035","_type":"block","asset":null,"children":[{"_key":"162eed2770c5","_type":"span","marks":[],"text":"Identify whether you're dealing with "},{"_key":"16f0894442ef","_type":"span","marks":["strong"],"text":"growth"},{"_key":"089bb9c6f1ba","_type":"span","marks":[],"text":" (compound interest) or "},{"_key":"1c8f49fb8f4c","_type":"span","marks":["strong"],"text":"reduction"},{"_key":"c3fc7eb4b6ca","_type":"span","marks":[],"text":" (depreciation)"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"f9e7ccff14f6","_type":"block","asset":null,"children":[{"_key":"a8c6418a2827","_type":"span","marks":[],"text":"Determine the"},{"_key":"78a5bfd736de","_type":"span","marks":["strong"],"text":" time period"},{"_key":"42a37df3374c","_type":"span","marks":[],"text":" and "},{"_key":"e1abeedcd748","_type":"span","marks":["strong"],"text":"frequency"},{"_key":"a9145c67e87a","_type":"span","marks":[],"text":" of compounding/depreciation"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"776c2c4ea73e","_type":"block","asset":null,"children":[{"_key":"0c8b1e67a03e","_type":"span","marks":[],"text":"Gather all "},{"_key":"001233cf8a3c","_type":"span","marks":["strong"],"text":"given values"},{"_key":"840ede611553","_type":"span","marks":[],"text":" and match them to the appropriate formula"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"1a3533898e61","_type":"block","asset":null,"children":[{"_key":"199b70ea4fca","_type":"span","marks":[],"text":"Pay attention to "},{"_key":"91a3b973f0f1","_type":"span","marks":["strong"],"text":"units"},{"_key":"725ec36a1224","_type":"span","marks":[],"text":" and "},{"_key":"a9d6962dbe5a","_type":"span","marks":["strong"],"text":"decimal places"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"5f1db33e0bbe","_type":"block","asset":null,"children":[{"_key":"93cac2b6a657","_type":"span","marks":[],"text":"Check if your answer makes "},{"_key":"6263e640cfd7","_type":"span","marks":["strong"],"text":"logical sense"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"}] [{"_key":"Uk7kePxaUXt2OXbBhw25Nr","_type":"block","asset":null,"children":[{"_key":"Uk7kePxaUXt2OXbBhw25Oc","_type":"span","marks":[],"text":"When solving financial problems, always round your final answer to "},{"_key":"17af4d60ffcb","_type":"span","marks":["strong"],"text":"two decimal places"},{"_key":"45f76168c5d3","_type":"span","marks":[],"text":", as this represents currency accurately."}],"markDefs":[],"style":"normal"}]