#set document(title: "6.6 Methods of Savings", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 6.6#h(0.6em)Methods of Savings #figure(figph[Four bundles of currency notes are stacked one above the other.], alt: "Four bundles of currency notes are stacked one above the other.", caption: [Money wisely invested grows over time.]) === Learning Objectives After completing this section, you should be able to: + Distinguish various basic forms of savings plans. + Compute return on investment for basic forms of savings plans. + Compute payment to reach a financial goal. The stock market crash of 1929 led to the Great Depression, a decade-long global downturn in productivity and employment. A state of shock swept through the United States; the damage to people’s lives was immeasurable. Americans no longer trusted established financial institutions. By October 1931, the banking industry’s biggest challenge was restoring confidence to the American public. In the next 10 years, the federal government would impose strict regulations and guidelines on the financial industry. The Emergency Banking Act of 1933 created the Federal Deposit Insurance Corporation (FDIC), which insures bank deposits. The new federal guidelines helped ease suspicions among the general public about the banking industry. Gradually, things returned to normal, and today we have more investment instruments, many insured through the FDIC, than ever before. In this section, we will first look at the different types of savings accounts and proceed to discuss the various types of investments. There is some overlap, but we will try to differentiate among these financial instruments. Saving money should be a goal of every adult, but it can also be a difficult goal to attain. === Distinguish Various Basic Forms of Savings Plans There are at least three types of savings accounts. Traditional savings accounts, certificates of deposit (CDs), and money market accounts are three main savings account vehicles. ==== Savings Account A savings account is probably the most well-known type of investment, and for many people it is their first experience with a bank. A #strong[savings account] is a deposit account, held at a bank or other financial institution, which bears some interest on the deposited money. Savings accounts are intended as a place to save money for emergencies or to achieve short-term goals. They typically pay a low interest rate, but there is virtually no risk involved, and they are insured by the FDIC for up to \$250,000. Savings accounts have some strengths. They are highly flexible. Generally, there are no limitations on the number of withdrawals allowed and no limit on how much you can deposit. It is not unusual, however, that a savings account will have a minimum balance in order for the bank to pay maintenance costs. If your account should dip below the minimum, there are usually fees attached. #notebox("Who Knew?", rgb("#183B6F"), rgb("#183B6F"), rgb("#EFF1F5"))[ Many banks are covered by FDIC insurance. The FDIC is the Federal Deposit Insurance Corporation and is an independent agency created by the U.S. Congress. One of its purposes is to provide insurance for deposits in banks, including savings accounts. Be aware, not all banks are FDIC insured. The FDIC insures up to \$250,000 for a savings account, so you do not want your balance to exceed that federally insured limit. ] Having your savings account at the same bank as your checking account does offer a real advantage. For example, if your checking account is approaching its lower limit, you can transfer funds from your savings account and avoid any bank fees. Similarly, if you have an excess of funds in your checking account, you can transfer funds to your savings account and earn some interest. Checking accounts rarely pay interest. #notebox("People in Mathematics", rgb("#183B6F"), rgb("#183B6F"), rgb("#EFF1F5"))[ #emph[J.P. Morgan] J.P. Morgan was a wealthy banker around the turn of the 20th century. His business interests included railroads and the steel industry. However, it was in 1907 that a financial crisis, caused by poor banking decisions and followed by such great distrust in the banking system that a frenzy of withdrawals from banks occurred, that J.P. Morgan and other wealthy bankers lent from their own funds to help stabilize and save the system. ] There are some weaknesses to savings accounts. Primarily, it is because savings accounts earn very low interest rates. This means they are not the best way to grow your money. Experts, though, recommend keeping a savings account balance to cover 3 to 6 months of living expenses in case you should lose your job, have a sudden medical expense, or other emergency. Around tax time, you will receive a 1099-INT form stating the amount of interest earned on your savings, which is the amount that must be reported when you file your tax return. A #strong[1099 form] is a tax form that reports earnings that do not come from your employer, including interest earned on savings accounts. These 1099 forms have the suffix INT to indicate that the income is interest income. Savings accounts earn interest, and those earnings can be found using the interest formulas from previous sections. The final value of these accounts is sometimes called the future value of the account. #examplebox("Example 1")[Single Deposit in a Savings Account][ Violet deposits \$4,520.00 in a savings account bearing 1.45% interest compounded annually. If she does not add to or withdraw any of that money, how much will be in the account after 3 years? #solutionbox[ To find the compound interest, use the formula from Compound Interest, #math.equation(block: false, alt: "A equals P open parenthesis 1 plus the fraction r over n close parenthesis to the power n t")[$A = P attach(( 1 + frac(r, n) ), t: n t)$], where #math.equation(block: false, alt: "A")[$A$] represents the amount in the account after #math.equation(block: false, alt: "t")[$t$] years, with initial deposit (or principal) of #math.equation(block: false, alt: "P")[$P$], at an annual interest rate, in decimal form, of #math.equation(block: false, alt: "r")[$r$], compounded #math.equation(block: false, alt: "n")[$n$] times per year. Violet has a principal of \$4,520.00, which will earn an interest of #math.equation(block: false, alt: "r")[$r$] = 0.0145, compounded yearly (so #math.equation(block: false, alt: "n")[$n$] = 1), for #math.equation(block: false, alt: "t")[$t$] = 3 years. Substituting and calculating, we find that Violet’s account will be worth #math.equation(block: true, alt: "A, equals, P open parenthesis 1 plus the fraction r over n close parenthesis to the power n t; equals, $ 4,520.00 open parenthesis 1 plus the fraction 0.0145 over 1 close parenthesis to the power 1 times 3; equals, $ 4,520.00 open parenthesis 1.0145 close parenthesis cubed; equals, $ 4,719.48")[$A & = & P attach(( 1 + frac(r, n) ), t: n t) \ & = & "$" 4,520.00 attach(( 1 + frac(0.0145, 1) ), t: 1 × 3) \ & = & "$" 4,520.00 attach(( 1.0145 ), t: 3) \ & = & "$" 4,719.48$] Or, Violet will have \$4,719.48 after 3 years. ] ] #notebox("Who Knew?", rgb("#183B6F"), rgb("#183B6F"), rgb("#EFF1F5"))[ Banks have not always offered interest on savings accounts. An 1836 publication from Indiana noted that banks in other states allow small interest on deposits. It specifically says that in these other states, these deposits are what business transactions are based upon. And that giving interest would encourage deposits, and thus increase the business that banks can do. #link("https://openstax.org/r/onepage")[Journal of the House of Representatives of the Sate of Indiana] ] ==== Certificates of Deposit, or CDs We discussed #strong[certificates of deposit] (CDs) in earlier sections. CDs differ from savings accounts in a few ways. First, the investment lasts for a fixed period of time, agreed to when the money is invested in the CD. These time periods often range from 6 months to 5 years. Money from the CD cannot be withdrawn (without penalty) until the investment period is up. Also, money cannot be added to an existing CD. Certificates of deposit have features similar to savings accounts. They are insured by the FDIC. They are entirely safe. They do, though, offer a better interest rate. The trade-off is that once the money is invested in a CD, that money is unavailable until the investment period ends. #examplebox("Example 2")[5-Year CD][ Silvio deposits \$10,000 in a CD that yields 2.17% compounded semiannually for 5 years. How much is the CD worth after 5 years? #solutionbox[ This also uses the compound interest formula from Compound Interest, #math.equation(block: false, alt: "A equals P open parenthesis 1 plus the fraction r over n close parenthesis to the power nt")[$A = P attach(( 1 + display(display(frac(r, n))) ), t: italic("nt"))$], Substituting the values #math.equation(block: false, alt: "P")[$P$] = \$10,000, #math.equation(block: false, alt: "r")[$r$] = 0.0217, #math.equation(block: false, alt: "n")[$n$] = 2 (semiannually means twice per year), and #math.equation(block: false, alt: "t")[$t$] = 5, we find the account will be worth #math.equation(block: true, alt: "A, equals, P open parenthesis 1 plus the fraction r over n close parenthesis to the power nt; equals, $ 10 , 000.00 open parenthesis 1 plus the fraction 0.0217 over 2 close parenthesis to the power 2 times 5; equals, $ 10,000.00 open parenthesis 1.01085 close parenthesis to the power 10; equals, $ 11 , 239.53")[$A & = & P attach(( 1 + display(display(frac(r, n))) ), t: italic("nt")) \ & = & "$" 10 , 000.00 attach(( 1 + display(display(frac(0.0217, 2))) ), t: 2 × 5) \ & = & "$" 10,000.00 attach(( 1.01085 ), t: 10) \ & = & "$" 11 , 239.53$] The CD will be worth \$11,219.53 after 5 years. ] ] ==== Money Market Account A #strong[money market account] is similar to a savings account, except the number of transactions (withdrawals and transfers) is generally limited to six each month. Money market accounts typically have a minimum balance that must be maintained. If the balance in the account drops below the minimum, there is likely to be a penalty. Money market accounts offer the flexibility of checks and ATM cards. Finally, the interest rate on a money market account is typically higher than the interest rate on a savings account. #examplebox("Example 3")[Single Deposit to a Money Market Account][ Marietta opens a money market account, and deposits \$2,500.00 in the account. It bears 1.76% interest compounded monthly. If she makes no other transactions on the account, how much will be in the account after 4 years? #solutionbox[ This, once again, uses the compound interest formula from Compound Interest: #math.equation(block: false, alt: "A equals P open parenthesis 1 plus the fraction r over n close parenthesis to the power nt")[$A = P attach(( 1 + display(display(frac(r, n))) ), t: italic("nt"))$], Substituting the values #math.equation(block: false, alt: "P")[$P$] = \$25,000, #math.equation(block: false, alt: "r")[$r$] = 0.0176, #math.equation(block: false, alt: "n")[$n$] = 12, and #math.equation(block: false, alt: "t")[$t$] = 4, we find the account will be worth #math.equation(block: true, alt: "A, equals, P open parenthesis 1 plus the fraction r over n close parenthesis to the power nt; equals, $ 2,500.00 open parenthesis 1 plus the fraction 0.0176 over 12 close parenthesis to the power 12 times 4; equals, $ 2,500.00 open parenthesis 1.0014 6 ¯ close parenthesis to the power 48; equals, $ 2,682.20")[$A & = & P attach(( 1 + display(display(frac(r, n))) ), t: italic("nt")) \ & = & "$" 2,500.00 attach(( 1 + display(display(frac(0.0176, 12))) ), t: 12 × 4) \ & = & "$" 2,500.00 attach(( 1.0014 overline(6) ), t: 48) \ & = & "$" 2,682.20$] The money market account will be worth \$2,682.20 after 4 years. ] ] ==== Return on Investment If we want to compare the profitability of different investments, like savings accounts versus other investment tools, we need a measure that evens the playing field. Such a measure is #strong[return on investment]. #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ The return on investment, often denoted ROI, is the percent difference between the initial investment, #math.equation(block: false, alt: "P")[$P$], and the final value of the investment, #math.equation(block: false, alt: "F V")[$F V$], or #math.equation(block: false, alt: "ROI equals the fraction F V minus P over P")[$"ROI" = display(display(frac(F V − P, P)))$], expressed as a percentage. ] #notebox("Checkpoint", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ #emph[The length of time of the investment is not considered in ROI.] ] #examplebox("Example 4")[Calculating Return on Investment][ + Determine the return on investment for the 5-year CD from Example 2. Round the percentage to two decimal places. + Determine the return on investment for the money market account from Example 3. Round the percentage to two decimal places. #solutionbox[ + The initial deposit in the CD was \$10,000, so #math.equation(block: false, alt: "P")[$P$] = \$10,000. The value at the end of 5 years was \$11,239.53. so #math.equation(block: false, alt: "F V")[$F V$] = \$11,239.53. Substituting and computing we find the return on investment. #linebreak() #math.equation(block: true, alt: "ROI, equals, the fraction F V minus P over P; equals, the fraction $ 11,239.53 minus $ 10,000 over $ 10,000; equals, the fraction $ 1,239.53 over $ 10 , 000; equals, 0.123953")[$"ROI" & = & display(display(frac(F V − P, P))) \ & = & display(display(frac("$" 11","239.53 − "$" 10","000, "$" 10","000))) \ & = & display(display(frac("$" 1","239.53, "$" 10 "," 000))) \ & = & 0.123953$] #linebreak() The ROI is 12.40%. #linebreak() + The initial deposit in the money market was \$2,500, so #math.equation(block: false, alt: "P")[$P$] = \$2,500. The value at the end of 4 years was \$2,682.20. so #math.equation(block: false, alt: "F V")[$F V$] = \$2,682.20. Substituting and computing we find the return on investment. #linebreak() #math.equation(block: true, alt: "ROI, equals, the fraction F V minus P over P; equals, the fraction $ 2,682.20 minus $ 2,500 over $ 2,500; equals, the fraction $ 182.20 over $ 2,500; equals, 0.07288")[$"ROI" & = & display(display(frac(F V − P, P))) \ & = & display(display(frac("$" 2","682.20 − "$" 2","500, "$" 2","500))) \ & = & display(display(frac("$" 182.20, "$" 2","500))) \ & = & 0.07288$] #linebreak() The ROI is 7.29%. ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/This_video")[Return on Investment, ROI] ] ==== Annuities as Savings In Compound Interest, we talked about the future value of a single deposit. In reality, people often open accounts that allow them to add deposits, or #emph[payments], to the account at regular intervals. This agrees with the 50-30-20 budget philosophy, where some income is saved every month. When a deposit is made at the end of each compounding period, such a savings account is called an #strong[ordinary annuity]. The formula for the future value of an ordinary annuity is #math.equation(block: false, alt: "F V equals pmt times the fraction open parenthesis 1 plus r / n close parenthesis to the power n times t minus 1 over r / n")[$F V = italic("pmt") × display(display(frac(attach(( 1 + r / n ), t: n × t) − 1, r / n)))$], where #math.equation(block: false, alt: "F V")[$F V$] is the future value of the annuity, #math.equation(block: false, alt: "pmt")[$italic("pmt")$] is the payment, #math.equation(block: false, alt: "r")[$r$] is the annual interest rate (in decimal form), #math.equation(block: false, alt: "n")[$n$] is the number of compounding periods per year, and #math.equation(block: false, alt: "t")[$t$] is the number of years. #notebox("Checkpoint", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ #emph[It is important to note that the number of deposits per year and the number of periods per year are the same.] ] #notebox("Checkpoint", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ #emph[Another form of annuity if the annuity due, which has deposits at the start of each compounding period. This other annuity type has different formulas and is not addressed in this text.] ] #examplebox("Example 5")[Future Value of an Ordinary Annuity][ Jill has an account that bears 3.75% interest compounded monthly. She decides to deposit \$250.00 each month, at the end of the compounding period, into this account. What is the future value of this account, after 8 years? #solutionbox[ These are regular payments into an account bearing compound interest. She is depositing them at the end of each compounding period. This makes this an ordinary annuity. Substituting the values #math.equation(block: false, alt: "pmt")[$italic("pmt")$] = 250, #math.equation(block: false, alt: "r")[$r$] = 0.0375, #math.equation(block: false, alt: "n")[$n$] = 12, and #math.equation(block: false, alt: "t")[$t$] = 8 into the formula, we find the future value of the account. #math.equation(block: true, alt: "F V, equals, pmt times the fraction open parenthesis 1 plus r / n close parenthesis to the power n times t minus 1 over r / n; equals, 250 times the fraction open parenthesis 1 plus 0.0375 / 12 close parenthesis to the power 12 times 8 minus 1 over 0.0375 / 12; equals, 250 times the fraction open parenthesis 1.003125 close parenthesis to the power 96 minus 1 over 0.003125; equals, 250 times the fraction 1.34922752406 minus 1 over 0.003125; equals, 250 times the fraction 0.34922752406 over 0.003125; equals, 250 times 111.752807699; equals, 27 , 938.202")[$F V & = & italic("pmt") × display(display(frac(attach(( 1 + r / n ), t: n × t) − 1, r / n))) \ & = & 250 × display(display(frac(attach(( 1 + 0.0375 / 12 ), t: 12 × 8) − 1, 0.0375 / 12))) \ & = & 250 × display(display(frac(attach(( 1.003125 ), t: 96) − 1, 0.003125))) \ & = & 250 × display(display(frac(1.34922752406 − 1, 0.003125))) \ & = & 250 × display(display(frac(0.34922752406, 0.003125))) \ & = & 250 × 111.752807699 \ & = & 27 , 938.202$] The account, after 8 years, will contain \$27,938.20. ] ] #notebox("Who Knew?", rgb("#183B6F"), rgb("#183B6F"), rgb("#EFF1F5"))[ #emph[Setting Savings Account Interest Rates] There are a number of factors that contribute to the amount a bank gives for savings accounts. The interest rate reflects how much the bank values deposits. It also reflects the money that the bank will earn when they lend out money. Finally, interest rates are impacted by the Federal Reserve Bank. When the Fed raises interest rates, so do banks. ] #notebox("People in Mathematics", rgb("#183B6F"), rgb("#183B6F"), rgb("#EFF1F5"))[ #emph[The Federal Reserve Chairperson] The Federal Reserve Board monitors the risks in the financial system to help ensure a healthy economy for individuals, companies, and communities. The Board oversees the 12 regional reserve banks. The Chairperson of the Federal Reserve Board testifies to Congress twice per year, meets with the secretary of the Treasury, chairs the Federal Open Market Committee, and is the face of federal monetary policy. Currently, the Fed Chair is Jerome Powell, who has served since 2018. ] #examplebox("Example 6")[Saving for College][ When Yusef was born, Rita and George began to save for Yusef’s college years by investing \$2,500 each year in a savings account bearing 3.4% interest compounded annually. How much will they have saved after 18 years? #solutionbox[ To find the future value of the account, we use the ordinary annuity formula #math.equation(block: false, alt: "F V equals pmt times the fraction open parenthesis 1 plus r / n close parenthesis to the power n times t minus 1 over r / n")[$F V = italic("pmt") × display(display(frac(attach(( 1 + r / n ), t: n × t) − 1, r / n)))$]. The payment is \$2,500, rate is 0.034, the number of compounding periods is 1, and the number of years is 18. Substituting these values and computing, we have #math.equation(block: true, alt: "F V, equals, pmt times the fraction open parenthesis 1 plus r / n close parenthesis to the power n times t minus 1 over r / n; equals, $ 2,500 times the fraction open parenthesis 1 plus 0.034 / 1 close parenthesis to the power 1 times 18 minus 1 over 0.034 / 1; equals, $ 2,500 times the fraction open parenthesis 1.034 close parenthesis to the power 18 minus 1 over 0.034; equals, $ 2,500 times the fraction 1.82544897331 minus 1 over 0.034; equals, $ 2,500 times 24.2779109798; equals, $ 60,694.77")[$F V & = & italic("pmt") × display(display(frac(attach(( 1 + r / n ), t: n × t) − 1, r / n))) \ & = & "$" 2,500 × display(display(frac(attach(( 1 + 0.034 / 1 ), t: 1 × 18) − 1, 0.034 / 1))) \ & = & "$" 2,500 × display(display(frac(attach(( 1.034 ), t: 18) − 1, 0.034))) \ & = & "$" 2,500 × display(display(frac(1.82544897331 − 1, 0.034))) \ & = & "$" 2,500 × 24.2779109798 \ & = & "$" 60,694.77$] After saving for 18 years, Rita and George will have \$60,694.77 for Yusef’s college. ] ] #notebox("Tech Check", rgb("#183B6F"), rgb("#183B6F"), rgb("#EFF1F5"))[ Google Sheets offers a function to calculate the future value of an ordinary annuity. To get Google Sheets to calculate the future value, you use the following: =fv(rate,number\_of\_periods, payment, present\_value, end\_or\_beginning). To explain, the rate is the rate per compounding period. From our formula, that is #math.equation(block: false, alt: "r / n")[$r / n$]. Also, the number of periods must be entered. From our formula, that is #math.equation(block: false, alt: "n times t")[$n × t$]. The payment is the amount deposited each period. Present value is 0 if we begin with no money and rely only on the payments to be made. However, if some money is available to put in the account before the payments start, that amount, an initial deposit, would be the value of #math.equation(block: false, alt: "P V")[$P V$]. Finally, for an ordinary annuity, enter 0 for end or beginning. Using the values for Jill, the payment amount is \$250, #math.equation(block: false, alt: "r")[$r$] = 0.0375, #math.equation(block: false, alt: "n")[$n$] = 12, #math.equation(block: false, alt: "t")[$t$] = 8, and that there is no initial deposit, #math.equation(block: false, alt: "P V")[$P V$] = 0, the Google Sheets formula is =fv(0.0375/12,12\*8,250,0,0). shows the formula in Google Sheets. #figure(figph[A Google Sheet spreadsheet. The first-row cell reads a formula, = f v (0.0375/12, 12\*8, 250, 0) and the equation is highlighted.], alt: "A Google Sheet spreadsheet. The first-row cell reads a formula, = f v (0.0375/12, 12*8, 250, 0) and the equation is highlighted.", caption: [Google Sheets formula]) Hitting the enter key shows the payment value . #figure(figph[A Google Sheet spreadsheet. The first-row cell reads \$27,938.20 and the formula is mentioned as,= f v (0.0375/12, 12\*8, 250, 0).], alt: "A Google Sheet spreadsheet. The first-row cell reads $27,938.20 and the formula is mentioned as,= f v (0.0375/12, 12*8, 250, 0).", caption: [Payment value]) Notice that the future value is negative, since it is a payment leaving an account. ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/cell_references")[Future Value Using Google Sheets] ] === Compute Payment to Reach a Financial Goal The formula used to get the future value of an ordinary annuity is useful, finding out what the final amount in the account will be. However, that isn’t how planning works. To plan, we need to know how much to put into the ordinary annuity each compounding period in order to reach a goal. Fortunately, that formula exists. #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ The formula for the amount that needs to be deposited per period, #math.equation(block: false, alt: "pmt")[$italic("pmt")$], of an ordinary annuity to reach a specified goal, #math.equation(block: false, alt: "FV")[$italic("FV")$], is #math.equation(block: false, alt: "pmt equals the fraction F V times open parenthesis r / n close parenthesis over open parenthesis 1 plus r / n close parenthesis to the power n times t minus 1")[$italic("pmt") = display(display(frac(F V × ( r / n ), attach(( 1 + r / n ), t: n × t) − 1)))$], where #math.equation(block: false, alt: "r")[$r$] is the annual interest rate (in decimal form), #math.equation(block: false, alt: "n")[$n$] is the number of periods per year, and #math.equation(block: false, alt: "t")[$t$] is the number of years. ] With this formula, it is possible to plan the amount to be saved. #examplebox("Example 7")[Saving for a Car][ Yaroslava wants to save in order to buy a car, in 3 years, without taking out a loan. She determines that she’ll need \$35,500 for the purchase. If she deposits money into an ordinary annuity that yields 4.25% interest compounded monthly, how much will she need to deposit each month? #solutionbox[ Yaroslava has a goal and needs to know the payments to make to reach the goal. Her goal is #math.equation(block: false, alt: "FV")[$italic("FV")$] = \$35,500, with an interest rate #math.equation(block: false, alt: "r")[$r$] = 0.0425, compounded per month so #math.equation(block: false, alt: "n")[$n$] = 12, and for 3 years, making #math.equation(block: false, alt: "t")[$t$] = 3. Substituting into the formula, Yaroslava finds the necessary payment. #math.equation(block: true, alt: "pmt, equals, the fraction F V times open parenthesis r / n close parenthesis over open parenthesis 1 plus r / n close parenthesis to the power n times t minus 1; equals, the fraction 35,500 times open parenthesis 0.0425 / 12 close parenthesis over open parenthesis 1 plus 0.0425 / 12 close parenthesis to the power 12 times 3 minus 1; equals, the fraction 35,500 times open parenthesis 0.003541 6 ¯ close parenthesis over open parenthesis 1.003541 6 ¯ close parenthesis to the power 36 minus 1; equals, the fraction 125.7291 6 ¯ over 0.13572901696; equals, 926.325")[$italic("pmt") & = & display(display(frac(F V × ( r / n ), attach(( 1 + r / n ), t: n × t) − 1))) \ & = & display(display(frac(35","500 × ( 0.0425 / 12 ), attach(( 1 + 0.0425 / 12 ), t: 12 × 3) − 1))) \ & = & display(display(frac(35","500 × ( 0.003541 overline(6) ), attach(( 1.003541 overline(6) ), t: 36) − 1))) \ & = & display(display(frac(125.7291 overline(6), 0.13572901696))) \ & = & 926.325$] To reach her goal, Yaroslava would need to deposit \$926.33 in her account each month. ] ] #notebox("Checkpoint", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ #emph[This has been rounded up, so that the deposits don’t fall short of the goal. However, some round off using the standard rounding rules: if the last digit is 1, 2, 3, or 4, the number is rounded down; if the last digit is 5, 6, 7, 8, or 9 the number is rounded up.] ] #notebox("Tech Check", rgb("#183B6F"), rgb("#183B6F"), rgb("#EFF1F5"))[ Google Sheets offers a function to calculate the payment necessary to reach a goal using ordinary annuities. To get Google Sheets to calculate the payment, you use the following: =pmt(rate,number\_of\_periods, present\_value, future\_value, end\_or\_beginning). To explain, the rate is per compounding period. From our formula, that is #math.equation(block: false, alt: "r / n")[$r / n$]. Also, the number of periods must be entered. From our formula, that is #math.equation(block: false, alt: "n times t")[$n × t$]. The present value is the amount of money that the account begins with. If we begin with no money and rely only on the payments to be made, then this number is 0. However, if some money is available to put in the account before the payments start, that amount, an initial deposit, would be the value of #math.equation(block: false, alt: "P V")[$P V$]. Next, enter the future value, #math.equation(block: false, alt: "F V")[$F V$]. Finally, for an ordinary annuity, enter 0 for end or beginning. Using the values for Yaroslava, #math.equation(block: false, alt: "r")[$r$] = 0.0425, #math.equation(block: false, alt: "n")[$n$] = 12, #math.equation(block: false, alt: "t")[$t$] = 3, and that there is no initial deposit, #math.equation(block: false, alt: "P V")[$P V$] = 0, the Google Sheets formula is =pmt(0.0425/12,12\*3,0,35500,0). shows the formula in Google Sheets. #figure(figph[A Google Sheet spreadsheet. The first-row cell reads a formula, = p m t (0.0425/12, 12\*3, 35500, 0) and the formula is highlighted.], alt: "A Google Sheet spreadsheet. The first-row cell reads a formula, = p m t (0.0425/12, 12*3, 35500, 0) and the formula is highlighted.", caption: [Google Sheets formula]) Hitting the enter key shows the payment value . #figure(figph[A Google Sheet spreadsheet. The first-row cell reads \$926.32 and the formula is mentioned as,= p m t (0.0425/12, 12\*3, 35500, 0).], alt: "A Google Sheet spreadsheet. The first-row cell reads $926.32 and the formula is mentioned as,= p m t (0.0425/12, 12*3, 35500, 0).", caption: [Payment value]) Notice that the payment is negative, since it is a payment leaving an account. Additionally, the payment is \$926.32. We rounded that up, but Google Sheets rounded off. ] === Key Terms - Savings account - 1099 form - Certificate of deposit - Money market account - Return on investment - Ordinary annuity === Key Concepts - There are three main types of savings accounts, saving accounts, certificates of deposit (CD), and money market accounts. - Savings account are very risk free, and so yield low interest rates. - The differences in the three types of savings accounts relate to their convenience. - Savings account typically have a lower interest rate that money market accounts, which typically have lower interest rates than CDs. - Ordinary annuities more accurately reflect how we save, in that money is deposited repeatedly over time. - Spreadsheet software, such as Google Sheets, have built in functions that can be used to quickly calculate both the future value of an ordinary annuity account, but also the payment necessary to reach a goal using an ordinary annuity. === Videos - #link("https://openstax.org/r/This_video")[Return on Investment, ROI] - #link("https://openstax.org/r/cell_references")[Future Value Using Google Sheets] === Formulas #math.equation(block: true, alt: "A equals P open parenthesis 1 plus the fraction r over n close parenthesis to the power n t")[$A = P attach(( 1 + frac(r, n) ), t: n t)$] #math.equation(block: true, alt: "ROI equals the fraction F V minus P over P")[$"ROI" = frac(F V − P, P)$] #math.equation(block: true, alt: "F V equals p m t times the fraction open parenthesis 1 plus r / n close parenthesis to the power n times t minus 1 over r / n")[$F V = p m t × frac(attach(( 1 + r / n ), t: n × t) − 1, r / n)$] #math.equation(block: true, alt: "p m t equals the fraction F V times open parenthesis r / n close parenthesis over open parenthesis 1 plus r / n close parenthesis to the power n times t minus 1")[$p m t = frac(F V × ( r / n ), attach(( 1 + r / n ), t: n × t) − 1)$]