Bank Interest
Regina has $\$10{,}000$ saved from her mom's year-end bonus (aguinaldo), and two banks are offering her an account. Bank A pays 6% per year and adds the interest every quarter. Bank B pays 6.1% per year and adds it only once a year. 6.1 is the bigger number, but is it the better deal? By the end of the compounding section you will know, because comparing percentages is not enough: you have to know how often the interest is calculated.
Simple interest
When you deposit money in a bank or take out a loan, the money has a "price": the interest. With simple interest, the interest is always calculated on the initial principal, so you earn the same amount every year.
Simple interest grows arithmetically: the same amount is added each period. That is why its graph is a straight line.
To buy a state-of-the-art tablet costing $\$12\,000$, you take out a small loan from a financial institution that charges a simple interest rate of $6 \%…
To buy a state-of-the-art tablet costing $\$12\,000$, you take out a small loan from a financial institution that charges a simple interest rate of $6 \%$ per year. If you agree to repay the money together with the interest in a single payment after 8 months, calculate:
- a) The rate ($r$) expressed as a decimal.
- b) The time ($t$) expressed in years.
- c) The total interest you will have to pay for the loan.
Calculate what initial capital ($C_0$) we should deposit in a bank account so that, at a simple interest rate of $4 \%$ per year, we obtain exactly $\$120…
Calculate what initial capital ($C_0$) we should deposit in a bank account so that, at a simple interest rate of $4 \%$ per year, we obtain exactly $\$120$ in interest after one year.
An online microcredit platform offers fast loans at a monthly simple interest rate of $1 \%$. If a user requests $\$8\,000…
An online microcredit platform offers fast loans at a monthly simple interest rate of $1 \%$. If a user requests $\$8\,000$ to repair their motorbike and plans to repay everything at the end of a quarter (3 months):
- a) What is the interest they will pay each month?
- b) What is the total amount (capital + interest) they will repay at the end of the three months?
Compound interest
With compound interest, at the end of each period the interest is added to the principal, and in the next period the interest is already calculated on the principal plus what was earned. We say the interest is compounded: the money earns interest on the interest.
Your grandparents have given you $\$40\,000$ for your 16th birthday and you decide to invest it in a bank product at compound interest of $3 \%…
Your grandparents have given you $\$40\,000$ for your 16th birthday and you decide to invest it in a bank product at compound interest of $3 \%$ per year for 3 years. Since it is compound interest, the interest is compounded annually (added to the capital to generate new interest).
Calculate the final capital ($C_f$) you will have after 3 years using the formula:
$$C_f = C_0 \times (1 + r)^t$$Marcos has received a prize of $6\,000$ pesos and decides to invest it for 4 years. His bank offers him two different options to earn a return on his money: Option A: A…
Marcos has received a prize of $6\,000$ pesos and decides to invest it for 4 years. His bank offers him two different options to earn a return on his money:
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Option A: A simple interest deposit with an annual rate of $4.5$ %.
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Option B: A compound interest savings account with an annual rate of $4.2$ % and annual compounding of interest.
Calculate the final amount he would obtain with each option and explain which one is more financially advantageous for him at the end of the 4 years.
Simple versus compound: table and graph
To compare two options, it helps to make a table of the amount at the end of each year. Let's take $\$10{,}000$ at 6% per year:
| Year | Simple: $10{,}000\,(1+0.06\,t)$ | Compound: $10{,}000\times 1.06^{t}$ |
|---|---|---|
| 0 | $\$10{,}000.00$ | $\$10{,}000.00$ |
| 1 | $\$10{,}600.00$ | $\$10{,}600.00$ |
| 2 | $\$11{,}200.00$ | $\$11{,}236.00$ |
| 3 | $\$11{,}800.00$ | $\$11{,}910.16$ |
| 4 | $\$12{,}400.00$ | $\$12{,}624.77$ |
| 5 | $\$13{,}000.00$ | $\$13{,}382.26$ |
With simple interest, the amounts go up by $\$600$ each year. With compound interest, each year it goes up a little more than the one before.
Graph the growth of the accumulated interest of a $\$1000$ investment at a simple interest rate of $5 \%…
Graph the growth of the accumulated interest of a $\$1000$ investment at a simple interest rate of $5 \%$ per year over the first 4 years.
First complete the table of values (accumulated interest at the end of each year) before drawing the graph:
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Year 1: $I = \dots$
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Year 2: $I = \dots$
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Year 3: $I = \dots$
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Year 4: $I = \dots$
Then, draw a set of axes: time (in years, from 0 to 4) on the horizontal axis and interest (in pesos) on the vertical axis. Choose a suitable scale, mark the points from the table (plus the point for year 0) and join them.
You can do it on paper and send a photo with your answer.
A bank offers two types of accounts for young people: Account A: Simple interest of $3.1 \%$ per year. Account B: Compound interest of $3 \%$ per year. If you are going…
A bank offers two types of accounts for young people:
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Account A: Simple interest of $3.1 \%$ per year.
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Account B: Compound interest of $3 \%$ per year.
If you are going to deposit $\$20\,000$ for just one year, which of the two accounts will give you a greater profit at the end of that year? Justify your answer by showing the necessary calculations.
Fractional compounding and effective rate
Banks quote an annual rate, but they can add the interest every month, every two months, every quarter, or every six months. That annual rate is called the nominal interest rate ($j$). If it is compounded $m$ times a year, in each period the rate $\frac{j}{m}$ is applied.
| Compounding | Monthly | Bimonthly | Quarterly | Semiannual |
|---|---|---|---|---|
| $m$ | 12 | 6 | 4 | 2 |
Go back to the beginning. Bank A: $1.015^4-1\approx 6.14\,\%$ effective. Bank B: $6.1\,\%$ effective. Even though B's nominal rate is higher, Bank A is the better deal, because it compounds every quarter.
A financial institution offers an investment plan with a nominal annual interest rate of $3.6$ %. To attract different saver profiles, it allows customers to choose the…
A financial institution offers an investment plan with a nominal annual interest rate of $3.6$ %. To attract different saver profiles, it allows customers to choose the frequency with which interest is paid into the account. If a person invests an initial capital of $240\,000$ pesos for 5 years, determine the difference between the final capital obtained if the compounding is:
- a) Semiannual.
- b) Monthly.
Express the results rounded to the nearest cent of a pesos.
The "Futuro Seguro" bank advertises a savings account with a nominal annual interest rate of $5$ %, paying interest quarterly. Calculate the Annual Equivalent Rate (AER)…
The "Futuro Seguro" bank advertises a savings account with a nominal annual interest rate of $5$ %, paying interest quarterly. Calculate the Annual Equivalent Rate (AER) of this financial product. If the bank decided to change the interest payment frequency to monthly while maintaining the same AER, what should the new nominal annual interest rate be?
A young person who has just got their first job applies for a loan of $15,000$ pesos to buy a car. The bank offers two options: Option A: Nominal interest rate of…
A young person who has just got their first job applies for a loan of $15,000$ pesos to buy a car. The bank offers two options:
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Option A: Nominal interest rate of $6.5\%$ per year with monthly compounding and an arrangement fee of $2\%$ that is added to the initial capital.
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Option B: Nominal interest rate of $7.2\%$ per year with quarterly compounding, with no arrangement fees, but with a fixed quarterly maintenance fee of $15$ pesos.
Calculate the real APR of both options for a period of one year and give a reasoned answer as to which is more advantageous if the loan is repaid in full at the end of the first year.
Solving for the time and the rate with logarithms
Sometimes you know the amount and the question is: how long?, what rate? It is the same formula, with a different unknown.
Simple interest. This is a linear equation. From $I=C\,i\,t$ we get $C=\dfrac{I}{i\,t}$, $i=\dfrac{I}{C\,t}$, and $t=\dfrac{I}{C\,i}$. For example, to earn $\$300$ in 1 year at 5%: $C=\dfrac{300}{0.05\times 1}=\$6{,}000$.
Compound interest, the time. The unknown is in the exponent. You solve for it with logarithms, because the logarithm is the inverse of the exponential.
Present value and promissory notes
The present value is what an amount that will be received or paid in the future is worth today. You get it by solving for $C$ in the amount formula:
$$C=\frac{M}{(1+i)^{n}}=M\,(1+i)^{-n}$$A promissory note is a document in which someone commits to paying a fixed amount, the face value, on a specific date. If you want to collect it earlier, the bank gives you its present value: the face value discounted at the agreed rate.
All these calculations can be done with a scientific or financial calculator, with a spreadsheet (for example, the formula =C*(1+i)^n and dragging down the time column to build the table), or with GeoGebra, graphing $M(t)$ to compare simple and compound.
Sara wants to go on an international volunteering trip in a few years. The estimated total cost of the project is $4\,500$ pesos. Currently she has $3\,200$ pesos…
Sara wants to go on an international volunteering trip in a few years. The estimated total cost of the project is $4\,500$ pesos. Currently she has $3\,200$ pesos deposited in an account offering $3.75$ % annual compound interest. Using logarithms, calculate how long (expressed in years and months) she will need to keep the money invested to reach exactly the amount needed for her trip.
We invest $300\,000$ pesos in a financial product offering compound interest compounded quarterly. After two years, the capital has grown to $325\,089,40$ pesos.…
We invest $300\,000$ pesos in a financial product offering compound interest compounded quarterly. After two years, the capital has grown to $325\,089,40$ pesos. Determine the annual rate (nominal interest percentage) at which the operation was carried out. Round the final percentage to an integer.
A young woman opens a savings deposit of $200\,000$ pesos with the aim of saving for the deposit on her first apartment. The institution agrees to pay back the capital…
A young woman opens a savings deposit of $200\,000$ pesos with the aim of saving for the deposit on her first apartment. The institution agrees to pay back the capital plus accrued interest in a single payment when the total amount has tripled. If the agreed interest rate is $12\%$ per year compounded semiannually, how many years and months must pass before this payment is made? Use logarithms in the process.
Saving with periodic deposits
Many people save a fixed amount each month, for example, for a tanda (a rotating savings group) or for retirement. Each deposit earns interest from the day it is made. The first one accumulates more time than the last.
To find out in how many periods you reach a goal $M$, you solve for $N$ with logarithms:
$$N=\frac{\ln\!\left(1+\dfrac{M\,i}{A\,(1+i)}\right)}{\ln(1+i)}$$A Year 12 student decides to open a youth pension plan. To do so, they commit to depositing $2\,000$ pesos at the beginning of each month for the next 10 years. If the…
A Year 12 student decides to open a youth pension plan. To do so, they commit to depositing $2\,000$ pesos at the beginning of each month for the next 10 years. If the financial institution guarantees an annual interest rate of $3$ % with monthly compounding of interest, what total capital will they have accumulated at the end of the 10 years?
A family decides to open a pension plan for their newborn daughter. They deposit $2\,000$ pesos at the beginning of each month into an account with a nominal interest…
A family decides to open a pension plan for their newborn daughter. They deposit $2\,000$ pesos at the beginning of each month into an account with a nominal interest rate of $4.8\%$ per year compounded monthly. They want to know when the accumulated capital will exceed $1\,000\,000$ pesos. Use logarithms to calculate the exact number of months required and express the final result in years and months.
An investment fund guarantees an amount of $600\,000$ pesos in $10$ years through fixed annual contributions at the beginning of each year. The interest rate is $3.5\%$…
An investment fund guarantees an amount of $600\,000$ pesos in $10$ years through fixed annual contributions at the beginning of each year. The interest rate is $3.5\%$ per year. However, due to a financial crisis, at the end of the fifth year the interest rate drops to $2\%$ per year for the remainder of the period. If the investor maintains their annual contributions equal to those originally calculated, what capital will they actually have at the end of the $10$ years?
Loans with a fixed monthly payment
When you take out a loan (a car, a mortgage), the normal thing is to pay equal monthly payments at the end of each month. In each payment, part is interest on the balance you still owe and the rest amortizes (reduces) the debt.
To finance the purchase of an electric car costing $24\,000$ pesos, a family pays a deposit of $6\,000$ pesos and finances the rest through a bank loan at a nominal…
To finance the purchase of an electric car costing $24\,000$ pesos, a family pays a deposit of $6\,000$ pesos and finances the rest through a bank loan at a nominal interest rate of $6$ % per year. The loan must be repaid over 5 years in equal monthly installments paid at the end of each month.
- a) Calculate the amount of the monthly installment the family must pay.
- b) What is the total amount of interest the family will end up paying to the bank after 5 years?
A young entrepreneur applies for a microcredit loan of $160\,000$ pesos to buy IT equipment. The bank offers two repayment methods: Option 1: A single payment at the end…
A young entrepreneur applies for a microcredit loan of $160\,000$ pesos to buy IT equipment. The bank offers two repayment methods:
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Option 1: A single payment at the end of 3 years with compound interest of $7$ % per year.
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Option 2: Repayment through equal annual installments over 3 years at a nominal interest rate of $7.5$ %.
Calculate how much extra money is paid in interest under the option that turns out to be more expensive.
To finance the purchase of a hybrid vehicle costing $28,000$ pesos, an environmentally conscious person takes out a $6$-year loan at an interest rate of $5.5\%$ per…
To finance the purchase of a hybrid vehicle costing $28,000$ pesos, an environmentally conscious person takes out a $6$-year loan at an interest rate of $5.5\%$ per year. After paying exactly the first $36$ monthly installments, they receive an inheritance and decide to make a partial repayment of $8,000$ pesos. If they decide to keep the remaining duration of the loan, how much money will they save in total interest compared to the original plan?
Special cases: inflation, grace period, and perpetuity
Inflation. Your money growing in pesos does not mean it will buy more things, because prices also go up. If the effective annual rate is $i$ and inflation is $\pi$, the real return is
$$i_{real}=\frac{1+i}{1+\pi}-1$$For example, with $i=8\,\%$ and $\pi=4\,\%$: $\dfrac{1.08}{1.04}-1\approx 3.85\,\%$, not $4\,\%$. Another way to look at it: compare the amount with what the original principal should be worth, $C(1+\pi)^{t}$, to have the same purchasing power.
Grace period. During a few months only the interest is paid: $C\cdot i$ each month, and the balance does not go down. Afterward, the normal monthly payment is calculated with the same principal and the remaining months.
Perpetuity. If you want to collect a monthly income $R$ forever without touching the principal, the principal must generate exactly $R$ in interest each month: $C\cdot i=R$, so
$$C=\frac{R}{i}$$with $i$ the rate per month. For example, for $\$2{,}000$ per month at 0.5% monthly: $C=\dfrac{2{,}000}{0.005}=\$400{,}000$.
An investor analyses a savings product that offers compound interest of $3.2\%$ per year compounded daily (assume years of $360$ days). If the average inflation forecast…
An investor analyses a savings product that offers compound interest of $3.2\%$ per year compounded daily (assume years of $360$ days). If the average inflation forecast for the next $5$ years is $2.5\%$ per year, calculate the net real yield (purchasing power gained) of an initial investment of $200\,000$ pesos after that time. Hint: Compare the final amount obtained with the value the money would need to have to maintain its purchasing power against inflation.
A mortgage of $3\,600\,000$ pesos is taken out over $25$ years at an interest rate of $2.25\%$ per year. The contract includes an "interest-only grace period" clause of…
A mortgage of $3\,600\,000$ pesos is taken out over $25$ years at an interest rate of $2.25\%$ per year. The contract includes an "interest-only grace period" clause of $2$ years, during which the customer only pays the interest generated each month, without repaying any capital. Calculate:
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The monthly installment during the first two years.
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The final monthly installment for the remaining $23$ years.
Calculate the initial capital that a person must deposit in a financial institution offering a nominal annual interest rate of $4.2\%$ compounded monthly, so that their…
Calculate the initial capital that a person must deposit in a financial institution offering a nominal annual interest rate of $4.2\%$ compounded monthly, so that their heirs can receive a monthly income of $30\,000$ pesos in perpetuity (consider the limit as the number of years tends to infinity, or simply the capital that generates that interest each month without decreasing). What would be the AER of this operation?