Simple and Compound Interest
Ximena puts $\$5{,}000$ into CETES (Mexican government savings bonds), and her cousin Diego leaves his in a savings account. They both earn the same interest rate, but after several years she has more money than he does. Why? The answer lies in how interest is calculated: simple (you earn the same amount every year) or compound (interest also earns interest). At the end of this section you will solve that mystery.
Simple interest
When you borrow or save, the starting money is called the principal ($C$). What you pay or earn for using it is the interest ($I$). The percent applied each year is the interest rate ($r$ %). The principal plus the interest is the amount ($M$).
Calculate the simple interest produced by a principal of $60\,000$ pesos invested at $4\%$ per annum for $5$ years. What will be the final amount at the end of this…
Calculate the simple interest produced by a principal of $60\,000$ pesos invested at $4\%$ per annum for $5$ years. What will be the final amount at the end of this period?
A bank grants a loan of $30\,000$ pesos at $6\%$ annual simple interest. If the loan is repaid over a period of $8$ months, determine the total amount that must be paid…
A bank grants a loan of $30\,000$ pesos at $6\%$ annual simple interest. If the loan is repaid over a period of $8$ months, determine the total amount that must be paid to the bank as interest.
Calculate the simple interest generated by $200\,000$ pesos invested at $2.5\%$ per annum for $146$ days. Assume a year has $365$ days to perform the calculations.
Calculate the simple interest generated by $200\,000$ pesos invested at $2.5\%$ per annum for $146$ days. Assume a year has $365$ days to perform the calculations.
Compound interest
With compound interest, at the end of each year the interest is added to the principal. The next year, the interest is calculated on a larger amount.
The same formula works for anything that grows by a fixed percent each period, such as prices with inflation. If a family spends $\$1{,}000$ a month and prices rise 3% per year, in 2 years they will spend $1{,}000 \cdot 1.03^2 = \$1{,}060.90$.
Calculate the final amount obtained by depositing $100\,000$ pesos into a savings account offering $3\%$ annual compound interest for $4$ years.
Calculate the final amount obtained by depositing $100\,000$ pesos into a savings account offering $3\%$ annual compound interest for $4$ years.
A family spends $24\,000$ pesos per month on food. If the average inflation is estimated at $2.5\%$ per year, how much will they spend per month in $4$ years to maintain…
A family spends $24\,000$ pesos per month on food. If the average inflation is estimated at $2.5\%$ per year, how much will they spend per month in $4$ years to maintain exactly the same level of consumption? Assume that price increases follow the compound interest formula.
The following graph shows the evolution of the same initial principal invested at simple interest (blue bars) and compound interest (green bars) with the same annual…
The following graph shows the evolution of the same initial principal invested at simple interest (blue bars) and compound interest (green bars) with the same annual interest rate over the first three years. By observing the exact values on the graph, deduce the initial principal and the interest rate. Next, calculate what the exact difference in pesos between both methods will be after $10$ years.
Comparing simple and compound interest, and chaining investments
To compare two options, calculate the final amount for each one with its formula and subtract. If the money moves from one product to another, the final amount of the first is the starting principal of the second.
Compare the return on a principal of $160\,000$ pesos invested for $6$ years at a rate of $5\%$ per year under two different methods: simple interest and compound…
Compare the return on a principal of $160\,000$ pesos invested for $6$ years at a rate of $5\%$ per year under two different methods: simple interest and compound interest. What is the exact difference in pesos between the interest generated by both methods?
An amount of $80\,000$ pesos is invested at a simple interest rate of $3\%$ per annum for $4$ years. After that time, the total final amount is withdrawn and reinvested…
An amount of $80\,000$ pesos is invested at a simple interest rate of $3\%$ per annum for $4$ years. After that time, the total final amount is withdrawn and reinvested in another financial product offering $4\%$ annual compound interest for $2$ more years. Calculate the overall final amount that will be obtained at the end of the $6$ years.
To finance a project, a company evaluates two loan options for an amount of $200\,000$ pesos to be repaid in a single payment after $3$ years. Bank A offers a simple…
To finance a project, a company evaluates two loan options for an amount of $200\,000$ pesos to be repaid in a single payment after $3$ years. Bank A offers a simple interest rate of $6\%$ per year, but charges an origination fee of $1.5\%$ on the principal, paid at the time of granting. Bank B offers a compound interest rate of $5\%$ per year with no fee. Determine which option is cheaper at the end of the period and what the difference in pesos is between both.
Solving for unknowns in simple interest
In simple interest, $C$, $r$, and $t$ appear multiplied together, so you solve for them with a division.
Find the principal initially invested at a simple interest rate of $5\%$ per annum if after $3$ years it generated $9\,000$ pesos in profit.
Find the principal initially invested at a simple interest rate of $5\%$ per annum if after $3$ years it generated $9\,000$ pesos in profit.
A principal of $40\,000$ pesos has grown to $44\,800$ pesos after remaining invested for $4$ years at simple interest. Find the annual interest rate applied.
A principal of $40\,000$ pesos has grown to $44\,800$ pesos after remaining invested for $4$ years at simple interest. Find the annual interest rate applied.
Determine how many years a principal must remain invested at a simple interest rate of $8\%$ per annum for its value to double.
Determine how many years a principal must remain invested at a simple interest rate of $8\%$ per annum for its value to double.
Principal and rate in compound interest
In $M = C\,(1+i)^t$, with $i = r/100$, the principal is multiplied and the rate sits inside a power.
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Initial principal: divide by the factor: $C = \dfrac{M}{(1+i)^t}$.
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Rate: take the $t$-th root: $1+i = \sqrt[t]{M/C}$, and then $r = 100\,i$.
A person wishes to have $200\,000$ pesos in $5$ years to buy a vehicle. If their bank offers a deposit with $4\%$ annual compound interest, how much money must they…
A person wishes to have $200\,000$ pesos in $5$ years to buy a vehicle. If their bank offers a deposit with $4\%$ annual compound interest, how much money must they deposit today to achieve their goal exactly?
An initial principal of $120\,000$ pesos has grown to $142\,922$ pesos after $3$ years of investment. Knowing that the interest has been reinvested each year (compound…
An initial principal of $120\,000$ pesos has grown to $142\,922$ pesos after $3$ years of investment. Knowing that the interest has been reinvested each year (compound interest), determine the annual interest rate applied.
Calculate what annual compound interest rate must be applied to a principal for its value to triple in exactly $15$ years. Express the result as a percentage rounded to…
Calculate what annual compound interest rate must be applied to a principal for its value to triple in exactly $15$ years. Express the result as a percentage rounded to two decimal places.
Time and two principals
When the unknown is the time in compound interest, it is in the exponent and can't be solved with a division or a root. The method for this course is trial and error: build a table year by year and see when the condition is met.
When two investments must become equal, you set up an equation with the shared unknown.
Salma invests $3000$ pesos at $4\%$ per annum and Bruno invests $2500$ pesos at $6\%$ per annum, both at simple interest. Set up an equation and calculate in how many…
Salma invests $3000$ pesos at $4\%$ per annum and Bruno invests $2500$ pesos at $6\%$ per annum, both at simple interest. Set up an equation and calculate in how many years they will have exactly the same accumulated final amount.
Set up the corresponding equation and use logarithms to determine how many years must elapse for a principal of $100\,000$ pesos to grow to $140\,000$ pesos if invested…
Set up the corresponding equation and use logarithms to determine how many years must elapse for a principal of $100\,000$ pesos to grow to $140\,000$ pesos if invested at an annual compound interest rate of $3.5\%$.
Carlos invests $80\,000$ pesos at an annual compound interest rate of $5\%$. On the same day, Diana invests $100\,000$ pesos at an annual compound interest rate of…
Carlos invests $80\,000$ pesos at an annual compound interest rate of $5\%$. On the same day, Diana invests $100\,000$ pesos at an annual compound interest rate of $3\%$. Use logarithms to determine after how many years the accumulated amount for Carlos will equal Diana's.
Compounding periods and several deposits
Interest isn't always added once a year. If it is compounded every month, the annual rate is divided among the periods.
Several deposits. Each deposit grows only from the moment it is made. Calculate the amount of each one using its own time and add them up.
An investor deposits $100\,000$ pesos at an annual compound interest rate of $6\%$, but with monthly compounding periods. Calculate the final amount they will have in…
An investor deposits $100\,000$ pesos at an annual compound interest rate of $6\%$, but with monthly compounding periods. Calculate the final amount they will have in their account after $2$ years.
A person deposits $40\,000$ pesos into an account offering $4\%$ annual compound interest. After exactly $2$ years, they make a new deposit of $30\,000$ pesos into the…
A person deposits $40\,000$ pesos into an account offering $4\%$ annual compound interest. After exactly $2$ years, they make a new deposit of $30\,000$ pesos into the same account. If they make no withdrawals, calculate the total accumulated amount they will have after $5$ years from the first deposit.
An investor wishes to accumulate a final amount of $400\,000$ pesos in $4$ years. To achieve this, they plan to make an initial deposit of an amount $C$ today, and a…
An investor wishes to accumulate a final amount of $400\,000$ pesos in $4$ years. To achieve this, they plan to make an initial deposit of an amount $C$ today, and a second deposit of the same amount $C$ in exactly $2$ years. If the bank account offers a yield of $3\%$ annual compound interest, set up the corresponding equation and find the value of the contribution $C$.
Depreciation and inflation: compound interest in reverse
A cell phone, a computer, or a car loses value every year. If they lose a fixed percent, the same idea is used, but subtracting.
Inflation is also a percent loss: money buys less. If purchasing power drops 2% per year, each year you multiply by $0.98$. To get a "real" value, multiply the two factors.
A new motorbike has a price of $80\,000$ pesos. If it loses $10\%$ of its value each year, calculate what its exact value will be after $3$ years.
A new motorbike has a price of $80\,000$ pesos. If it loses $10\%$ of its value each year, calculate what its exact value will be after $3$ years.
A cell phone battery loses $15\%$ of its charge for every hour of gaming. If you start playing with $100\%$ charge, determine what percentage of battery will remain…
A cell phone battery loses $15\%$ of its charge for every hour of gaming. If you start playing with $100\%$ charge, determine what percentage of battery will remain after $5$ hours.
A high-end computer system currently costs $30\,000$ pesos. Due to technological obsolescence, its nominal value depreciates by $15\%$ each year. At the same time,…
A high-end computer system currently costs $30\,000$ pesos. Due to technological obsolescence, its nominal value depreciates by $15\%$ each year. At the same time, inflation causes money to lose $2\%$ of its purchasing power annually. Calculate what the real value (adjusted for the loss of purchasing power) of the system will be after $4$ years.
Summary
Go back to the beginning: Ximena earned compound interest, so each year her interest was added to the principal and produced more interest. Diego earned only simple interest, always on $\$5{,}000$.