Calculate annuity balance, interest earned, and payout
Calculate loan payments, balance, and interest using the loan formula
Compare loans in real-world applications
Savings Annuity
The Main Idea
A savings annuity allows an individual to save money and earn interest on a regular basis, typically over a long period of time. The annuity is usually offered by banks, insurance companies, or other financial institutions.
To start a savings annuity, an individual typically needs to make an initial investment, which can be a lump sum or regular payments over a period of time. This money is invested and earns interest over time, usually at a fixed rate.
The interest earned is added to the principal amount, which means that the individual’s investment grows over time. This process is called compounding interest, and it allows the investment to grow faster than it would with simple interest.
Once the predetermined period of time has ended, the individual can choose to receive regular payments from the annuity, usually in the form of a monthly or yearly income stream. The amount of the payments is determined by the initial investment, the interest rate, and the length of the payout period.
[latex]P_t[/latex] is the balance in the account after [latex]t[/latex] years.
[latex]d[/latex] is the regular deposit (the amount you deposit each year, each month, etc.)
[latex]r[/latex] is the annual interest rate in decimal form.
[latex]n[/latex] is the number of compounding periods in one year.
If the compounding frequency is not explicitly stated, assume there are the same number of compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, [latex]n = 12[/latex].
If you make your deposits every year, use yearly compounding, [latex]n = 1[/latex].
If you make your deposits every quarter, use quarterly compounding, [latex]n = 4[/latex].
Etc.
A conservative investment account pays [latex]3\%[/latex] interest. If you deposit [latex]$5[/latex] a day into this account, how much will you have after [latex]10[/latex] years? How much is from interest?
The account will be worth [latex]$21,282.07[/latex] after [latex]10[/latex] years. How much of that is from interest earned?
You deposited [latex]$5[/latex] per day for [latex]10[/latex] years. That’s [latex]5\text{ dollars }\ast 365\text{ days } \ast 10\text{ years }=18250\text{ dollars}[/latex].
Subtract the amount you deposited, [latex]$18,250[/latex], from the account balance, [latex]$21,282.07[/latex]. You earned [latex]$3,032.07[/latex] from interest.
Payout Annuities
The Main Idea
A payout annuity allows an individual to receive regular payments, usually on a monthly or yearly basis, for a predetermined period of time.
The payout period can be for a fixed number of years, such as 10 or 20 years, or for the rest of the individual’s life. In the case of a life annuity, the payments will continue until the individual’s death, no matter how long that may be. The amount of the payments is based on a number of factors, including the individual’s age, gender, and life expectancy.
Payout annuities are often used as a source of retirement income, as they provide a guaranteed income stream for the individual’s lifetime. They can also be used to provide income for a spouse or other beneficiary after the individual’s death, depending on the terms of the annuity.
[latex]P_0[/latex] is the balance in the account at the beginning (starting amount, or principal).
[latex]d[/latex] is the regular withdrawal (the amount you take out each year, each month, etc.)
[latex]r[/latex] is the annual interest rate (in decimal form. Example: [latex]5\% = 0.05[/latex]).
[latex]n[/latex] is the number of compounding periods in one year.
[latex]t[/latex] is the number of years we plan to take withdrawals.
You know you will have [latex]$500,000[/latex] in your account when you retire. You want to be able to take monthly withdrawals from the account for a total of [latex]30[/latex] years. Your retirement account earns [latex]8\%[/latex] interest. How much will you be able to withdraw each month?
In this example, we’re looking for [latex]d[/latex].
[latex]r = 0.08[/latex]
[latex]8\%[/latex] annual rate
[latex]n = 12[/latex]
since we’re withdrawing monthly
[latex]t = 30[/latex]
[latex]30[/latex] years
[latex]P_0 = $500,000[/latex]
we are beginning with [latex]$500,000[/latex]
In this case, we’re going to have to set up the equation, and solve for [latex]d[/latex].
Note: This video uses [latex]k[/latex] for [latex]n[/latex] and [latex]N[/latex] for [latex]t[/latex].
A donor gives [latex]$100,000[/latex] to a university, and specifies that it is to be used to give annual scholarships for the next [latex]20[/latex] years. If the university can earn [latex]4\%[/latex] interest, how much can they give in scholarships each year?
Solving for [latex]d[/latex] gives [latex]$7,358.18[/latex] each year that they can give in scholarships. It is worth noting that usually donors instead specify that only interest is to be used for scholarship, which makes the original donation last indefinitely. If this donor had specified that, [latex]$100,000(0.04) = $4,000[/latex] a year would have been available.
Conventional Loans
The Main Idea
A conventional loan is a type of loan that is backed by private lenders, such as banks and credit unions, rather than a government agency. Borrowers typically need a higher credit score, a lower debt-to-income ratio, and a larger down payment in order to qualify for a conventional loan.
Conventional loans offer more flexibility in terms of loan amounts, repayment periods, and interest rates than government-backed loans. Conventional loans require borrowers to make monthly payments that include both principal and interest. The goal is to pay off the loan over time.
[latex]P_0[/latex] is the balance in the account at the beginning (the principal, or amount of the loan).
[latex]d[/latex] is your loan payment (your monthly payment, annual payment, etc)
[latex]r[/latex] is the annual interest rate in decimal form.
[latex]n[/latex] is the number of compounding periods in one year.
[latex]t[/latex] is the length of the loan, in years.
You want to take out a [latex]$140,000[/latex] mortgage (home loan). The interest rate on the loan is [latex]6\%[/latex], and the loan is for [latex]30[/latex] years. How much will your monthly payments be?
In this example, we’re looking for [latex]d[/latex].
[latex]r = 0.06[/latex]
[latex]6\%[/latex] annual rate
[latex]n = 12[/latex]
since we’re paying monthly
[latex]t = 30[/latex]
[latex]30[/latex] years
[latex]P_0 = $140,000[/latex]
the starting loan amount
In this case, we’re going to have to set up the equation, and solve for [latex]d[/latex].
You will make payments of [latex]$839.37[/latex] per month for [latex]30[/latex] years.
You’re paying a total of [latex]$302,173.20[/latex] to the loan company: [latex]$839.37[/latex] per month for [latex]360[/latex] months. You are paying a total of [latex]$302,173.20 - $140,000 = $162,173.20[/latex] in interest over the life of the loan.
Note: This video uses [latex]k[/latex] for [latex]n[/latex] and [latex]N[/latex] for [latex]t[/latex].
Janine bought [latex]$3,000[/latex] of new furniture on credit. Because her credit score isn’t very good, the store is charging her a fairly high interest rate on the loan: [latex]16\%[/latex]. If she agreed to pay off the furniture over [latex]2[/latex] years, how much will she have to pay each month?
[latex]\begin{array}{r@{\hfill}l} d= && \text{unknown}\\ r=0.16 && \text{16% annual rate} \\ n=12 && \text{since we're making monthly payments} \\ t=2 && \text{2 years to repay} \\ P_0=3,000 && \text{we're starting with a } \$3,000 \text{ loan} \\ \end{array}[/latex]
Solve for d to get monthly payments of [latex]$146.89[/latex]. Two years to repay means [latex]$146.89(24) = $3525.36[/latex] in total payments. This means Janine will pay [latex]$3525.36 - $3000 = $525.36[/latex] in interest.
Which Formula to Use?
Which Formula to Use?
Compound interest: One deposit
Annuity: Many deposits
Payout Annuity: Many withdrawals
Loans: Many payments
In the following video, we present more examples of how to use the language in the question to determine which type of equation to use to solve a finance problem.