Compound Interest

What is Compound Interest?

Compound Interest

• It is interest that is computed from the principal value and also from the previous accumulated interests.

• When we compute compound interest, not only we compute the interest from the current year, we also need to compute the interest from the previous years.

  • For example, suppose that I placed Php on a savings account with interest that compounds annually. How much will I get after 5 years?
  • We can use a table to compute the interest each year.

Years after Initial Deposit	Computation	Savings
0
1
2
3
4
5

Computing Compound Interest that Compounds Annually

• To compute the maturity value that compounds annually, we use the following formula:

• where

  • future/maturity value
  • present/principal value
  • interest rate
  • time/length of term in years

• Take note that this formula computes the maturity/future value and not the interest.

  • To compute the interest, we simply subtract the interest from the maturity value.

• where stands for the compound interest.
• Since we already have a formula for the maturity value, we can simply substitute it to calculate the interest without calculating the maturity value first.

- We can practice by solving a compounding interest problem.

Problem

Wendell placed Php in his savings account with interest that compounds annually. How much is in his savings account after years?

Solution

• The principal value is .
• The interest rate is or .
• the length of term is years

Since the interest is compounding, we use the compounding interest formula.
Substituting these values, we get:

Problem

Sophie placed her Php worth of lottery winnings in a high-yield savings account that compounds yearly for interest rate. Find the interest and her maturity value after 5 years.

Solution

• The principal value is .
• The interest rate is or .
• The length of term is years.

Since the interest is compounding, and we are tasked to find the maturity value, we use the formula for finding the future value for compound interest, which is:

Substituting our values, we get:

To find for the interest, we simply subtract the principal from the maturity value, namely:

Computing Compound Interest that Compounds More Than a Year

• In the previous example, we showed how compound interest can be computed if the interest compounds once a year.
• If it compounds more frequently, then we have to modify our formula a bit.

• where:

  • is the future value
  • is the principal
  • is the annual interest rate
  • is how often interest compounds a year
  • is the length of the term in years

• The value of depends on how often the interest compounds per year.

• For example, if interest compounds semi-annually, then it compounds 2 times a year

  • Since it compounds twice a year, therefore .

• You may want to refer to the table below to know what to use depending on how often it compounds.

Frequency		Formula
annually
semi-annually
quarterly
semi-quarterly
monthly
weekly
daily

Problem

Carl placed his $ for his retirement savings account that compounds monthly for . Find the maturity value after:

1. 9 months

2. 2 years

Solution

• The principal is .
• The annual interest rate is or .
• The length of term is in 9 months or years, and in 2 years.
• The interest compounds every month, so .
  (since there are months in a year)

Since the interest is compounding and it compounds monthly, we use the formula:

Applying and substituting the variables, we get:

• For months:

• For years:

Problem

Maya Bank Inc. offers a savings plan that compounds daily for up to a maximum term length of months. Find the maximum interest if you invested in Philippine pesos in that account.

Solution

• The principal is .
• The annual interest rate is or .
• The term length is months or years.
• The interest compounds every day, so .
  (Since there are 365 days in one year)

Since the interest is compounding and the interest compounds daily, we use the formula:

Substituting values, we get:

Since we are asked to find the interest, we simply subtract the principal from the future value.

Solving Problems Involving Principal in Compound Interest

• Suppose that you want to solve for the initial value deposited in a compound interest problem.

• We can simply divide by to solve for the principal .
• Therefore, we get the formula for the principal:

• where:
  • is the principal value/present value
  • is the future value/maturity value
  • is the annual interest rate
  • is how frequent the interest compounds per year
  • is the length of time

Problem

How much should you invest in a fund earning compounding quarterly if you want to accumulate Php in 2 years?

Solution

• The future value is
• The interest rate is or .
• The length of the term is years
• The interest compounds quarterly, so
  (since there are 4 quarters in a year; quarter = in fours)

Since the interest is compounding, and we are asked to find the principal, we use the formula:

Applying the formula, we get:

Checking our answer by substituting it back to the original compound interest formula, we get:

Solving Problems Involving Time in Compound Interest

• What is we’re instead interested in when will our money reach a certain amount?
• In that case, we are solving for time or the length of term.
• However, solving for the length of term in a compound interest scenario may be a little complicated.
• Consider the compound interest formula:

• To solve for time, we apply the common logarithm on both sides.

• Product rule of logarithms say that is , therefore:

• Product rule of logarithms say that is

• To solve for time , we divide both sides by , to get:

• Therefore:

Problem

How long will it take to double my money if I invest in a fund that compounds semi-annually for interest?

Solution

• The principal is .
• The interest rate is or .
• The future value is double the principal, so .
• The interest compounds semi-annually, so every half a year, therefore
  (since there are 2 halves in a single year)

Since the interest is compounding, and we need to look for the time, the formula is:

Substituting the values, we get: