1. Time value of money - applications

2. Frequent compounding
In practice, interest is often compounded more frequently than once a year (annually).
Common compounding periods are:
semi-annually with interest added twice a year,
quarterly with interest added 4 times a year,
monthly with interest added 12 times a year,
weekly with interest added 52 times a year and
daily with interest added 365 times a year.

3. Stated annual rate
Standard practice is to quote interest rates (and rates of return) as an annual rate, with the compounding period stated immediately afterwards.
Often called the nominal rate or annual percentage rate (APR).
Example: ‘6%, compounded monthly’.
‘6%’ is the nominal annual rate, which is the interest rate based on simple interest for 1 year; i.e. rate excluding compounding effect.
‘Compounded monthly’ means that interest is added every month. For savings accounts, this is sometimes expressed as ‘interest paid monthly’.

4. Compounding period effects
Scenario 1: Rate quoted as 6%. Deposit $100 for 1 year.
Scenario 2: Rate quoted as 6%, compounded semi-annually. Deposit $100 for 1 year. Year and rate are divided into two:
3% is the periodic rate = r/m, where m = number of compounding periods per year.

5. Frequent compounding
If interest is compounded more frequently than annually, we must adjust the time value of money formulas as follows:
Instead of r, use r/m (i.e. the periodic rate)
Instead of n, use n x m (i.e. the total number of compounding periods)
For example, the formula for finding the future value of a lump sum becomes:
FV = PV× 𝟏+ 𝒓 𝒎 𝒏×𝒎 PV× 𝟏+ 𝒓 𝒎 𝒏×𝒎

6. More frequent compounding: Lump sum application example
You want $4,000 in 10 years. You can invest money in a savings account that will pay 6.9%, compounded monthly. How much should you invest now to meet your goal?
PV = ?
FV = $4,000
r = 6.9%
n = 10
m =12
PV = 𝑭𝑽 𝟏+ 𝒓 𝒎 𝒏×𝒎

7. Effective annual rate (EAR)
Compare interest rates with different stated rates, compounding periods or both by using the EAR.
The EAR discloses the true ‘benefit’ (from the perspective of a lender) or ‘cost’ (from the perspective of a borrower) of interest because the EAR allows for the compounding of interest.
To convert an APR (r) to an EAR, we use the following formula where m = number of compounding periods in a year:
EAR = 𝟏+ 𝒓 𝒎 𝒎 −𝟏

8. Effective annual rate (EAR)
You have two savings account options: one that pays 7% and one that pays 6.9%, compounded monthly. Which one is better?
The account paying 7% has an EAR of 7%. (If compounding periods are not stated, assume annual, so APR = EAR.)
The account paying 6.9%, compounded monthly has an EAR of 7.12%. This is better (as an investor) than the account paying 7%.

9. Finding payments to accumulate a future sum
These applications focus on finding payments to accumulate a future sum, rather than finding a lump sum PV needed now.

10. Finding payments to accumulate a future sum: Example
You want $4,000 in 10 years. You can invest money in a savings account that will pay 7.12% pa. How much would you need to invest in the account at the end of each year to meet your goal?
PMT = ?
FV = $4,000
r = 7.12%
n = 10
PV = 0
You would need to deposit $ at the end of every year for 10 years to have $4,000 if the account pays 7.12%.

11. Loan repayment agreements
There are 3 basic types of loan repayment agreements:
A discount loan: principal and interest paid at the maturity date (end of the loan term). E.g. a zero coupon bond.
An interest-only loan: interest paid each period throughout loan term and principal repaid at maturity date, along with final interest payment. E.g. government and corporate bonds.
An amortising loan (also called a principal-and-interest loan): equal payments made each period and these payments include interest and part repayment of principal. This results in a regular principal reduction on the loan, which is called amortisation. E.g. many home mortgages and car loans.

12. Amortising loans: Example
$200,000 loan at 6% p.a., compounded monthly, over 25 years with monthly payments.
With each payment the outstanding principal is reduced such that at the end of 25 years the balance is zero.

13. Finding an amortised loan payment
When equal payments are made at the end of each interest period, amortisation is an example of an ordinary annuity.
We can find the equal payment if we know the interest rate, principal and time period.
Example:
$200,000 loan at 6% p.a., compounded monthly, over 25 years with monthly payments.

14. Amortising loans
In this example, there are 300 equal monthly payments of $ Initially, the payment is mostly interest but as the principal falls, the interest component falls leaving more of the payment to go towards repaying principal.

15. Finding the rate on amortising loan
We can find the interest rate if we know the payment, principal and time period.
Example:
$200,000 loan with monthly payments of $1, over 25 years.
The rate is 6%, compounded monthly.