Rate and time
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1) The document discusses formulas for calculating interest rate and time given various factors like principal, interest rate, and future value in compound interest calculations. 2) It provides examples of using the formulas to find the interest rate if the principal, future value, and time period are given, and to find the time period if the principal, future value, and interest rate are given. 3) The examples demonstrate calculating interest rates compounded semi-annually and quarterly, as well as calculating time periods for investments compounded semi-annually and quarterly.
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![Time (t) can be computed, using
logarithms, if rate (r), amount (F) and
present value (P) of the investment are
given. The following formula will be used:
time
=
log
[log(1 + )]
where t is derived from the formula
= (1 + ) and = ＄.](https://image.slidesharecdn.com/rateandtime-150713112753-lva1-app6891/85/Rate-and-time-5-320.jpg)
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Here are the solutions:
1a) Simple Interest = Principal x Rate x Time
= P 60,000 x 8% x 1
= P 60,000 x 0.08
= P 4,800
1b) Compounded annually:
Interest = Principal x (1 + Rate)^Time - Principal
= P 60,000 x (1 + 0.08)^1 - P 60,000
= P 60,000 x 1.08 - P 60,000
= P 4,800
1c) Compounded semi-annually:
Rate per period = Annual rate / Times compounded per year = 8% / 2 = 4%
Interest = Principal x (1 +Simple & compound interest
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Effective vs. Nominal Interest Rates
Interest rates play a crucial role in finance and investments. Two key types are nominal interest rate and effective interest rate, each serving different purposes.
1. Nominal Interest Rate
The nominal interest rate (also called the stated rate) is the annual rate of interest without accounting for compounding within the year. It is typically used in loan agreements, bonds, and savings accounts.
Expressed as an annual percentage.
Does not consider the effect of compounding.
Commonly used in banking and financial contracts.
Example: If a bank advertises a 12% annual interest rate, compounded monthly, 12% is the nominal rate, but the actual return will be higher due to monthly compounding.
2. Effective Interest Rate (EIR)
The effective interest rate (also called the annual equivalent rate, EAR, or effective annual rate) represents the real return on an investment or cost of a loan after considering compounding.
Takes into account the effect of compounding frequency (monthly, quarterly, daily, etc.).
Shows the actual interest paid or earned over a year.
More accurate for comparing different financial products.
Formula:
=
(
1
+
)
1
EIR=(1+
n
r
)
n
1
Where:
r = nominal interest rate (decimal form)
n = number of compounding periods per year
Example Comparison
Suppose a bank offers a nominal rate of 12% per year, compounded monthly:
=
(
1
+
0.12
12
)
12
1
=
12.68
%
EIR=(1+
12
0.12
)
12
1=12.68%
Even though the nominal rate is 12%, the effective rate is 12.68% due to monthly compounding.
Key Takeaways
The nominal rate is the stated annual rate without compounding.
The effective rate considers compounding and reflects the true cost of a loan or return on an investment.
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Rate and time
- 1. Compound Interest The Interest Rate and Time
- 2. The interest rate can be determined given the final amount (F), present value (P), conversion period (m) and the total number of periods (n) by using the following formula: Interest rate = 1 ()(100%) where r is derived from the formula = (1 + ) and = .
- 3. Example 1: Find the rate compounded semi- annually if P 9,000 accumulates to P 15, 500 in 4 years and 5 months.
- 4. Example 2: At what rate compounded quarterly will P 15, 000 accumulate to P 20,000 in 4 years and 6 months?
- 5. Time (t) can be computed, using logarithms, if rate (r), amount (F) and present value (P) of the investment are given. The following formula will be used: time = log [log(1 + )] where t is derived from the formula = (1 + ) and = ＄.
- 6. Example 1: How long will it take P 16,000 to amount to P 19,200 if the interest rate is 10% compounded semi-annually?
- 7. Example 2: When will P 5, 500 accumulate to P 10,500 if invested at 15% compounded quarterly?