Interest Rate Equivalency and Effective Rate

Simple and Compound Interest Rates: Comparison and Equivalency Concept

Comparison of Simple and Compound Interest

Simple Interest (SI) and Compound Interest (CI) are the two primary ways interest is calculated, leading to fundamentally different patterns of money growth over time. Understanding their differences is crucial in financial analysis.

Simple Interest (SI):

• Basis of Calculation: Interest is calculated *only* on the original principal amount ($P$) for the entire duration of the investment or loan.

• Interest Amount Per Period: The amount of interest earned or paid in each time period (e.g., each year) remains constant, assuming the principal and rate are unchanged. The interest earned previously is not added to the principal for future interest calculations.

• Growth Pattern: The growth of the accumulated amount over time is linear. A graph of Amount vs. Time for simple interest is a straight line.

• Formula for Amount (A): $A = P(1 + rt)$, where $r$ is the annual rate (decimal) and $t$ is time in years.

Compound Interest (CI):

• Basis of Calculation: Interest is calculated on the original principal *plus* all the accumulated interest from previous periods. This phenomenon is known as "interest on interest".

• Interest Amount Per Period: The amount of interest earned or paid typically increases with each successive period because the base on which interest is calculated (the principal + accumulated interest) is growing.

• Growth Pattern: The growth of the accumulated amount over time is exponential. A graph of Amount vs. Time for compound interest is an upward-curving line.

• Formula for Amount (A): $A = P(1 + i)^n$, where $i$ is the periodic rate and $n$ is the total number of periods.

Key Consequence of the Difference: For any positive interest rate and principal, and for any time period greater than one compounding period (or one year if comparing annual rates), the accumulated amount under compound interest will be greater than the amount under simple interest. The longer the time period or the higher the interest rate or compounding frequency, the larger the difference between compound and simple interest becomes. They are equal only at the end of the first period (if the rate is the same for that period) or if the time period is zero, or if the interest rate is zero.

The graph visually demonstrates how the power of compounding leads to significantly faster growth over longer durations compared to the steady, linear growth of simple interest.

Equivalency Concept Between Simple and Compound Rates

Given that simple interest and compound interest grow money differently, a simple interest rate cannot be universally "equivalent" to a compound interest rate for all possible investment horizons or loan durations.

However, we can determine an "equivalent" simple interest rate for a *specific* compound interest rate over a *specific* time period. This is the simple interest rate that would produce the same final accumulated amount over that exact period as the given compound interest rate.

Let:

• $P$ = Principal amount
• $t$ = Specific time period in years
• $r_{simple}$ = Annual simple interest rate (as a decimal)
• $r_{compound}$ = Annual compound interest rate (as a decimal), compounded annually for simplicity here.

The Amount accumulated under simple interest is $A_{simple} = P(1 + r_{simple} t)$.

The Amount accumulated under compound interest (compounded annually) is $A_{compound} = P(1 + r_{compound})^t$.

To find the simple interest rate ($r_{simple}$) that is equivalent to the compound rate ($r_{compound}$) over the specific time period $t$, we set the final amounts equal to each other:

$A_{simple} = A_{compound}$

$P(1 + r_{simple} t) = P(1 + r_{compound})^t$

Since the principal $P$ is common and non-zero, we can divide both sides by P:

$1 + r_{simple} t = (1 + r_{compound})^t$

Now, we solve for $r_{simple}$:

$r_{simple} t = (1 + r_{compound})^t - 1$

Divide by $t$ (assuming $t \neq 0$):

This formula gives the simple annual interest rate that is equivalent to an annual compound interest rate $r_{compound}$ over exactly $t$ years. Note that the equivalent simple rate depends on the time period $t$.

Answer:

This dependency on time $t$ means that an 8% compound rate would be equivalent to a different simple rate over a different time period (e.g., over 1 year, the equivalent simple rate is exactly 8%; over 2 years, it would be $\frac{(1.08)^2-1}{2} = \frac{1.1664-1}{2} = \frac{0.1664}{2} = 0.0832 = 8.32\%$). This highlights why comparing quoted rates requires understanding the type of interest and the period.

Nominal vs. Effective Interest Rate: Definitions

When interest is compounded more frequently than annually (e.g., monthly, quarterly), the stated annual interest rate, known as the nominal rate, does not fully capture the actual rate of growth over a year due to the effect of compounding within the year. This leads to an important distinction between the nominal and effective annual interest rates.

Nominal Annual Interest Rate ($r$ or $j_m$)

The Nominal Annual Interest Rate is the stated or quoted annual rate for an investment or loan. It is typically expressed "per annum" but is accompanied by the frequency of compounding if it is more often than annual.

• It is simply the annual rate *before* considering how many times it will be compounded within the year.
• It serves as the basis for calculating the periodic interest rate ($i$) by dividing the nominal rate by the number of compounding periods per year ($m$).
• For example, "12% per annum compounded monthly" means the nominal annual rate is 12%. It *doesn't* mean you earn exactly 12% interest total over the year relative to the principal, because interest is added back monthly.
• The nominal rate does not truly reflect the year's percentage increase unless compounding is annual ($m=1$).
• Notation: Often $r$, $R$, or sometimes $j_m$, where $m$ is the compounding frequency. For example, $j_{12}$ might denote a nominal rate compounded monthly.

Effective Annual Interest Rate (EAR or $r_e$, $i_{eff}$, $i$)

The Effective Annual Interest Rate (EAR), also known as the Annual Equivalent Rate (AER) in some regions, is the actual annual rate of return earned on an investment or paid on a loan after taking into account the effect of compounding over a one-year period.

• It represents the total amount of interest accumulated in one year, expressed as a percentage of the original principal at the beginning of the year.
• The EAR is the simple interest rate that would yield the same accumulated amount as the compound rate when applied for one year.
• The EAR allows investors and borrowers to make a true comparison between different financial products that may have different nominal rates and different compounding frequencies.
• For any given nominal annual rate, the EAR will be equal to or greater than the nominal rate. They are equal only if the compounding frequency is annual ($m=1$). If $m > 1$, the EAR will always be greater than the nominal rate because of the interest earned on interest within the year.
• Notation: Often EAR, $r_e$, $i_{eff}$, or sometimes just $i$ when it refers specifically to the effective annual rate.

Illustration of Nominal vs. Effective Rate

Let's consider a principal of $\textsf{₹}\$ 100$ invested for one year at a nominal annual rate of 10%, compounded semi-annually.

• Nominal Annual Rate ($r$) = 10% or 0.10.
• Compounding Frequency ($m$) = Semi-annually (2 times per year).
• Periodic Rate ($i$) = $\frac{r}{m} = \frac{0.10}{2} = 0.05$ (5% per half-year).
• Time ($t$) = 1 year.
• Total periods ($n = mt$) = $2 \times 1 = 2$.

Let's calculate the amount after one year using the compound interest formula $A = P(1 + i)^n$:

$A = 100 (1 + 0.05)^2$

$A = 100 (1.05)^2$

$A = 100 (1.1025)$

$A = \textsf{₹}\$ 110.25$

The total interest earned in one year is the Amount minus the Principal:

$Interest\$ Earned = A - P = 110.25 - 100 = \textsf{₹}\$ 10.25$

To find the Effective Annual Rate (EAR), we express this total interest earned as a percentage of the original principal for one year:

$EAR = \frac{Interest\$ Earned}{Principal} \times 100\% = \frac{10.25}{100} \times 100\% = 10.25\%$

So, in this example:

• Nominal Annual Rate = 10%
• Effective Annual Rate = 10.25%

The EAR (10.25%) is higher than the nominal rate (10%) because the interest earned in the first half-year ($\textsf{₹}\$ 5$) itself earned interest in the second half-year ($\textsf{₹}\$ 5 \times 0.05 = \textsf{₹}\$ 0.25$). This extra $\textsf{₹}\$ 0.25$ is the effect of compounding within the year.

Understanding this distinction is crucial for comparing interest rates quoted with different compounding frequencies.

Effective Rate of Interest: Calculation and Formula

Concept Revisited

As discussed in the previous section, the Effective Annual Interest Rate (EAR) is the single annual rate that is equivalent to a nominal rate compounded $m$ times per year. It represents the true percentage increase in the principal over a one-year period, taking into account the effect of compounding. The EAR allows for a meaningful comparison of different interest rates with different compounding frequencies.

Derivation of the Formula for Effective Annual Rate (EAR)

Let's derive the formula for the EAR, denoted by $r_e$. We want to find the rate $r_e$ such that investing a principal $P$ at a simple annual rate $r_e$ for one year results in the same amount as investing $P$ at a nominal annual rate $r$, compounded $m$ times per year, for one year.

Let $P$ be the initial Principal amount.

Let $r$ be the nominal annual interest rate, expressed as a decimal (e.g., 12% p.a. means $r = 0.12$).

Let $m$ be the number of times interest is compounded per year (the compounding frequency).

The time period we are considering is exactly 1 year ($t=1$).

Using the compound interest formula $A = P\left(1 + \frac{r}{m}\right)^{mt}$, for a time period of $t=1$ year, the Amount (A) accumulated after 1 year is:

$A = P\left(1 + \frac{r}{m}\right)^{m \times 1}$

The total interest earned during this one-year period is the difference between the Amount and the Principal:

$Interest = A - P$

$Interest = P\left(1 + \frac{r}{m}\right)^{m} - P$

We can factor out the Principal $P$ from the right side:

$Interest = P \left[ \left(1 + \frac{r}{m}\right)^{m} - 1 \right]$

The Effective Annual Rate ($r_e$) is defined as the total interest earned in one year divided by the original principal. It's essentially the total interest earned per unit of principal over one year.

$r_e = \frac{Total\$ Interest\$ Earned\$ in\$ 1\$ Year}{Original\$ Principal}$

$r_e = \frac{Interest}{P}$

Substitute the expression for the total interest earned in one year:

Assuming $P \neq 0$, we can cancel $P$ from the numerator and denominator:

$\mathbf{r_e = \left(1 + \frac{r}{m}\right)^{m} - 1}$

This formula gives the Effective Annual Rate as a decimal.

To express the EAR as a percentage, we multiply the decimal value by 100:

$\mathbf{EAR (\%)= \left[ \left(1 + \frac{r}{m}\right)^{m} - 1 \right] \times 100}$

Formula Summary

The core formula for calculating the Effective Annual Rate (EAR) from a nominal annual rate and its compounding frequency is:

• Effective Annual Rate (EAR) as a Decimal ($r_e$):

  $r_e = \left(1 + \frac{r}{m}\right)^{m} - 1$

  Where:

  • $r$ = the nominal annual interest rate (as a decimal, e.g., $12\% = 0.12$)
  • $m$ = the number of times interest is compounded per year (e.g., 1 for annually, 2 for semi-annually, 4 for quarterly, 12 for monthly, 365 for daily)

• Effective Annual Rate (EAR) as a Percentage:

  $EAR (\%) = r_e \times 100$

  $EAR (\%) = \left[ \left(1 + \frac{r}{m}\right)^{m} - 1 \right] \times 100$

Special Case: Continuous Compounding

When interest is compounded continuously, the number of compounding periods per year ($m$) approaches infinity. The formula for the Amount under continuous compounding is $A = Pe^{rt}$. For $t=1$ year, $A = Pe^r$.

The interest earned in one year is $I = Pe^r - P = P(e^r - 1)$.

The Effective Annual Rate ($r_e$) is $\frac{I}{P}$: $r_e = \frac{P(e^r - 1)}{P} = e^r - 1$.

• Effective Annual Rate (EAR) for Continuous Compounding:

  $r_e = e^r - 1$

  Where $r$ is the nominal annual interest rate (as a decimal) and $e \approx 2.71828$ is the base of the natural logarithm.

Worked Example

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Problems involving Effective Rate of Interest

Problems involving the Effective Annual Rate (EAR) typically require you to calculate the EAR given a nominal rate and compounding frequency, use the EAR to compare different investment or loan options, or work backward to find a nominal rate or compounding frequency needed to achieve a specific EAR.

The core formula used is $r_e = \left(1 + \frac{r}{m}\right)^{m} - 1$, where $r_e$ is the effective annual rate (decimal), $r$ is the nominal annual rate (decimal), and $m$ is the number of compounding periods per year.

Worked Examples

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