## What is the relationship between effective interest rate and compound interest?

## 3. Effective interest rate calculation

__Effective Interest Rate Example 1__:

Company ABC invests $100,000. The investment earns 12% per year compounded semiannually (i.e., two times per year).

**EAR = (1
+ 0.12 ÷ 2) ^{2} – 1 = 0.1236 or 12.36%**

The EAR for
this investment is 12.36% per year.** **

__Effective Interest Rate Example 2__:

Company ABC invests $100,000. The investment earns 12% per year compounded quarterly

(i.e., four times per year).

**EAR = (1
+ 0.12 ÷ 4) ^{4} – 1 = 0.1255 or 12.55%**

The EAR for
this investment is 12.55% per year.** **

We can see from these two examples that the effective interest rate increases as the frequency of compounding increases.

__Effective Interest Rate Example 3__:

Company ABC invests $10,000 at a nominal rate of 10%. Interest is compounded continuously. What will the investment be worth in two years? What is the EAR?

**FV =
$10,000 x (e ^{0.10 x 2}) = 10,000 x 1.2214 = $12,214**

**EAR = e ^{0.10
x 2} – 1 = 1.2214 – 1 = 0.2214 or 22.14%**

The EAR on this investment is 22.14%.

__Effective Interest Rate Example 4__:

Company ABC will receive $100,000 as a lump-sum payments in five years. The stated interest rate is 10%. Interest is compounded continuously. What is the present value of the future payment?

**PV =
$100,000 x (e ^{-0.10x 5}) = $100,000 x (0.60653) = $60,653**

**EAR = e ^{0.10
x 5} – 1 = 0.6487**