5.6: Quoted versus Effective Rates

Question 1

What is the definition of the effective rate?

Answer 1

The effective rate is the rate at which a dollar invested grows over a given period, usually stated in percentage terms based on an annual period.

Question 2

How is the future value (FV) calculated for an investment of $1,000 at a quoted annual rate of 16% compounded annually?

Answer 2

FV = $1,000 * (1.16)^1 = $1,160

Question 3

How is the future value (FV) calculated for an investment of $1,000 at a quoted annual rate of 16% compounded quarterly?

Answer 3

FV = $1,000 * (1.04)^4 = $1,170 (rounded)

Question 4

What is the formula to determine the effective annual rate (k) for any quoted annual rate (QR), given the compounding interval (m)?

Answer 4

k = (1 + QR / m)^m - 1

Question 5

Calculate the effective annual rate (k) for a quoted rate (QR) of 16% compounded annually (m = 1).

Answer 5

k = (1 + 0.16 / 1)^1 - 1 = 16%

Question 6

Calculate the effective annual rate (k) for a quoted rate (QR) of 16% compounded quarterly (m = 4).

Answer 6

k = (1 + 0.16 / 4)^4 - 1 = 17%

Question 7

What is the effective annual rate for 12% compounded annually?

Answer 7

k = (1 + 0.12 / 1)^1 - 1 = 12%

Question 8

What is the effective annual rate for 12% compounded semi-annually?

Answer 8

k = (1 + 0.12 / 2)^2 - 1 = 12.36%

Question 9

What is the effective annual rate for 12% compounded quarterly?

Answer 9

k = (1 + 0.12 / 4)^4 - 1 = 12.55%

Question 10

What is the effective annual rate for 12% compounded monthly?

Answer 10

k = (1 + 0.12 / 12)^12 - 1 = 12.68%

Question 11

What is the effective annual rate for 12% compounded daily?

Answer 11

k = (1 + 0.12 / 365)^365 - 1 = 12.747%

Question 12

What is the effective annual rate for 12% compounded continuously?

Answer 12

k = e^0.12 - 1 = 12.75%

Question 13

What are the keystrokes to solve for the effective annual rate using a financial calculator for daily compounding?

Answer 13

1. Press 2ND, then ICONV.
2. Press 2ND, then CLR WORK.
3. Enter NOM = 12.
4. Enter C/Y = 365.
5. Press ENTER, then ↓, ↓.
6. Press CPT to get 12.747%.

Question 14

How do you calculate the effective monthly rate (k_monthly) given an annual effective rate (k_annual)?

Answer 14

k_monthly = (1 + k_annual)^(1/12) - 1

Question 15

Calculate the effective monthly rate for an annual effective rate of 12.36%.

Answer 15

k_monthly = (1.1236)^(1/12) - 1 = 0.0097588 or 0.97588%

Question 16

What is the equation to determine the effective rate for any period, given any quoted rate?

Answer 16

k = (1 + QR / m)^f - 1

Question 17

Why can effective rates often be very different from quoted rates?

Answer 17

Effective rates take into account the frequency of compounding, whereas quoted rates do not.

Question 18

Explain how to calculate the effective rate for any period.

Answer 18

1. Identify the quoted rate (QR) and the compounding frequency (m).
2. Determine the frequency of payments per year (f) for the desired effective rate period.
3. Use the formula: k = (1 + QR / m)^f - 1.