Question

when compounded annually an interest rate is 11%. what is the rate when expressed with (a) semiannual compounding, (b) quarterly compounding, (c) monthly compounding, (d) weekly compounding, and (e) daily compounding.

Answer 1

The interest rates with different compounding periods are: (a) 10.3025%, (b) 10.3908%, (c) 10.5826%, (d) 10.6647%, and (e) 10.7407%.

Step-by-step explanation:

(a) Semiannual compounding:

The interest rate can be expressed as:

Effective Annual Rate (EAR) = (1 + periodic interest rate)number of periods - 1

For semiannual compounding, the periodic interest rate is half the annual rate:

Periodic interest rate = 11% / 2 = 5.5%

Substituting the values into the formula:

EAR = (1 + 5.5%)2 - 1

Calculating the EAR:

EAR = 11.3025% - 1 = 10.3025%

(b) Quarterly compounding:

For quarterly compounding, the periodic interest rate is a quarter of the annual rate:

Periodic interest rate = 11% / 4 = 2.75%

Substituting the values into the formula:

EAR = (1 + 2.75%)4 - 1

Calculating the EAR:

EAR = 11.3908% - 1 = 10.3908%

(c) Monthly compounding:

For monthly compounding, the periodic interest rate is a twelfth of the annual rate:

Periodic interest rate = 11% / 12 = 0.9167%

Substituting the values into the formula:

EAR = (1 + 0.9167%)12 - 1

Calculating the EAR:

EAR = 11.5826% - 1 = 10.5826%

(d) Weekly compounding:

For weekly compounding, the periodic interest rate is 1/52 of the annual rate:

Periodic interest rate = 11% / 52 = 0.2115%

Substituting the values into the formula:

EAR = (1 + 0.2115%)52 - 1

Calculating the EAR:

EAR = 11.6647% - 1 = 10.6647%

(e) Daily compounding:

For daily compounding, the periodic interest rate is 1/365 of the annual rate:

Periodic interest rate = 11% / 365 = 0.0301%

Substituting the values into the formula:

EAR = (1 + 0.0301%)365 - 1

Calculating the EAR:

EAR = 11.7407% - 1 = 10.7407%

Answer 2

The equivalent interest rate for a given annual interest rate changes when the compounding frequency changes from annually to semiannual, quarterly, monthly, weekly, or daily. The effective rate for each compounding frequency requires a specific calculation using the formula provided.

Step-by-step explanation:

When an interest rate is compounded annually at 11%, it means that the interest is added to the initial amount once a year. For different compounding frequencies, the equivalent interest rate changes because interest is being compounded more frequently.

To convert an annual interest rate to a different compounding period, you can use the following formula:

Effective Rate = (1 + annual rate/n)n - 1

Where:

(a) For semiannual compounding (n=2), the rate would be: (1 + 0.11/2)2 - 1

(b) For quarterly compounding (n=4), the rate would be: (1 + 0.11/4)4 - 1

(c) For monthly compounding (n=12), the rate would be: (1 + 0.11/12)12 - 1

(d) For weekly compounding (n=52), the rate would be: (1 + 0.11/52)52 - 1

(e) For daily compounding (n=365), the rate would be: (1 + 0.11/365)365 - 1

Note: To find the actual percentages, calculate the expressions and then convert the resultant decimal back to a percentage.