Question

Your bank is offering you an account that will pay 22 % interest​ (an effective​ two-year rate) in total for a​ two-year deposit. Determine the equivalent discount rate for the following​ periods: a. Six months b. One year c. One month ​(Note: Be careful not to round any intermediate steps less than six decimal​ places.) a. Six months The equivalent discount rate for a period length of six months is. ​(Round to two decimal​ places.)

Answer 1

Solution :

The formula for calculating the equivalent discount rate for given effective rate

= ( 1 + r ) ⁿ - 1

Where

r = effective rate for a given period

n = Period Length / Effective rate period

a. Calculation of equivalent discount rate for six months :

As per the information given in the question we have

r = 22 % = 0.22 ( for a period of two years = 24 months )

Period length = 6 months ; Effective rate period = 2 year = 24 months

Thus n = 6 / 24 = 0.25

Applying the above information in the formula we have

= ( 1 + 0.22 ) ^{(0.25 )} - 1

= ( 1.22 ) ^{( 0.25 )} – 1

= 1.050969 – 1

= 0.050969

= 5.0969 %

= 5.10 % ( when rounded off to two decimal places )

Thus the equivalent annual discount rate for a period of six months = 5.10 %

NOTE : The value of ( 1.22 ) ^{( 0.25 )}   has been calculated using the excel function =POWER(Number,Power). Thus =POWER(1.22,0.25) = 1.050969

b. Calculation of equivalent discount rate for one year :

As per the information given in the question we have

r = 22 % = 0.22 ( for a period of two years = 24 months )

Period length = one year = 12 months ; Effective rate period = 2 years = 24 months

Thus n = 12 / 24 = 0.5

Applying the above information in the formula we have

= ( 1 + 0.22 ) ^{(0.5 )} - 1

= ( 1.22 ) ^{( 0.5 )} – 1

= 1.104536 – 1

= 0.104536

= 10.4536%

= 10.45 % ( when rounded off to two decimal places )

Thus the equivalent annual discount rate for a period of one year = 10.45 %

NOTE : The value of ( 1.22 ) ^{( 0.5 )}   has been calculated using the excel function =POWER(Number,Power). Thus =POWER(1.22,0.5) = 1.104536

c. Calculation of equivalent discount rate for one month :

As per the information given in the question we have

r = 22 % = 0.22 ( for a period of two years = 24 months )

Period length = 1 month ; Effective rate period = 2 years = 24 months

Thus n = 1/ 24 = 0.041667

Applying the above information in the formula we have

= ( 1 + 0.22 ) ^0.041667 - 1

= ( 1.22 ) ^{( 0.041667 )} – 1

= 1.008320 – 1

= 0.008320

= 0.8320 %

= 0.83 % ( when rounded off to two decimal places )

Thus the equivalent annual discount rate for a period of one month = 0.83 %

NOTE : The value of ( 1.22 ) ^{( 0.041667 )}   has been calculated using the excel function =POWER(Number,Power). Thus =POWER(1.22,0.041667) = 1.008320

Thus

a. The equivalent discount rate for a period length of six months is = 5.10 %

b. The equivalent discount rate for a period length of one year is = 10.45 %

c. The equivalent discount rate for a period length of one month is = 0.83 %