Question

Suppose that you put $5,000 in your saving account and your bank offers an interest rate of 5% p.a. What will be the compounded amount in five years and the corresponding annual effective rate (AER)?

a) If it was compounded quarterly?

b) If it was compounded monthly?

c) If it was compounded fortnightly?

d) If it was compounded weekly?

e) If it was compounded daily?

Answer 1

As per compound interest accumulated value can be computed as:

A = P x (1+r/m) m x t

A = Final amount

P = Principal amount

r = Interest rate

m = Number of compounding in a year

t = Number of years

Formula for effective rate, i = (1+r/m) m - 1

a)

P = $ 5,000, r = 0.05, t = 5, m = 4

A = $ 5,000 x (1+ 0.05/4) 4 x 5

= $ 5,000 x (1+ 0.0125) 20

= $ 5,000 x (1.0125) 20

= $ 5,000 x 1.28203723170859

= $ 6,410.18615854293 or $ 6,410.19

Effective rate = (1+ 0.05/4)4 – 1

                        = (1+ 0.0125)4 – 1

                        = (1.0125)4 – 1

                        = 1.05094533691406 – 1

                        = 0.05094533691406 or 5.095 %

b)

P = $ 5,000, r = 0.05, t = 5, m = 12

A = $ 5,000 x (1+ 0.05/12) 12 x 5

= $ 5,000 x (1+ 0.0041666667) 60

= $ 5,000 x (1.0041666667) 60

= $ 5,000 x 1.28335868105958

= $ 6,416.7934052979 or $ 6,416.79

Effective rate = (1+ 0.05/12)12 – 1

                        = (1+ 0.0041666667)12 – 1

                        = (1.0041666667)12 – 1

                        = 1.05116189830045 – 1

                        = 0.05116189830045 or 5.116 %

c)

P = $ 5,000, r = 0.05, t = 5, m = 26

A = $ 5,000 x (1+ 0.05/26) 26 x 5

= $ 5,000 x (1+ 0.0019230769) 130

= $ 5,000 x (1.0019230769) 130

= $ 5,000 x 1.28371718503674

= $ 6,418.5859251837 or $ 6,418.59

Effective rate = (1+ 0.05/26)26 – 1

                        = (1+ 0.0019230769)26 – 1

                        = (1.0019230769)26 – 1

                        = 1.05122061978266 – 1

                        = 0.05122061978266 or 5.122 %

d) P = $ 5,000, r = 0.05, t = 5, m = 52

A = $ 5,000 x (1+ 0.05/52) 52 x 5

= $ 5,000 x (1+ 0.0009615385) 260

= $ 5,000 x (1.0019230769) 260

= $ 5,000 x 1.28387120765671

= $ 6,419.3560382836 or $ 6,419.36

Effective rate = (1+ 0.05/52)52 – 1

                        = (1+ 0.0009615385)52 – 1

                        = (1.0009615385)52 – 1

                        = 1.05124584402767 – 1

                        = 0.05124584402767 or 5.125 %

e) P = $ 5,000, r = 0.05, t = 5, m = 365

A = $ 5,000 x (1+ 0.05/365) 365 x 5

= $ 5,000 x (1+ 0.0001369863) 1825

= $ 5,000 x (1.0001369863) 1825

= $ 5,000 x 1.28400342893772

= $ 6,420.0171446886 or $ 6,420.02

Effective rate = (1+ 0.05/365)365– 1

                        = (1+ 0.0001369863)365 – 1

                        = (1.0001369863)365 – 1

                        = 1.05126749594194 – 1

                        = 0.05126749594194 or 5.127 %