Question

for a 10 years deposit , what interest rate payable annually is equivalent to 5% interest payable quarterly? ​

Answer 1

Given :

For 10 years deposit ,

Interest rate payable annually is equivalent to 5% interest payable quarterly

To Find :

Annual Interest rate for 10 years

Solution :

From Compound Interest method

∵Amount = Principal × $(1+\dfrac{rate}{100})^{time}$

Let The interest rate for 10 years annually = r%

At Annual rate of interest

Amount = Principal × $(1+\dfrac{r}{100})^{10}$

At quarterly rate of interest

Amount =  Principal × $(1+\dfrac{5}{400})^{4 \times 10}$

∵ Both rate of interest are equivalent

∴  Principal × $(1+\dfrac{r}{100})^{10}$ =   Principal × $(1+\dfrac{5}{400})^{4 \times 10}$

Or,  $(1+\dfrac{r}{100})^{10}$ =    $(1+\dfrac{5}{400})^{4 \times 10}$

Or,  $(1+\dfrac{r}{100})^{10}$ = $(1.0125)^{40}$

Now, Taking Log base 10 both side

i.e $Log_1_0$  $(1+\dfrac{r}{100})^{10}$ = $Log_1_0$  $(1.0125)^{40}$

Or,  10  $Log_1_0$ ( 1 + $\dfrac{r}{100}$ ) = 40 ×  $Log_1_0$1.0125                ( ∵ $Log_1_0 a^{b}$ = b $Log_1_0$ a )

Or, 10  $Log_1_0$ ( 1 + $\dfrac{r}{100}$ ) = 40 × 0.00539

Or,  $Log_1_0$ ( 1 + $\dfrac{r}{100}$ ) = $\dfrac{0.2156}{10}$

∴  $Log_1_0$ ( 1 + $\dfrac{r}{100}$ ) = 0.0215

1 + $\dfrac{r}{100}$ = anti$Log_1_0$   0.0215                               ( taking anti Log )

Or,  1 + $\dfrac{r}{100}$ = 1.05089

Or,  $\dfrac{r}{100}$ = 1.05089 - 1

or,  $\dfrac{r}{100}$ = 0.05089

∴  r = 0.05089 × 100

i.e r = 5.08%

Hence, The rate of interest for equivalent condition is 5.08 %    Answer

Answer 2

Given :  a 10 year deposit   5% interest payable quarterly ​

To find :  equivalent  interest rate payable annually

Solution:

Let say amount = P

Rate of interest = R

Time  n = 10 Years

A =  P(1 + R/100)ⁿ

=> A = P(1 + R/100)¹⁰

R = 5%

payable quarterly ​

=> Quarterly interest rate = 5/4 = 1.25 %

Time = n = 10 * 4 = 40 Quarters

A = P(1 + 1..25/100)⁴⁰

=> A = P( 1.0125)⁴⁰

P(1 + R/100)¹⁰ = P( 1.0125)⁴⁰

=> (1 + R/100)¹⁰ = ( 1.0125)⁴⁰

=>  (1 + R/100)¹⁰  = ((1.0125)⁴)¹⁰

=> 1 + R/100 = 1..050945

=> R/100 = 0.050945

=> R = 5.0945

5.0945 % interest rate payable annually is equivalent to 5% interest payable quarterly ​ for a 10 year deposit

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