An account begins with $200. After one month it contains $203.75. Assuming the i

Question

An account begins with $200. After one month it contains $203.75. Assuming the interest rate remains the same, find the annual effective rate of interest and the annual nominal rate of interest if compounding occurs each month. Same question if interest is compounded four times a month at equal intervals. An account begins with $200. After one month it contains $203.75. Assuming the interest rate remains the same, find the annual effective rate of interest and the annual nominal rate of interest if compounding occurs each month. Same question if interest is compounded four times a month at equal intervals.

Explanation / Answer

a)

Consider the given problem here if “i” be the nominal rate of interest and “n” be the number of compounding periods, => the annual effective rate of interest is given below.

=> Annual effective rate of interest = (1+i/n)^n – 1.

Now, if the accounts balance of “$200” become “$203.75” after 1 month, => its monthly compounding.

So, let’s assume that “x” be the effective monthly interest rate, => $200*(1+x) = $203.75, => 1+x = 203.75/200.

=> 1+x = 1.01875, => x = 0.01875, => x = 1.875%.

So, we the “annual effective rate of interest rate” is “x*12”, => “22.5%”.

So, we have “annual effective interest rate”, now we can derive the “nominal rate of interest” by using the above expression.

=> Annual effective rate of interest = (1+i/n)^n – 1, => 22.5% + 1 = (1+i/n)^n, => 1.225 = (1+i/n)^n.

=> (1+i/n)^n = 1.225, => (1+i/12)^12 = 1.225, => (1+i/12) = 1.225^(1/12) = 1.0 171.

=> 1 + i/12 = 1.0171, => i/12 = 1.0171 – 1 = 0.0171, => i = 0.0171*12 = 0.2052.

So, the nominal rate of interest is “i=20.52%”.

b)

Consider the 2nd case.

Now, if the accounts balance of “$200” become “$203.75” after 1 month, => its compounded 4 times a month.

So, let’s assume that “x” be the effective interest rate, => $200*(1+x)^4 = $203.75, => (1+x)^4 = 203.75/200.

=> (1+x) = 1.01875^(1/4) = 1.0047, => x = 0.0047, => x = 0.47%.

So, we the “annual effective rate of interest rate” is “x*4*12 = 0.0047*4*12=0.2256”, => “22.56%”.

So, we have “annual effective interest rate”, now we can derive the “nominal rate of interest” by using the above expression.

=> Annual effective rate of interest = (1+i/n)^n – 1, => 22.56% + 1 = (1+i/n)^n, => 1.2256 = (1+i/n)^n.

=> (1+i/48)^48 = 1.2256, => (1+i/48) = 1.2256^(1/48) = 1.0042.

=> 1 + i/48 = 1.0042, => i/48 = 1.0042 – 1 = 0.0042, => i = 0.0042*12 = 0.2016.

So, the nominal rate of interest is “i=20.16%”.

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