Question

47. Method A assumes simple interest over final fractional periods, while Method B assumes simple discount over final fractional periods. The annual effective rate of interest is 20%. Find the ratio of the present value of a payment to be made in 1.5 years computed under Method A to that computed under Method B.

Answer 1

Simple Interest vs. Simple Discount

In case of Simple Interest, the Interest is calculated on Initial Amount or Principal in following manner –

I = Prt

A = P + I = P(1+rt)

Where,

I = Simple Interest

P = Principal

r = Interest rate

t = time

A = Accumulated value or Amount.

In case of Simple Discount, the Discount is calculated on Maturity value or Face value in following manner –

D = Mrt

P = M – D = M(1-rt)

Where,

D = Simple Discount

M = Maturity Value

P = Proceeds or Present value of M

r = interest rate

t = time

Now suppose, In our question, Payment to be made in 1.5 years is $ 100.

For simple interest, Payment $ 100 is accumulated value(A = P + I ) and For simple discount, Payment $ 100 is Maturity Value.

Calculation of Present Value of Payment as per Method A i.e Simple Interest -

A = P(1+rt)

Where,

A = $ 100

P = ?

r = 0.2

t = 1.5

P = A/(1+rt)

P = 100/(1+0.2*1.5)

P = 100/1.3

P = $ 76.92

Thus, Present value of Payment as per Method A is $ 76.92

Calculation of Present Value of Payment as per Method B i.e Simple Discount -

P = M(1-rt)

Where,

P = ?

M = $ 100

r = 0.2

t = 1.5

P = 100(1-0.2*1.5)

P = 100*0.7

P = $ 70

Thus, Present value of Payment as per Method B is $ 70

Therefore, ratio of Present value of Payment under Method A and Method B -

= Present value under method A / Present value under method B

= 76.92/70

= 1.099

Thus, ratio is 1.1 (rounded off)

Answer 2

Method A assumes simple interest over final fractional period

if present value for method A=x;

Amount(3/2)= x(1+20%)(1+20%/2)=1.32x

Method B assumes simple discount over fractional period

if present value for method B=y;

Amount(3/2)= y(1+20%)(1/(1-20%/2))=1.33y

if 1.32x=1.33y;

Then x/y=1.33/1.32

Correct me if I am wrong