Question

Question 1 (4 points) Income from the recycling paper and cardboard at Hamden recycling center has averaged $9,000 per quarter for 5 years. What is the future worth of the income at year 5 at an interest rate of 4%, compounded semiannually? a) Assume that interperiod cash flows earn prorated compound interest. b) Assume there is no interperiod compounding.

Answer 1

As mentioned in question, average income earned from Recycling paper and cardboard at Hamden Recyccling center has averaged $9000;

Therefore, we will take principal amount for compounding as $9,000 in a) and $18,000 in b)

Rate of compounding is 4% per year comounded semi-annualy;

Period is not mentioned and there for we are assuming period as 1 year of 4 quarter i.e. at the end of 5th year

Future vaklue with compound interest is given by:-

(FV) = C0   X (1 + r)n

Where,

FV is future Value

C0 is principal amount
r is rate of interest

n is the period for which amount will be compounded

Now let us solve requirements mentioned in question -

a) Assuming inter-period income earns prorated compound interest i.e. $9,000 earned quarterly will earn compound interest.

As rate mentioned in question is 4% compounded semi-annually, we assume rate is annual rate of interest and therefore for calculating quarterly income interest we need to convert this rate into quarterly rate of interest i.e. 4% / 4 (interest divided by number of quarter in a year) i.e. 1%

Let us assume period of compounding asked in question as at the end of year(in absence of information) therefore here n will be 3 for Q1 income, 2 for Q2 income, 1 for Q3 income and 0 for Q4 Income [Another assumption here is income is earned on at the end of quarter.]

FV(Q1)         = $9,000 X (1 + 0.01)3

                         =$9,000 X 1.0303

=$9,273

FV(Q2)         = $9,000 X (1 + 0.01)2

                         =$9,000 X 1.0201

=$9,181

FV(Q3)         = $9,000 X (1 + 0.01)1

                         =$9,000 X 1.01

                     =$9,090

FV(Q3)         = $9,000 X (1 + 0.01)0

                         =$9,000 X 1

                     =$9,000

Therefore, annual value of 5th year income will be 36,636

b) Assuming inter-period income does not earns prorated compound interest i.e. $9,000 earned quarterly will earn compound interest during half year only.

Accordingly, as calculated in a) rate of interest here will be 4% / 2 i.e. 2% with tenure as 1 for Q1 and Q2 income and 0 for Q3 and Q4 income. Also, here principal in use will be $18,000 (being $9,000 earned per quarter)

Therefore, FV will be calculated as follows: -

Q1 & Q2             = $18,000 X (1 + 0.02)1

   = $18,000 X (1.02)

                            = $18,360

Q3 & Q4             = $18,000 X (1 + 0.02)0

                                 = $18,000 X (1)

                            = $18,000

Therefore, annual value at the year 5 end will be $33,360.