Help with Effective Rate of Discount- Theory of interest

Question

I am just beggining Financial Mathematics.

One of my assignment questions are as follows:

(Q) Find the amount of interest earned from the principal of $1000 during the fourth period

If the effective rate of discount is dn = 0.02n + 0.005 for n = 1,2,3,4

I know dn = (a(n) - a(n-1)) / a(n) and interest recieved in period n In = k (a(n) - a(n-1)) but how to figure out the accumulative function? If I can't find this, then I can't find the interest recieved. And it doesn't mention simple discount rate or anything.

Help much appreciated!!

Answer 1

The discount rate is defined as

$$DR = \frac{IR}{1 + IR}$$

where $IR$ is the interest rate. Thus in the fourth period $DR = .08 + .005 = .085$

I think you can carry on to solve for IR and then multiply 1000 by it.

Please note that this answer only pertains to the question - interest from the principal amount in the fourth year, i.e., not asking for compounding of previously earned interest.

Answer 2

Convert the discount rate to the interest rate using $$IR = \frac{DR}{(1-DR)}$$ so that, for year $n$, the interest rate is given $$IR_n = \frac{0.02 \cdot n + 0.005} { 1 - (0.02 \cdot n +0.005})$$.

Then calculate the total receivable at the end of year three and four as $$\$1000 \cdot (1 + IR_1) \cdot (1 + IR_2) \cdot (1+IR_3) $$

and similarily for year four.

EDIT:

Finally, subtract the year three result from the year four result to get the interest earned in year four.