Comprehensive Study on Interest Rates and Discount Rates

Comprehensive Study on Interest Rates and Discount Rates

1. Interest Rates are the Fundamental Part of the Actuarial Work.
2. In What Situations will Banks act as a Borrower?
- accepts money from savers
- issues own shares to investors
- it sells Fixed interest Securities

So, how does a lender charge interest rates. Let’s see 2 Scenarios:
Scenario 1:  If you lend money to US government and to a Businessman, then you will probably demand a higher rate of interest from businessman as he is more likely not to repay the loan or interest amount.
Scenario 2: If a lender expects higher inflation over the term of loan, it may demand higher rate of interest to increase its real return

INTEREST:
1.Simple Interest: Here the interest once credited, does not itself earn further interest. So, the accumulation factor will be: (1+ni) = here “n” represents no. of years and “i” represents simple interest rate per annum.

This is not the right approach that further interest rate is not earned on earlier interest because you as an investor withdraws your money with interest and then invest somewhere to get more return. Even by using this technique it leads to heavy and unnecessary additional transaction expenses. Let’s see an example:
a.)I invested 1000rs today at 10% p.a. for 2 years simple interest. My amount at the end will be rs.1200
b.)I invested 1000rs today for 1 year at 10%p.a. and I will get after 1 year rs.1100 and then reinvested rs.1100 again for one more year and get rs.1210.
Then there comes Compound Interest:

2.Compound Interest: Here the interest itself earns interest. So accumulation factor will be: (1+i)n where “i” represents compound interest rate

Principle of Consistency (From Accumulation Context): A(t0 , tn) = A(t0,t1)*A(t1,t2)…..*A(tn-1,tn)
Important point: Nominal rates are those where interest rates are paid more frequently than once per unit time. Bank accounts normally use nominal rates. They quote the annual interest rate but interest is actually added at the end of each month. Thus, here interest is paid more frequently than once per unit time year.

DISCOUNT:
1.Simple Discount: Accumulation factor will be (1-nd).
2. Compound Discount: Accumulation Factor will be (1-d)n .
Where is simple discount rate used normally? = It is often quoted for treasury bills which are short term loans made by the government. Rather than quoting the amount they wish to borrow they quote the amount of repauyment.

Let’s revise now the accumulated values using accumulation factors:
Simple interest= P(1+ni) = A
Compound Interest = P(1+i)n = A
Simple Discount = A(1-nd) = P
Compound Discount = A(1-d)n = P

So, what is the difference between interest rate and discount rate? = Discount rate is interest paid at the start of the year. Interest rate is the interest that is paid at the end of the year.
Example= I have Rs. 100 to invest @10% interest rate p.a. Then I will get Rs.110 at the end of the year.  Now I want 100Rs. Loan at 10% discount rate p.a. Then in this case I will get Rs.90 in the beginning and I have to pay Rs.100 at the end of the year.

Case Study Question: If you are a borrower and go to bank for a loan of rs.100. Now banker say that you have 2 choices that either you can get the loan at 10% discount rate or get the loan at 10% interest rate. So, which option will you choose and why?
Case Study Answer: As a borrower I will choose to get loan at 10% interest rate because when I convert 10% discount rate into interest rate it will be more than 10% interest rate; and as a borrower I am looking for loan at minimum interest rate.

Last point : We can equate two rates using their accumulation factors. For reference, see ch-2 of CT1.