# Consistency of Textbook Problems Involving Discount Rate

A book on Theory of Interest explains about the discount rate as a rate when the interest is paid at the beginning of period:

If a person borrows \$100 for 1 year and with discount rate of 10% anually, then he will get \$90 at beginning and must pay \\$100 at end of one year.

Problem example:

• Find $$d_{5}$$ if the rate of simple discount is 10%. Assuming unit amount of investment.

My initial solution: $$d_{5}$$ is the effective discount rate between end of year 4 and end of year 5. Based on the explanation about discount rate, I assume the $$1$$ is the amount have to be paid at end of year 1, and borrower gets $$0.9$$ at time $$t=0$$ year. Using this I get the amount have to be paid at $$t=4$$ and $$t=5$$ is $$1.4$$ and $$1.5$$. So $$d_{5} = 1/13$$. But the answer key is $$1/6$$.

2nd Attempt: Now I try that the amount borrowed at $$t=0$$ is $$1$$. (I think of it as $$1$$ invested by the lender). Then the amount that have to be paid by the borrower at $$t=4$$ and $$t=5$$ is $$\frac{1}{1-(0.1)4} = 10/6$$ and $$\frac{1}{1-(0.1)(5)}=10/5$$. So $$d_{5} = 1/6$$.

Sometimes the problems in textbook about interest theory don't explain situation clear enough. Is my assumption in the 1st attempt reasonable, or normal? or there is a general consensus that if investment is unit amount, then at $$t=0$$ is $$1$$ (unless stated otherwise).