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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).

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