# Confused about Effective Rate of Discount- Theory of Interest

I'm currently reading Kellison's book, The Theory of Interest. I've reached the chapter on Effective Rate of Discount and it's somewhat confusing. The book explains it as a loan where interest is paid up-front and simply deducted from the Principal. That makes sense over one period, but for time > 1, the concept seems rather non-intuitive. How does one calculate the accumulated value of an investment with a given rate of discount?

Looking at discount as a function of interest, I can see how one would calculate the effective rate of discount. The book gives the formula as the ratio of interest earned during a period to the amount invested at the end of the period, which makes sense. This is in contrast to the effective rate of interest which is the ratio of interest earned during a period to the principal. But, given only a rate of say simple discount, how would one calculate the effective rate of discount during some arbitrary period n? That doesn't really make sense to me.

Would an investment using effective rate of discount over a period greater than 1 look like a treasury bond?

http://en.wikipedia.org/wiki/United_States_Treasury_security#Treasury_bond

tl;dr:

Answering a few specific questions would really help me. One is, how would you calculate the effective rate of discount during some arbitrary period n given a simple rate of discount? And, how can one calculate the accumulated value of an investment with a given rate of discount (simple or compound?)

For the first question above, here's an example from the problems in the book. "Given a rate of 10% simple discount, calculate the effective rate of discount during period 5."

Any explanations of what exactly this concept of discount is, or pointers to non-confusing resources would be appreciated. It seems very unintuitive.

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I was completely confused by that same explanation for a while. Don't even think of it like that. Think of it like this.

Definition: The effective rate of interest during the $n$th time period is $$i_n = \frac{A(n) - A(n-1)}{A(n-1)}$$

Definition: The effective rate of discount during the $n$th time period is $$d_n = \frac{A(n) - A(n-1)}{A(n)}$$

where $A(n)$ is the amount function (as defined in Kellison), the amount of money you have at time $n$. So, all you really need to understand here is that the rate of interest is a rate based on what you start with during the period. The rate of discount is a rate based on what you end up with. It's just two ways of looking at the same situation. There aren't a whole lot of real world situations where you borrow a bunch of money and immediately give some of it back. You would just borrow less.

By the way, that formula is all you need to calculate the effective rate of discount during period $n$ no matter what your $A(n)$ function is. So, in particular, it would work for your specific question of simple discount.

Question: Given a rate of 10% simple discount, calculate the effective rate of discount during period 5.

Answer: If we have 10% simple discount, then we know our accumulation function is $a(t) = \frac{1}{1 - 0.1t}$ for $0 \leq t < \frac{1}{d} = 10$. This is basically the definition of simple discount. If you have simple discount, this is your accumulation function. Memorize that. Then use it.

Therefore $$d_5 = \frac{a(5) - a(4)}{a(5)} = \frac{2-10/6}{2} = \frac{1}{6} = 16.666666... \%$$

If you wanted to calculate the effective rate of interest when you are given the effective rate of simple discount, you can do that too. For example, in this same example,

$$i_5 = \frac{a(5) - a(4)}{a(4)} = \frac{2-10/6}{10/6} = \frac{1}{5} = 20 \%$$

Nothing changed. We're just looking at the same problem differently. In the discount case, how much money did we earn that period relative to how much we had at the end? In the interest case, how much money did we earn that period relative to how much we had at the beginning.

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What happened to the answer? The book gives 1/6th as the answer which I haven't been able to reach by any means... – Josh Infiesto Sep 18 '12 at 17:19
Wow I'm dumb. I tried inverting the function in the book to get the accumulation function, but I didn't get the right answer. I think I made a computational mistake and that's what's had me tripped up this whole time. Thanks. – Josh Infiesto Sep 18 '12 at 17:25
Also, really awesome answer. Thanks. – Josh Infiesto Sep 18 '12 at 17:26
@Josh No problem. Feel free to ask more questions here. Tag them with "actuarial-science", which I just created. finance is good too. – Graphth Sep 18 '12 at 17:27
Yeah... I was surprised there wasn't an actuarial science tag. I don't have enough rep to make tags obviously... – Josh Infiesto Sep 18 '12 at 19:47