 From the nominal rate i, we can calculate the effective annual interest rate under continuous compounding. What if we had our accumulation function a of t equal e to something and wanted to find the nominal rate? So we note that after one year, $1 would grow to e to power delta. Since this amount will be $1 plus the interest i on $1 we have, and we can solve this for delta and so we'd have delta equal log of 1 plus i. This value delta is known as the force of interest and the way to think about it is the force of interest delta is the effective rate you'd need to get a nominal rate of i compounded continuously. For example, let's find the force of interest delta that would produce a nominal rate of 5% annually and then let's interpret the results. But remember, don't memorize formulas, understand concepts. The force of interest delta is the effective rate you'd need to get a nominal rate of i compounded continuously. In other words, we want to find delta where our interest is e to power delta. So we know the interest rate will hit both sides with a log and we find delta is approximately, so what this means is that 5% annual interest compounded once per year gives the same yield as about 4.88% annual interest compounded continuously. So remember, the force of interest delta is the rate we'd need under continuous compounding to have effective rate i. Now if our accumulation function looks like 1 plus i to the t then delta is log 1 plus i. But what if a of t has some other form? Again, we'll need a little more calculus. So first of all, interest can be viewed as the rate of change of the amount, but this is just the derivative a prime of t. Now the interest rate is the ratio of this amount to the present amount a of t and so this means that the force of interest delta t at time t is this ratio a prime t divided by a of t. So let's find the force of interest at time t if a of t is e to the power delta t. So we find the derivative, we find the quotient and simplify, and we find the force of interest is delta which is what it should be. Because our accumulation function is what we would get if we had interest rate delta compounded continuously. What about simple interest? So simple interest at rate s has accumulation function a of t equal 1 plus s t. So our derivative will be, and so delta t will be, now notice that as t goes to infinity this force of interest tends to zero and one way we might look at this is that our force of interest gets weaker which reflects the reality that the effective rate does decrease over time. Or let's take an accumulation function like this, let's find the force of interest. So we could find the force of interest as a ratio between the derivative and the function, and differentiating we have, yeah, while we could find the derivative it would be painful and tedious. So let's see if we can simplify this problem. Recall that if f of x was a product or quotient of powers or roots you could simplify the problem of finding the derivative by hitting everything with a log. Suppose a of t is our accumulation function. The derivative of log of a of t will be, which is our force of interest. This gives a very useful result. Let a of t be our accumulation function, the force of interest is the derivative of the log of a of t. So returning to our problem we can find the derivative of log of a of t. So the derivative of log of a of t, well, we'll apply the chain rule, that's one over times the derivative of a of t, which isn't very helpful. The thing to remember here is that hitting something with a log only works if it actually breaks the thing apart. So let's try that again, we have the derivative of the log, but we can use our rules of logs to simplify. This is a product, so we can write it as a sum. We have a power, and we can remove the power to the front. And so now we have a much simpler expression to differentiate, and we find, given the accumulation function a of t, we can find the force of interest as the derivative of the log of the accumulation function, but if we treat the force of interest as a function itself, then given the force of interest we can find the accumulation function. So if our force of interest is the derivative of the log of the accumulation function, then integrating the fundamental theorem of calculus gives us, but remember that the accumulation function at zero is one, and the log of one is zero. And so this will just be the log of the accumulation function. And so the accumulation function itself will be e to the power of the integral. So suppose we want our force of interest to have this form, let's find the accumulation function and the effective interest rate during the first year. Remember the force of interest is the derivative of the log, and so the log will be the integral. So the accumulation function itself will be e raised to this power, which we can simplify. And we note that after t equals one year, a dollar will grow to two dollars forty. This gives an effective interest rate of one hundred forty percent.