Another example that we touched on last class, I told you it’s possible to switch between continuous growth models and discrete growth models. For example, if you have this equation right here, V equals A times e to the rt, you’re assuming continuous growth. You’re assuming that V is growing continuously r percent per instant. It’s growing continuously. Actually, the interest rate would be per year, but it’s growing stantaneously at that rate per year. In a discrete model, the form is A times 1 plus i raised to the t power, where i is the annually compounded interest rate. So, for any given annually compounded interest rate, you can find the continuously compounded interest rate and vice versa. For any given value of r, you can find the equivalent value of i. Suppose you want to solve for the continuously compounded interest rate that’s equivalent to a given value of i. Then, what you would do is you would take this equation here. you would want them both to have the same value of V. Both to have the same future value. Divide both sides of this by A and you get e to the rt equals 1 plus i raised to the t power. Now, take both sides of that equation and raise it to the 1 over t power. Doing that, you get e to the r equals 1 plus i. So, i, the annually compounded interest rate, is just e to the r minus 1. So, suppose – Oh, I forgot. We were doing r first, assuming you knew i. So, suppose – Oh, I forgot. We were doing r first, assuming you knew i. If you already know i, this is good enough. Only, see our goal is solve for r. So, if we know i, how are we going to solve this equation for r? [Student comment] Take the natural log of both sides. And taking the natural log of both sides, we get the natural log of e to the r equals the natural log of 1 plus i. And, one more thing, what is the natural log of e to the r? It’s just r. Because you could use rule #3 to bring this exponent down and write this as r times the natural log of e equals the natural log of 1 plus i. And the natural log of is what? 1. So, it’s just r star equals the natural log of 1 plus i. So, if I give you the value of i, you can solve for the equivalent value of r. For example, suppose the annually compounded interest rate is 0.09. What will be the equivalent continuously compounded interest rate? It would be r star. What is the natural log of 1.09? Anybody? [Student comment] 0.0862. So, in other words, there’s not a great difference between the two. As long as the interest rates aren’t too high, the difference between the two will be relatively small And it does make sense that the continuously compounded interest rate is always going to be lower than the annually compounded interest rate. Why is that? Assuming that the annually compounded interest rate is positive, why is the continuously compounded interest rate going to be always slightly lower? Why does that make sense? [Student comment] If the rates were the same but it was compounded continuously, you would end up with more money. So, the continuously compounded interest rate is always going to be slightly less than the annually compounded interest rate. Here, the annually compounded interest rate was 0.09. So, it makes sense that the continuously compounded interest rate would be slightly lower. Now, the bigger the interest rates get, the bigger the difference between the two. But if interest rates are relatively small, then the difference between the two won’t be that great. So, this means it’s possible, by using logarithms, to switch back and forth between the discrete growth models and continuous growth models. Now, let’s go the other way. that our value was 0.06. What would the equivalent i value be? We would want to solve for i. r would be known. So, in this case, it would be very easy. i star, taking this formula right here, would just be e to the r minus 1. And, if we assume that r is 0.06, then i star is just e to the 0.06 minus 1. So, what would this be? e raised to the 0.06 minus 1 is what? [Student comment] 0.0618. That’s far enough. So, again, it makes sense, doesn’t it, that the annually compounded interest rate is slightly higher than the continuously compounded interest rate. You need a slightly higher annually compounded rate to compensate for the fact that you’re only compounding once a year instead of continuously.