Welcome back to corporate finance. Last time we talked about some useful shortcuts to compute the present value and future value of cash flow streams that we commonly come across in practice. Streams like perpetuities and annuities, and growing perpetuities and growing annuities. This time, I want to shift gears and talk about taxes and their impact on our dollar returns. Let's get started. Hi everyone welcome back to corporate finance. Last time we talked about several useful shortcuts for computing the present value of a cash flow stream. We talked about annuities, and growing annuities, and perpetuities and growing perpetuities. And we also discussed that these cash flow streams were representative of cash flow streams we might find in practice. What I want to do now is I want to shift gears and talk about taxes. In particular, I want to talk about how taxes impact our discount rate, our cost of capital, and ultimately how they impact our dollar returns. So let's get started. I want to start with a little bit of motivation. I know we don't have a lot of time, but it's worth mentioning a few things. This is a picture of the top statutory tax rate on different sources of income. In particular, dividends which is blue line, capital gains which is the red line, and interest income which is the green line. And I don't want to make too much of this picture, but I just want to emphasize a couple of things. First, you should see that the tax rates are moving all over the place. There's a enormous amount of variation in taxes over time, number one. And number two, you can see that historically, right? Tax rates have gotten really high. Now, admittedly it's not clear to how many people those tax rates of 90, 95% actually applied, but taxes are substantial and they move around a lot over time. So it's important to understand how they impact our investment income. So lets do illustrate this by example. And I want to revisit an example we looked at in the past in particular, in our first lecture on discounting. So the question is how much do you have to save today to withdraw $100 at the end of each of the next four years if you can earn 5% per annum? Well, recall the first thing we do is we lay out a time line, and here are our cash flows, right? We're going to be withdrawing $100 a year over the next four years, and the question is asking, how much do we need today? So what we did was we discounted each cash flow, each of the $100 by the 5% discount rate to get the present value. So, for example, we took this first 100. 100 divided by 1 plus .05. And that got us $95.24 approximately. And we took the second cash flow, 100 divided by 1 plus .05 squared. That's a two. And that got us the second cash flow, and on and on. And then we could add these cash flows because they're all in the same time zero units and that produced our answer 354.60. Okay. Then when we thought about what was going on in the bank account recall, right? We would insert $354.60 into the account, it would then earn interest at 5%. We'd add that to the previous balance, then we'd pull out some money, and that would reduce our balance and we would continue that for four years. At the end of the four years, we would be left with nothing. Okay? All right. But now, consider what happens when we have some taxes. See, now we put our $354.60 into the account. That earns interest at 5%, but now we have to pay taxes on that interest income and I'm going to assume that tax rate we're facing is 35% which isn't too far off from top statutory rate currently prevailing. So that 35% of the 17.73 is 6.21 in taxes. That's going to reduce our pre-withdrawal balance to 366.12. Then we pull out $100.00, and then we're left with 266.12 at the end of the first year. Which by the way, the 266.12 is less than the 272.32 because of the taxes. Now, if we continue that process, we'll get down to zero but only because in the last year we can only pull out $83.06. That's all we have left in that last year. Okay? All right. So on net, we're $16.94 short. The taxes are reducing the funds available for withdrawl. We're running out of money early. So the lesson here is that taxes reduce the return on our investment. The dollar return on our investment. And one way to account for that is an after tax discount rate, I'll call it Rt. And that equals our usual discount rate times 1 minus the tax rate. Okay? So, for our example, the discount rate R, was 5%. And the tax rate, t, was 35%. For an after tax discount rate of 3.25%. Now let's revisit the problem using this after tax discount rate. So now when I discount all of these cash flows back to today, I'm going to use our t, which equals three and a quarter percent instead of the R which equals 5%. When we do that, when we do the arithmetic and add all these numbers up, we get 369.50. A number that's bigger than the 354.60 we originally found with the 5% discount rate. Now let's run through the exercise of looking at what happens to the saving account. So now we put in 369.50. That's going to earn interest at 5%. We're going to pay taxes on that interest at a rate of 35%. Which generates a pre-withdrawal balance in the first year of 381.51. We pull out the $100 and then we're left with 281.51. We continue that process for a few more years, and at the end of four years we're left with nothing in the bank account and we've been able to pull out $100 each year. So what's the implication? Well, we need to save more to withdraw $100 per year after taxes, right? In particular, we need to save 369.50 as opposed to the 354.60. And it's also interesting to note that the difference in how much we had to save, the difference between the 369.50 and the 354.60, which equals 14.90, that equals the present value of the taxes at 5%. Check this. Discount these cash flows at 5% and see what the present value is. It should be $14.90. Okay. Let's summarize this and bring it back together. Taxes reduce our dollar return. The after-tax return is less than the pre-tax return. And it's less by a factor of one minus the tax rate that you're facing. So we can discount by this after-tax return to see how much money we're going to have once we've swept out the effect of taxes. And given tax rates in the U.S., not to mention most of Europe and many other countries, taxes are significant and can have a huge impact on our cash flow streams, our savings behavior, and ultimately, our decision making. So looking ahead, I want you to dive into the problems, work on the problem set. And then we're going to look at inflation and investigate how inflation affects our cash flow stream and our decision making. Thank you and I look forward to seeing you in the next class.