**After-Tax Cost of Debt**

There is a part in the WACC formula – r (D) × (1 – t) that stands for *after-tax cost of debt*. The debt holders must earn this after-tax rate of return till the debt reaches to maturity. The ** cost of debt formula **of a company should be based on the capitulation of the related instruments. Whenever the capitulation of maturity is not available, the cost needs to be calculated using the current capitulation of the instrument.

*The after-tax cost of debt* is required for the EACC’s calculation since debt provides the essential tax shield which is ‘interest expense on debt reduces taxes.’ Therefore, if the tax rate is reduced, that is revealed in the reduction of cost of debt capital.

#### Cost of Debt Formula

After Tax cost of Debt = Kd (1-T)

Where

Kd = Rate of interest on debt

T = Tax rate

All right. Now we are going to deal with the cost of debt. And, remember, for, for the cost of debt is RD, and for what follows, we’re going to forget about taxes. We, we do not need taxes for this. We already understood the role that taxes play on the cost of debt. So, what we’re going to do, is to focus exclusively on the cost of debt. And, and we’re going to assume that, that this is a company that issues a bond. A bond is simply, a promise. You give me some money today, and I promise to pay you back over a number of years. And let’s consider a very simple example because of corporations issue bonds. Government issued bonds there, there are many issuers of bonds out there and, and all the bonds are more or less the same. The, they, so, so-called coupon bond, plain vanilla bond let us consider one in particular. As you see there let us consider a bond that is four years away from maturity. Has a face value of 1000. And has an interest rate of 10%. What does this mean? Well, being four years away from maturity means that in four years, the bond will expire. That might mean that the bond was issued today. And in four years it will expire, or that the bond was issued some time ago, and it has four years of life still to go. So it doesn’t really matter, whether it is issued today or it was issued before. The only thing that mattered is that four years from today. That bond will expire. What does it mean, that has a face value of 1000, $1000? Well, that means that is called, the face value of the, or the principal. And that means that the interest payments, that the company is going to be making over time. Are calculated, relative to that face value, relative to that principle. The company will return that amount, when the bond expires five years, four years down the road. So, $1,000 has two roles. It determines the annual payments. That investors are going to receive, and it determines, the final payment that investors are going to receive, four years in our case, four years down the road. And the interest rate and it’s a very important difference between the interest rate and the return. Those investors get and we will get to that in just a minute. But the interest rate, is that the only role, is to determine the annual payment. So if you have an interest rate of 10%, that basically means that on the four years that the bond has to go until it expires, you’re going to be paying 10% of the principle. So 10% of 1,000 is basically 100. So you’re going to be paying 100 or say 100 million, as time goes by, at once a year, for the four years of the life of the bond. So as you see there when we have a little, cash flows laid out there. So if you buy this bond which is four years away from maturity, and let’s say that it cost you $1,000, that means that you’re going to be pocketing $100 one year down the road, $100 two years down the road, $100 three years down the road, and $1,100 four years down the road, because you will get the 10% interest which is $100. Plus, the 1,000 of principle, all right? So the cash flows are very simple. You know whenever you buy a bond, you know exactly what you are going to get, and you know exactly when, you’re going to get it. Bonds are slightly more complicated, because they pay interest twice a year, typically, rather than once a year but to understand what a bond is all about this is more than enough. Now, once you issue a bond, once a company, once a government issues a bond, the bond trades in the market. And trading in the market basically means that people can buy the bond and sell the bond and buy the bond and sell the bond. And in the same way, that you’re familiar, with how stock prices change it over time. Well, bond prices also change over time. Which means, that the promise that the company has made, is not going to change. This company’s always going to be paying $100 in the next four years and $1000 at the end of the four years. What does change is the price that people are willing to pay for those promised cash flows. And why is going to pre, the price is going to change? Well, for the same reasons that we always exploring finance. If you think that risk goes up, you will be actually, willing to pay less for those cash flows, but if you think that risk goes down, and then you will be willing to pay more for those cash flows. Think; think about it in, in this way. Require returns and willingness to pay, are always go, in opposite directions, so if you perceive that a company becomes riskier, then you’re going to require more return and requiring more return and being willing to pay less for the cash flows. Are exactly the same thing, if you think that a company has become less risky than it was before, well your require return will go down, which means that we will be, you will be willing to pay more, for the cash flows that this bond promises, all right? So now let’s consider a particular example, let’s suppose that for whatever reasons they. The risk of this company has gone up. Maybe they have made bad investment decisions. Maybe they have hired a bad CEO. Whatever is the reason, the risk of this company is perceived in the market, as having gone up. Well, that means that, the cash flows of the bond, will not change, but now you will be willing to pay less, because you have more uncertainty. You have more risk and therefore, you’re willing to pay less. It’s kind of natural, you know. Let’s, let’s make a quick parenthesis here, if I promise you, to pay you $100 in each of the next four years and $1,000.00 at the end of that, and you thought that I was fully, fully reliable, then you’d be willing to pay me ache even money for those expected cash flows one, two, three, and four years down the road. But if I told you, that in each of those years, I’m going to flip a coin and if it’s heads, I’ll give you the promised cash flows but if it’s tails. I will not pay you, at that cash flow. Then you will say, but wait a minute, this is not the same proposition. Now I have a risk, 50% you pay me and 50% you don’t pay me. So now, you would be willing to pay me less, for that promise that I’m making to you. This is exactly, the same thing. That is, the more risk you perceive, then the less you had be willing to pay for those cash flows. So let me just in order, because we are going to be using these numbers, let me just say that for whatever reasons, the perceived risk of the company has gone up. Therefore, you’re willing to pay less. Instead of being, being willing to pay $1,000, which is what we said before. Let’s suppose now, you’re willing to pay $939 for those cash flows. Now, let’s go to the other side, let’s assume the opposite. And assuming the opposite is, from the Base Case, that is from when, from a situation which you’re willing to pay $1,000 let’s suppose that now the risk goes down. Moreover, the risk might go down because, the company just came up with a new product, and everybody thinks that it is going to be very profitable. Or maybe, because they’ve hire a fantastic CEO with a great reputation. For whatever reasons, everybody seems to perceive that the risk of this company has gone down. Now remember, the cash flows of the bond still do not change. But now you’d be willing to pay more. You bear less risk, so now you would be willing to pay more. And let’s just put an arbitrary number. Let us say that what you are willing to pay, is $1,000 and $66. Right so, from the base case, when risk goes up, you go from paying $1,000 to paying less, 9.30, 9.39. When risk goes down, you are going from paying $1000 to paying more, which is $1066 odd dollars. Now, let us think of this from the point of view of the investor. And let’s think of this from the point of view of taking cash out of your pocket and putting cash in your pocket. In the first case, if you pay $1000. And what you expect to get is 100, one year down the road.100, two years down the road, three 100, three years down the road. And 1100. Four years down the road. And there’s a way to calculate the return that you get by taking $1,000 out of your pocket and then pocketing 100 in the next four years and another 1,000 four years down the road. And we’ll see in a minute how we calculate that number. But, if you were to run the calculation. If you were to ask the question, if I take $1000 out of my pocket today and I receive 100, 1 year, 2 years and 3 years down the road and then, on 4 years down the road, I pocket $1100, my mean annual return would be 10%. And that is the number that you’re seeing there. So, if you take, again, $1,000.00 out of your pocket and you expect to get 100, 100, 100 and 1100 in one, two, three, and four years. Your mean annual return would be ten percent. Now remember, let us take the we look into two cases, one case in which the risk of the company went up and one case in which the risk of the company went down. Now let us think for a minute what happens when the risk goes up. We had agreed that you would be paying less for those cash flows. And let’s always compare everything to the base case. The base case, you pay $1000. And as we know now, you get a 10% return. Mean annual return. By buying this particular bond. But if you pay less for those cash flows, then you’re going to get more return. The cash flows do not change. But if instead of $1000, you pay $939. Because you pay less for exactly the same cash flows, your return is going to go up. And if you were to run a calculation and again in a minute we’ll see how to run that calculation, you’ll see that your mean annual return now goes up to 12%. Now compare. The column that starts with more risk with the column right underneath, that starts with minus 939 that number in parenthesis means minus 939. Well, what that says it brings together the two things that we said before when you perceive more risk you increase your require return. Notice that now our require return goes from 10% to 12%. And that translate into paying less for that particular bond. So, so this is the typical way that markets work. And they promise in terms of cash flows from the point of view of the company doesn’t change, what you expect to get does not change, but because risk has gone up, required return has gone up, and the bond price has gone down. Now let’s go to the other case. In the other case, and remember we always compare to the base case, and compared to the base case risk went down. Something happened to this company. They hire this fantastic CEO. For whatever reason risk went down. We said before that you are willing to pay more. You are willing to $1000.00 and 66 in order to get those same cash flows and remember the bonus cash flows actually never change but now you are paying more. Before you were paying 1000 now you are willing to pay 1066, which means that the return that you are going to get is going to be lower than the 10% that you get when you pay 1000 dollars. And if you actually were to run the numbers and once again. In a minute, we will run those numbers, and then your return would be 8%. So that means if I take $1066 out of my pocket today, and I expect to get 100 one, two, and three years down the road, and 1100. Four years down the road, my mean annual return would be 8%. So notice two important things about bonds. And, and it’s going to become important when we define Rd. When we define, the, the cost of debt. What important thing number one, when risk, the perceived risk of the company goes up, then your willingness to pay go down, and your required return goes up. More risk implies more required return, and you pay less for the cash flows. On the other hand when risk goes down then your require return goes down. And what you’re willing to pay goes up. So there’s always this negative relationship between how much you pay and the return that you get. And as it’s kind of natural to assume or to think about the more you pay the less return you get, The less return you, the, the less you pay the more return that you’re going to get. Now, how do we calculate those 10%, 12%, and 8%? Well, it’s not that complicated, but it is not that simple either. That is the general expression. C are coupons. Those are the interest payments. In our case. Those coupons are the 100, 100, 100 that the company promised to pay for the first four years until the bond expires. And P is the principal or the face value which in our case is $1,000. Now the Y that is in the denominators that is what we solve for. That is our mean annual return. So let us put some numbers in this to make sure that you understand this idea. We take 939 dollars out of our pocket. Remember this was the situation in which we perceived the risk of the companies going up. We still expect to get exactly the same cash flows 100, 100, 100, and 11,000. And the Y’s that you see in the denominator that’s what we need to solve for. Now you may be more. Unless mathematically inclined but that is not an easy thing to solve. It is not an easy thing that you can solve by hand. Excel actually solves these very quickly and there’s a technical note that I will recommend later on that you have to read that shows you how to actually calculate that Number, and were going to do it exactly as we did in sessions one and two, we’re going to do it mostly throughout these scores. That is everything that is formal, everything that is mostly a formal expression; we are going to delegate that to the technical notes. And then we’re going to do a little problem set, so that you can test whether you’re understanding the concept or not. Again, solving that equation is not that simple. You actually need either a scientific calculation, calculator or you need Excel. But if you asked Excel, if you threw those numbers into Excel and you said solve for x y, then Excel will find that 12% return. That we mentioned before. So that’s the way we calculate the return that we obtain from bonds. Now notice one important thing. The cash flows of the bond never change. So the numerator, so the expression that we have it will never change. The promise that the company makes never change. What does change is the market price. What does change is people’s willingness to pay. For those cash flows. When the company becomes riskier, they will be willing to pay less. When the company becomes less risky, they will be willing to pay more. And as that number on the left-hand side changes, the solution for those y’s will change. And so as the 939, you know, suppose now you see that risk is actually decreasing as you’re willing to pay more and more and more. The 939, then that is going to solve is going to go down and down and down. So the more you pay, the less return you are going to get and the less you pay, the more return you’re going to you’re going to get. This is, if you have never seen this before, this may not be entirely obvious. But if you think about it for a little bit it’s kind of natural that more dollars out of your pocket will simply, everything else equal, that’s very important. Everything else equal that will imply a lower return and the less dollars you take out of your pocket again, everything else equal, that will imply a higher return.