CF 9-2A – Bond Returns – Corporate Finance

Let's take a look at problem 9-2-A. We're being asked to compute bond returns, and to compute the bond return, we've got to actually be able to compute bond prices, which is a bit of a rehash of material we've covered earlier, but that happens in courses. You do something several weeks before, and you have to recall how to do it. So let's work through it. You can see I've got formulas up here. Just to remind you, bond is we lend a company some money, they pay us typically semi-annually some interest back, and then at the end, they pay us our money back. It's a loan between two parties. And of course, because those payments are happening way in the future, or can't be way in the future, we have to do the present value of all those future interest payments as well as the payment of the money back at the end that we loan them. So because of that, there's a present value of an annuity, which is just the present value of those interest payments, and the present value of a lump sum. We will solve this first in our financial calculator, we'll complete the whole problem just using the financial calculator, then we'll solve the present value part by hand just to show you it can be done, but we'll save that for the very end of the video. Let's jump into it. One year ago, you purchased a five-year, 10% coupon bond. You paid 1020 for the bond, and it had a face value of a thousand bucks. The bond makes annual coupon payments. So most bonds we've looked at in the course make semi-annual coupon payments. This one is annual, just makes our life easy. The math is a little bit easier. Today you sell the bond at a time when the bond, our required rate of return is 9%. So first comment here, I didn't mention, but it's a thousand dollar bond. We paid 1020, we paid a premium on this bond, and we're selling it. It's a 10% bond, and we're selling it when the required rate of return is 9%. We're also going to sell it at a premium. The question is, how much are we selling it for? It never told us. Are we selling it for exactly 1020? I don't think so. Are we selling it for exactly a thousand? I don't think so. We've got to compute the price of the bond when we sell it, because we want to know, did we make any money on this bond? We bought it for 1020. Did I sell it for more than 1020, less than 1020? What'd I sell it for? We need to know the price. So we'll use our financial calculator, figure out the present value of this bond. And so I'll just punch in some numbers. We know the number of periods. Well, we purchased it one year ago. It was a five-year bond one year ago. How many interest payments are left on the bond? Four, because we would have received one of the interest payments during the year. Our IY is the discount rate, which is 9%. Our PV is what we're solving for. We don't know the PV. Our PMT, well, it's a thousand dollar bond that has 10% coupons. So it means there's a hundred dollar payment every year. Often we're doing this semi-annually, we divide by two and we get $50 payments, but our end would be higher. Anyway, we're not doing any of that. So it makes our life easy. PMT is just a hundred. And our FV is how much we're going to get paid back at the end. And we loan them 1020, but they only got to pay us back a thousand. So a thousand is the future value or face value, but future value of the bond. So let's crunch those numbers. Four goes in as N, nine is IY, a hundred is PMT, a thousand is FV. Then I compute the PV and I get 1032.4. Let's just call it 1032. Okay. So I bought this bond for 1020. It's worth 1032 today. The bond price went up by 12 bucks. That's a part of my return, right? Not only do I get the a hundred dollar coupon payment, which is part of my return, I also get $12 just by bonds. The bond went up and the bonds don't always go up, right? If interest rates in the economy go way, way up, our bond price actually goes down. But in this case, our bond price went up a little bit. How much did it go up as a percentage? Well, it went up $12 when I invested 1020, went up by about 1.2%. Our investment yield is plus 1.2%. The coupon yield, well, it's a hundred dollar return on a bond that I paid 1020 for, a hundred dollar coupon. They sent me a hundred dollar payment. So again, a hundred dollars on a thousand dollar bond, it's 10%, but a hundred dollars on a 1020 investment is going to be less than 10%. It's going to be 9.8%. So overall my return on the bond is 1.2% because it went up and 9.8% because I got a hundred dollars. My overall return on the bond is 11% and there we have it. So those are just useful calculations to know and to understand where our money's coming from, where our yield is coming from. We have solved nine to eight. I will go back and do it by hand. But if you want to drop off now, hit the thumbs up on your way out of here. If you want to stick around, hit the thumbs up now anyway, and let's get into just calculating this by hand. So I mentioned the two streams of cash flows. One is every year we get a hundred dollar payment. That's this one. So let's calculate that one. This is the a hundred dollar payment that comes in every year. A hundred times one minus one over one plus R, so one point, and the R is our required rate of return. It's our 9% discount rate, 1.09 to the power of T, just to the power of four, because there's four more payments left. We're trying to figure out what the bond price is after a year of ownership. So one over 1.09 to the four divided by 0.09. All right. Let's compute that piece. Clear, clear, clear. 1.09, Y to the X, four. 1.41. I go one over X. I get 0.708. So then one minus 0.708. It's going to be 0.29 minus one, 0.2915, divide by 0.09, 3.23. So what's in this bracket is we're 3.23 in this big square bracket. Multiply that by a hundred and we're done. Let's compute that piece. Oops. I was going to multiply by 110. Multiply by a hundred and it's three, two, four. We'll call it three, two, four. So the present value of the a hundred dollar payments, remember I'm getting $400 payments over the next four years, the present value is 320 bucks, 324 bucks. I'm also getting a thousand dollars at the end of four years. So a thousand divided by 1.09 to the power of four. What's that worth to me? 1.09 to the power of four. I go one over X times a thousand. A thousand dollars in four years is worth $708 today. All right. So $708 lump sum plus $324 worth of payments here. 708 plus 324 is 1032. And that's exactly what we said the present value of our bond was. So you can calculate this by hand. You can calculate this in a calculator, but whatever you do, hit the thumbs up button before you go. Thanks for watching. See you in the next video. Bye-bye.