CF 4-3A – Interest Rate Risk – Corporate Finance

Let's run through problem 43A, another problem on interest rate risk. This time we're comparing a bond with a higher coupon rate to a bond with a lower coupon rate and seeing how those are affected by fluctuations in interest. Now here's what your intuition should say. If I have a bond where, you know, it's paying, as this one is, 4% interest and rates in the market go up by 1%, that's actually very significant for a low rate bond. A fluctuation interest is, you know, it's a big chunk of it. Where if my bond is 12%, a 1% fluctuation is not as significant. So my hypothesis here that we're going to put to the test is the lower rate coupon rate bond will be more profoundly affected by the change in interest rates. But we haven't even read the questions. Let's read the question and solve it and we'll test that hypothesis as we go. If you are, you, not if, you are considering two nearly identical bonds, bond high and bond low, both have par value of $1,000. Both have 10 years to maturity and both have a yield to maturity of 8%. The key difference between the bonds is their coupon rate. Bond high has a coupon rate of 12, bond low has a coupon rate of 4. So the first job is just to compute the price of the bonds. They're not the same. They're not $1,000. They're going to be some other amount. And in fact, we can say that bond high is, well, the yield to maturity is 8, bond high is 12. Bond high is going to be a very large premium and bond low is going to be a very significant discount because it's lower than the yield to maturity. But anyway, let's figure out the amount. So here's what I'll, I'll use the financial calculator. I'll do one of these by hand, but you know, so I have my formulas ready to go. But I'll use the financial calculator to answer the whole question. And then I'll just solve one by hand for fun. And we'll be out of here. Compute the price of each bond. Okay. The high number of periods, let's see, 10 years to maturity. So 10 years to maturity means there's 20 payments, there's 20 interest periods, and that's same for high and low. The IY is based on the yield to maturity, which is 8%, but of course this is semiannual. So we divide by two. We put 4% or just four into our calculator PV. We don't know in either case, PMT, this is the one that's going to be different between the two alternatives. Uh, 12% is what our bond high pace, $1,000 times, 12%, it's 120 bucks divided by two. Our PMT for our payment, our regular payment for bond high is $60 semiannually for bond low 4%. So again, 4% of 1,000 is 40 divided by two is 20. Our FV is $1,000. Our future value is $1,000. Okay. So in both cases, we're going to solve for PV. Let's do high first. And again, we'll use the financial calculator, clearing my memory. Uh, 20 goes in as N, four goes in as IY, PV we don't know, 60 goes in as PMT, 1,000 goes in as FV, and we compute the PV. We get 1,271.81, and this is at, uh, 8%, 1,271.81. And for low, it's going to be, uh, below a thousand. We're expecting it to be a discount. We just changed the PMT though. That's the only thing different between the two options. So I changed the PMT, I can compute PV. All the other numbers hold 728.19. Okay. So that's part one. We've got the prices of the bonds, you know, $270 discount and a $270 premium. Okay. Very exciting. Uh, if interest rates decrease by 1%, that's yields to maturity in the market decreased by 1%. So now it's 7% yield to maturity in the, for this bond. Uh, what does that look like? Well, we just changed the IY from four based on our 8% divided by two now to three and a half. So just change to 3.5% so, and it's again, the same for both. So 20, uh, let me just clear my memory and we'll do high 20 goes, Oh, what are we expecting to have happen? The interest rates in the market go down by a percent. Uh, well, our bonds that are paying the coupon rates they're paying are more attractive. Our bond prices are going to go up. That's what we're expecting to see. And well, that's what we will see. So 20 goes in as our N, IY goes in at 3.5 this time. That's the only difference. Uh, we don't know our PV, our PMT is 60. I'm doing the, uh, high bond first and a thousand goes in as FV. I compute PV, I get 1,355.31, 1,355.31. And for my low one, uh, let's see the change there. Um, so I just changed the PMT because everything else is the same. The PMT goes in at 20. Now I'm still expecting this to be a discount. It just won't be as severe of a discount. The price is going to go up. So I compute PV, I get 786.81. Okay. So let's check the swings here. Let's check the changes. There's been an increase in the price of both bonds, 1,355.31. I'll zoom right in here. Minus 1,271.81 means there's a change, an increase of $83.50 for our high bond. For our low bond, 786.81 minus 728.19, 58.62. And there's an increase of 58.62. Now, before I started, I said, I hypothesized that the low bond is going to swing more than the high bond, but the change is actually lower. What I should have said to be more precise was the percentage change is going to be higher. So let's, let's see if that's true. You divide the change by the earlier of the two years. So I'm going to divide 83 by 1,271 and 58 by 728. And we're going to get a percentage change. And I think the percentage swing is going to be bigger. I hope so. I got 58.62 in my calculator. I'll do the one on the right first. 58.62 divided by 728.19, I get 8.05% increase plus 83.5 divided by 1,271.81, 6.56% or 6.57%, I suppose. And that's an increase. So it did follow kind of what we were expecting, which is the low bond is more impacted by an interest rate change than the higher rate bond. And that's really the point of the exercise. And why would that be? And the reason is because, you know, a 1% change in the market is going to affect a bond with 4% interest, more than 12%. It should hit your intuition as being right, right? A change in something with a lower denominator is going to have a higher impact. And that's essentially what's happening here. Okay, I'll just solve one of these. I'm not going to do four present values by hand. If you made it to the here though, and you're done, great. Thank you so much for watching. And if you want to stick around, I'll do one of these PVs by hand, just to like show you that it can be done. And if you were asked to do it, you should be able to do this. So let's redo our baseline high. So we're going to shoot for this number 12, 71.81. I'll use all these baseline high figures. So PV of an annuity, C is the payment, 60 times 1 minus 1 over 1 plus R, which is 4%, 1.04 to the number of periods to the power of 20, all divided by 0.04. That's going to give us our present value of our interest flow. Now we're getting $1,000 payment at the end of this thing. So it's going to be 1,000 divided by 1.04 to the 20. 10 years from now, that's what the value of that's going to be. And we combine the two values, we'll get hopefully 12, 71.81 as our total. But let's run the numbers. 1.04 to the power of 20, it's 2.19. I go 1 over X, 0.45, 1 minus that. So minus 1, I hit the plus minus button. So 0.54 divided by 0.04, I'm at 13.59. I multiply by 60, 815.42, 815.42. Do the bottom one, 1,000 divided by 1.04 to the power of 20, 1.04 to the power of 20, 2.19, 1 over X times 1,000, 1 over X times 1,000, 456.39. So combine these two, 456.39 plus 815.42. And there's some rounding on here, but 12, 71.81. And the sun is shining, the birds are singing, our answers matched. We made it to the very end. Thanks so much for watching. Have a great day. See you in the next video. Bye-bye.