CF 4-2A – The Impact of Changing Interest Rates – Corporate Finance

Let's take a look at problem 4-2a. We're playing around with interest rates and how they affect bond prices. Remember, the price of the bond is just the present value of all of the interest payments you're going to get. Typically, every six months, you get some interest payments, so we do PV of an annuity. I'm going to do this all in my financial calculator, but I always think of it in terms of these formulas. Maybe this helps you. Maybe it doesn't, but that's how I think about it. And then at the end of the bond, you typically get your $1,000 back, or whatever it is you invested in the bond, you get the face value of the bond back, and so there's a present value of a lump sum coming back to you at the end of the period of the bond. So what we're going to do is we're going to look at a very short-term bond, well, a five-year bond, and compare it to a 20-year bond and say, hey, if there's an interest rate fluctuation, how does it impact the bond? If the interest rates in the market increase by 1%, you know, the Fed increases the interest rate, something like this, so the market rate of interest goes up by a percent, what does it do to our bond price? And the answer is the bond price goes down, right? When you can buy similar bonds at a higher interest rate, our bond becomes less attractive, less people want to buy it, and therefore our price will come down. So that's what we're expecting to see, but the question is, will it affect the five-year bond or the 20-year bond more, right? Which bond has higher interest rate risk? I'm going to answer that question for you right now. If you have a short bond, the shorter one here, which is five years, which is 10 payments, compared to a bond, the long one, which has 40 interest payments, well, the market is going to react to this and say, well, we're getting, our interest payments aren't that attractive anymore. So I can have 10 unattractive payments or I can have 40 unattractive payments. The answer is the rates of interest in the economy going up is going to hurt the longer term bond value more. Okay. Maybe I'm overthinking this or over speaking it at the start. Let's actually just solve the problem and then, you know, it'll come out in the numbers. You're considering two nearly identical bonds, bond short and bond long. Both have par values of $1,000 and both are priced at par value of $1,000. So they're both priced at $1,000. They both have the same coupon rate, 10%. And since they're both par value, the yield to maturity is 10% also because coupon rate, when it equals the yield to maturity, that's how bonds issue at par. There's no discount or premium to worry about. Both bonds pay interest semi-annually. So everything's the same about these bonds so far. The key difference is their time to maturity. Bond short is five years, bond long is 20. So we already know the price at 10%, right? When the yield to maturity is 10%, the bond prices, the PVs are $1,000 for both short and long. What about when the interest rates go up to 11%? What is the price going to be? The price of these bonds is both going to come down a little bit because again, the market rate of interest goes up. Our bonds that pay 10% interest, a coupon to 10% aren't as attractive. We're going to have to take a discount on our bonds if that were the case. Let's figure out the price. So let's do short first. Number of periods. Well, these are five-year bonds. So the number of periods or the number of payments is 10. Our IY was 10%, it's now 11%, but remember we divide by two for bonds. So 5.5%. PV is what we don't know. Our PMT, well, it's $1,000 bond times 10%. It's $100 a year, but again, remember bonds are all, well, in my class, all semiannual and that's pretty normal. The PMT is 50 and the FV is 1,000. Let's compute the price of this bond. So clear out 10 goes in as N, 5.5 goes in as my IY, 50 goes in as my PMT and 1,000 goes in as my FV. I will compute PV, I get 962.31, 962.31. Now for the bond that has 40 interest payments left, having a weaker interest rate is going to hurt the value of this bond more. In other words, a longer term bond has higher interest rate risk. Let's see what the numbers look like for our long bond. Everything's the same except for the number of payments, 40. Our IY is 5.5, same as the other, our present value, we don't know. Our payment's the same, our FV's the same. So the only thing that changes here is the N. And in fact, I think I can just put in 40 N and everything else stayed the same. So I don't have to mess with any of the other inputs here. And now I just compute PV and I get 919.77. So the question asks, if the interest rates increase by 1%, so go from 10 to 11%, what will the change be in the price of bond short and bond long? So what is the change? So 1,000 minus 962, 1,000 minus 919.77. I'll do the 919 because it's already in my calculator. So 1,000 minus that is 80.23. Our price went down by 80.23, $80.23. And in this case, our price went down 1,000 minus 962.31. Our price went down by $37.69. State your answer in both dollar and percentage change. So percentage change, you just divide the change by the earlier number, the 1,000. So 37.69 divided by 1,000 is 0.03769, which is 3.769%. And this one, and again, this is a negative, so it went down by 3.769. And this one went down by 8.023, whoops, it's a little bit messy, 8.023%. So as we speculated, the longer term bond has higher interest rate risk. And that should make sense, right? If you are going to get 40 interest period payments over a number of years, if interest rates fluctuate, that's risky for you holding the longer term bond. By the way, if interest rates in the market went down, what would you expect to happen? Well, short would start selling at a premium, long would sell at a much larger premium, because again, you're benefiting from having your locked in interest rate that is now higher than the market rate of interest. So this is something to contemplate. If you think interest rates are going down, you should be buying bonds. If you think interest rates are going up, bonds are not as good of an investment. Anyway, that's sort of random thought. I'm not an investor, I'm an accountant, but it's just my gut feeling. Okay, we've solved the problem. Congratulations. You've made it to the end. You can stop here if you want. If you want to do it by hand though, stay with me, we'll do one of these by hand. Let's do bond short at 11% by hand. Okay, so what are the details of bond short? The payment is $50 times one minus one over one plus that rate, 1.055, 5.5%, to the T, it's 10 years, all divided by 0.055, 5.5%. Okay, so that's the present value of those payments, those regular interest payments. So 1.055 to the power of 10, one over X, one minus that number, so minus one, hit the plus minus, and divide by 0.055, divide by 0.055, and I get 7.537 for what's in the brackets times 50, 376.88. Let's do the lump sum, the $1,000 we get at the end of 10 years, $1,000 divided by 1.055 to the power of 10, 1.055 to the power of 10, 1.708, and I go one over X times 1,000, 585.43, add them together, 585 plus 376, and I get 962.31, and that's exactly what we got from our financial calculator, 962.31. So you could do this both ways. It wouldn't be that complicated. The question's a bit of a jumble, but once you sort of figure out what it's talking about, the idea is longer term bonds have higher interest rate risks, all else being equal. In the real world, it's rare that all else is equal, but in a finance question, we can make all else equal. So it's just the idea that interest rate risk is higher for longer term bonds. Hope this video helped, and I hope you give me a thumbs up. Bye-bye.