Prices versus Interest Rates

Pay attention to the slide, I have written the generic formula, but with numbers in it. I shouldn't call it generic, with numbers in it of the pricing of a coupon paying bond. Now if I were to make it zero coupon, what would I do? I just remove the PMT, that 3,000 make it 0. What I'm going to do is something different. I want you to understand the relationship between the price of a bond and the discount rate or the interest rate. This is not coupon, I'm talking about coupon as in dollars, I'm talking about the rate of return. So if you stare up, it started off with an interest rate of 4%.

What I would like you to do with me is change.

This interest rate from 1% to 8%. What will this tell you? It will show you the sensitivity of the price of the bond as you change the interest rate. Before we even do that, this is one of the ones you'll read over and over in the press, if interest rates go up, bond prices have to go down. And, therefore, the relationship between interest rates and bond prices is one of the given of life are inverse. But more interesting is when you do this calculation, so let's do it. You do it on Excel, I'll write out the numbers, and hopefully all you're doing is changing this number, making it 0.01, return. 0.02, return. 0.03, return. You'll find something like this.

At 1%, you'll find a price of 111,168.93. At 2%, what will happen? The price of the bond will drop. 105 627.54. Okay, at 3%, what will you find? This is very interesting. Before even doing it, what should you find? You should find face value, should be 100,000. In other words, the bond should be selling at par, that's the terminology. Why, because think about it, if your coupon rate is 3% and your interest rate happens to be 3%, the denominator and numerator with compounding kinda cancel each other out and you're left with just face value. It's very intuitive. At 4%, how much? It'll be 94,712. Sorry, 742.24. Or actually, back up. 712.33. I jumped ahead to 5%.

So, last one to reconnect with you is 94,712.33. Please keep doing these, what it will do is it will give you a familiarity with doing present value. But also what you'll see is, I challenge you to now figure out why did I make you do it? And I made you do it because if you look at the relationship between price and interest rates, I'm going to draw this, you'll find a downward sloping but what is called a convex curve. It's not straight. So the percentage change is happening. Right? Or different. If it were a straight line, a change in interest rate would lead to the same percentage change in price. But that's not the case.

So what you'll find is, it's called convexity. You have a downward sloping curve, but the change in price is relative to change in 1% interest rate, as where this is P and this is R as marked straight line, right. It's not the same, okay. Enjoy this, and this is also something that you'll read and depress all the time. And you'll be able to figure out why is it that a longer term barn versus a shorter term barn behaves differently and so on and so forth. So, we'll take a little break here. Think about this. We'll come back and talk some more.