# Bonds: Prices, Yields and Confusion- a Visual Guide

Bonds have been the talk of the financial world lately. One minute it’s a thirty-year bull market, the next it’s a bondcano. Prices are up, yields are down and that’s bad. But then in the last couple of months, prices are down and yields are up and that’s bad too, apparently. I’m going to take some of the confusion out of these relationships and give you a visual guide to what’s been going on in the bond world.

The mathematical relationship between bond prices and yields can be a little complicated and I know very few people who think their lives would be improved by more algebra in it. So for our purposes, the fundamental relationship is that bond prices and yields move in opposite directions. If one is going up, the other is going down. But it’s not a simple 1:1 relationship and there are a few other factors at play.

There are several different types of bond yields that can be calculated:

• Yield to maturity: the yield you would get if you hold the bond until it matures.
• Yield to call: the yield you would get if you hold the bond until its call date.
• Yield to worst: the worst outcome on a bond, whether it is called or held to maturity.
• Running yield: this is roughly the yield you would get from holding the bond for a year.

We are going to focus on yield to maturity here, but a good overview of yields generally can be found at FIIG. Another good overview is here.

To explain all this (without algebra), I’ve created two simulations. These show the approximate yield to maturity against the time to maturity, coupon rate and the price paid for the bond. For the purposes of this exercise, I’m assuming that our example bonds have a face value of \$100 and a single annual payment.

The first visual shows what happens as we change the price we pay for the bond. When we buy a bond below face value (at, say \$50 when its face value is \$100), yield is higher. But if we buy that bond at \$150, then yield is much lower. As price increases, yield decreases.

The time the bond has until maturity matters a lot here, though. If there is only a short time to maturity then the differences between below/above face value can be very large. If there are decades to maturity, then these differences tend to be much smaller. The shading of the blue dots represent the coupon rate that might be attached to a bond like this- the darkest colours will have the highest coupon rate and the lighter colour will have the lowest coupon rates. Again, the differences matter more when there is less time for a bond to mature.

The second animation is a representation of what happens as we change the coupon rate (e.g. the interest rate the debtor is paying to the bond holder). The lines of dots represent differences in the price paid for the bond. The lighter colours represent a cheaper purchase below face value (better yields- great!). The darker colours represent an expensive purchase above face value (lower yields-not so great).

If we buy a bond cheaply, then the yield may be higher than the coupon rate. If we buy it over the face value, then the yield may be lower than the coupon rate. The difference between them is less the longer the bond has to mature. When the bond is very close to maturity those differences can be quite large.

When discussing bonds, we often mention something called the yield curve and this describes the yield a bond (or group of bonds) will generate over their life time.

If you’d like to have a go at manipulating the coupon rate and the price to manipulate an approximate yield curve, you can check out this interactive I built here.

Remember that all of these interactives and animations are approximate, if you want to calculate yield to maturity exactly, you can use an online calculator like the one here.

So how does this match the real data that gets reported on daily? Our last chart shows the data from the US Treasury 10-year bills that were sold on the 25th of November 2016. The black observations are bonds maturing within a year, the blue are those that have longer to run.  Here I’ve charted the “Asked Yield”, which is the yield a buyer would receive if the seller sold their bond at the price they were asking. Sometimes, however, the bond is bought at a lower bid, so the actual yield would be a little higher. I’ve plotted this against the time until the bond matures. We can see that the actual yield curve produced is pretty similar to our example charts.

This was the yield curve from one day. The shape of the yield curve will change on a day-to-day basis depending on the prevailing market conditions (e.g. prices). It will also change more slowly over time as the Federal Reserve issues bonds with higher or lower coupon rates, depending on economic conditions.

Data: Wall Street Journal.

Bond yields and pricing can be confusing, but hopefully as you’re reading the financial pages they’re a lot less so now.

A huge thanks to my colleague, Dr Henry Leung at the University of Sydney for making some fantastic suggestions on this piece.