# Why does the yield curve slope upwards?

In this post I give a short, but I think rather usefully direct reason for why the yield curve should slope upwards. All it requires is for you to put yourself in the shoes of an investor that has to lock up their money in a bond for a fixed amount of time (and a very simple piece of algebra).

Yes, you can find plenty of papers which give complicated economic reasons for why this should be the case, but I tend to think that it is the simple insights that explain most of what you see happening in financial markets.

As I was saying, put yourself in the shoes of an investor that has to lock up a quantity of money in the bond market for 5 years. Suppose that the options you have to chose from are:

1. buy a five-year bond, which is trading at a yield of Y5,
2. buy a ten-year bond, which is trading at a yield of Y10, which you will sell after five years.

At first sight option 2 would look attractive if Y10>Y5 and you would invest your money in the bond with the higher yield.

But of course it then occurs to you that with option 2 you also bear the risk that when you come to sell the bond in 5 years time, the general level of interest rates may have moved higher, impacting the sale price and therefore impacting the probability of getting back less than all your initial capital; differently, in option 1 you know that in year 5 the bond will definitely be worth 100 so you bear no risk to your capital (putting credit concerns aside obviously).

How much do you need option 2 to be cheapened therefore in order that it looks attractive even with this downside risk to your capital? Well, here enters the human element, but let´s suppose that you think that rates will not move higher by any more than 1% over the next 5 years. If this worst case happens then your bond will effectively lose 5% of its face value at the 5y maturity (at that point it would have a duration of about 5) .

To compensate for this you would require that option 2 pays you 5% extra over the next 5 years — meaning 1% per year.

In other words, you are saying that you will consider option 2 only if Y10=Y5+1% — the yield of the 10y bond must be 1% higher than the yield of the 5y bond.

So there you have it:

the yield curve slopes upward in order to compensate the holders of longer bonds for the risk that they lose money when they sell the bond at the end of their preferred holding period.

### Further comments: the convexity effect

If you look at the HJM modelling of a yield curve you can actually distinguish three effects that drive the shape of the yield curve:

1. expectations of future rates,
2. compensation for the risk of selling cheap, as above, which puts a discount in longer-dated bonds,
3. a convexity effect, which puts an increasing premium on longer-dated bonds.

The third component actually generates a downward, curving effect (proportional to maturity squared) in a yield curve, which explains why a ‘normal’ yield curve is upward sloping in the first section but then can actually double back and becoming downward sloping at the long end where the convexity benefit of long-dated bonds dominates the discounting for duration risk.

### Bull steepening and bear flattening

It can be shown from historical data that the yield curve generally steepens in a rally and flattens in a sell off.

If you read my post on mnemonics and trader jargon (click here) you will understand that both of these events are consistent with the following hypothesis:

Hypothesis: most of the buying and selling in the market is at the short end.

I’ll explain: if the short end rallies without much move at the long end you will see a steepening of the yield curve; and vice-versa.

When I put this together with the comments on curve steepness, we have a pair of driving actions that rationalise a lot of the activity in the market:

1. most buying and selling is in the shorter-maturity bonds (up to 10y would be my guess),
2. the steepness of the curve is driven by the risk of capital loss towards the end of a preferred shorter holding period: for example, I would suggest that 10s2s steepness is a function of activity by investors that need to lock their money up for about 2 years, buying 10s in the hope of selling at a good prices if the discount (steepness) is enough.