# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] Term structure of interest rates

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In a previous post we introduced the notion of the term structure of interest rates. Here we try and understand the shape of the term structure.

There is considerable evidence that historically the shape of the term structure has exhibited the following patterns:

• More often than not the term structure has been upward sloping, i.e: $r_1 \le r_2 \le r_3 \le ... \le r_n$
• Movement in short-term and long-term rates are positively correlated (that is, they tend to move together)
• The term structure tends to be downward sloping in recessionary times (i.e. in those circumstances typically short-term rates are higher than the long-term rates)
• Corporate bond rates are higher than the sovereign rates – the difference between corporate bonds and the sovereign rates for a given maturity is called the spread
• Spread tends to widen during crises

We focus on the first three facts in this post. As always, we use a two-period set-up to try and understand the problem.

Let the one-period rate be $r_1$ and the two-period rate be $r_2$. From our discussion on forward rates we already know that embedded in these two rates is an implied one year forward rate $f_{12}$ from year one to year two such that:

$(1 + r_2)^2 = (1 + r_1)(1 + f_{12})$

Ignoring higher order terms, the above simplifies as:

\begin{aligned} 1 + 2r_2 + r^2_2 &= 1 + r_1 + f_{12} + r_1 f_{12} \\ 1 + 2r_2 &\approx 1 + r_1 + f_{12} \\ \mbox{ or } r_2 &\approx \frac{r_1 + f_{12}}{2} \end{aligned}

First thing to notice here is to see that the two-period rate $r_2$ is an average of the today’s one-period rate, $r_1$ and the one-period  ‘forward rate’, $f_{12}$. Secondly, if the term-structure has to be upward sloping, then in our two-period world the only way $r_2$ can be greater than $r_1$ is if $f_{12} > r_1$. Similarly, the term structure can be downward sloping iff $f_{12} < r_1$.

So if we can understand the relationship between $f_{12}$ and $r_1$, we would have explained the shape of the term structure.

Expectations Hypothesis

The expectations hypothesis of the term structure says that the term structure is upward or downward sloping only because people expect it to be so. That is, the reason the term structure is upward sloping, i.e. $f_{12} > r_1$ is because $f_{12}$ is just the expected value of the future one-year rate, $\mathbb{E}[r_{12}]$ and people expect that to be higher than today’s one-year rate, i.e. $f_{12} =\mathbb{E}[r_{12}] > r_1$. That is, the theory says that:

$(1 + r_2)^2 = (1 + r_1)(1 + \mathbb{E}[r_{12}])$

So one gets an upward sloping term structure if $\mathbb{E}[r_{12}] > r_1$ and vice-versa, i.e.

$\mathbb{E}[r_{12}] > r_1 \Rightarrow r_2 > r_1$

and

$\mathbb{E}[r_{12}] < r_1 \Rightarrow r_2 < r_1$

According to the expectations hypothesis, then, the forward rate $f_{12}$ captures the expected future rate $\mathbb{E}[r_{12}]$. The economic story, as we talked about in the class, is just that as people expect the rates to go up, they flock to borrow long-term right now itself and in the process demand for 2 years loans go up, causing the two year rate $r_2$ to go up.

Horizon Matters

Consider now an Investor A who wants to invest money for two years. She has two choices:

1. Buy a two-year bond
2. Buy a one-year bond and ‘roll-over’ at the end of first year till year two

If A selects option 1, then for every unit of money invested today, she will get $(1 + r_2)^2$ at the end of two years. On the other hand, if she selects option 2, then for every unit of money invested today, she can expect to get $(1 + r_1)(1 +\mathbb{E}[r_{12}])$ at the end of two years.

Now consider another Investor B who wants to invest money for one year. She also has two choices:

1. Buy a two-year bond and sell that bond at the end of one year
2. Buy a one-year bond

If B selects option 2, then for every unit of money invested today, she will get $(1 + r_1)$ at the end of one year. On the other hand, if she selects option 1, then for every unit of money invested today, at the end of year one she expects to get whatever the price of a two year bond would be at the end of year one. The expected price of a two year bond at the end of one year is just the PV of $(1 + r_2)^2$ discounted to end of year one at the expected one year rate prevailing at that time, i.e. she can expect to get $\displaystyle \frac{(1 + r_2)^2}{(1 +\mathbb{E}[r_{12}])}$ at the end of year one if she selects option 2.

If expectations hypothesis is true, that is $f_{12} = \mathbb{E}[r_{12}]$, then it essentially means that both the investors should be indifferent to the available options (why?). The upshot is that if expectations hypothesis is true, bonds of different maturities are perfect substitutes for each other – that is, time horizon of investment is irrelevant as far as investors are concerned.

(Another implication of $f_{12} =\mathbb{E}[r_{12}]$ is that if the forward rate is just the expected value of the future one-year rate, then on an average $f_{12}$ is as likely to be less than $r_1$ as it is likely to be more than $r_1$. So if it is only the expectations driving the shape of the term structure, then it is as likely to be upward sloping as it is to be downward sloping . Then, on an average over long periods of time we should expect to encounter an upward sloping term structure as often as a downward sloping term structure. But the evidence has been that the term structure has been upward sloping for better part of the last 100 years. Clearly, the expectations hypothesis cannot be the whole story.)

The question to ask, then, is are investors really indifferent to time horizons? Well, clearly not.

For investor A, option 1 is completely risk free, as the two-period rate today is known to be $r_2$. Option 2, however, for investor A comes with the risk of not knowing what the one-period rate would be at the end of one year, expectations notwithstanding. That is, there is no guarantee that the future realized $r_{12}$ will turn out to be equal to $\mathbb{E}[r_{12}]$. Similarly for investor B, option 2 seems to be less risky, as the one-period rate today is known to be $r_1$. Option 1, however, for investor B, comes with the risk of not knowing what the one-period rate would be at the end of one year.

I believe the point has been made. If the horizon matches the maturity of the investment, there is no risk in holding a bond, but if the horizon is different from the maturity of the investment, there is an interest rate risk even in holding a (default) risk-free bond. That is, a risk-free bond is free of risk only if held to maturity, otherwise if an investor plans to sell it off before maturity he/she runs the risk of taking a loss (if interest rates are high at the time of the sale).

But then, in aggregate, which is riskier – investing for one year or investing for two years?

Answering the question whether a one year bond is more risky than a two year bond is an empirical matter. Which kind of investors dominate the real world – the ones with an investment horizon of two years, or the ones with an investment horizon of one year? Like many other original insights on macroeconomics and finance, the answer to this question was also first given by John Maynard Keynes.

Liquidity Preference

Keynes argued that most people like to invest their money only for short horizons and are averse to locking-in their money for long horizons. That is, given a choice, most people prefer having a liquid cash position. And their demand for liquidity, as you will learn in your first year macro, comes from needing money for:

1. Day-to-day transactions (the transactions demand)
2. Emergency reasons (the precautionary demand)
3. Taking advantage of possible arbitrage opportunities in the future (the speculative demand)

That is, a preference for liquidity means that most people would rather lock-in their money for one year than for two years. Investing for two years means taking the risk of selling the investment off at a lower price if the money is required either for reasons of transactions or emergency or speculation. So, for long-horizon investments risk-aversion kicks in.

For risk-averse people, as we have already seen, utility of a sure payoff is more than the expected utility from the payoff, i.e. $U(\mathbb{E}[X]) > \mathbb{E}[U(X)].$ So risk-aversion implies that people would want a (liquidity) premium for taking the risk of having to sell off the two year investment at the end of one year at a loss, i.e. they would invest for two years only if there was an extra incentive for locking-in money for two years, i.e. they would want $r_2$ such that:

\begin{aligned} (1 + r_2)^2 &= (1 + r_1)(1 +\mathbb{E}[r_{12}]) + \mbox{ Liquidity Premium } (> 0) \\ \mbox{ i.e. } (1 + r_2)^2 &> (1 + r_1)(1 +\mathbb{E}[r_{12}]) \\ \Rightarrow f_{12} &>\mathbb{E}[r_{12}] \end{aligned}

That is, even if the expected one-year rate at the end of one year is the same as today’s rate, i.e. $\mathbb{E}[r_{12}] = r_1$, the sign of Liquidity Premium (> 0) implies that even then $f_{12} >\mathbb{E}[r_{12}] = r_1$, i.e. even if the rates are not expected to change at all the term-structure would be upward sloping.

In fact, even if the expected one-year rate at the end of one year is less than today’s rate, i.e. $\mathbb{E}[r_{12}] < r_1$, even then it is possible that the term structure would still be flat or even upward sloping depending on the extent of the Liquidity Premium.

A stronger consequence is that for a term structure to be downward sloping, the expected one-year rate at the end of one year has to be a lot less than today’s rate to counter the effect of Liquidity Premium. That is, the term structure being downward sloping is a very strong signal of $\mathbb{E}[r_{12}] < r_1$, i.e. recessionary expectations.

Why do we call $\mathbb{E}[r_{12}] < r_1$ recessionary expectations? Expected future interest rates lot lower than today’s imply that there will be a reduced demand for loanable funds in the future, which, in turn, points towards a slowing economy.

But is this all?

Can we say that the these two theories together explain the term structure completely? Well, of course not. But expectations and liquidity premium are two of the bigger forces in play driving the shape of the term structure.

For the most part, to the extent one can’t explain all the movements in the term structure to either expectations hypothesis or the liquidity premium, one can attribute the changes to the simple fact of demand and supply. Sometimes companies and institutions have very specific needs linked to short or long horizons, for either the reasons of working capital and/or investment requirements, and those concerns can sometimes dominate short-term movements in the term-structure.

Also, very often central banks deliberately play around with the amount of funds available in the money market to suit their policy needs, and thereby influencing the shape of the term structure. (Come to think of it, when a central bank makes a policy intervention, one of the things it is really doing is playing around with expectations.) So, yes, of course, it’s not all, but the liquidity preference theory does help put the data in perspective.