BKM Chapter 15 - The Term Structure of Interest Rates

Question 1

Short Rate vs. Spot Rate

Answer 1

Assume for a moment that we knew precisely what the 1-year interest rates were going to be one year from now, two years from now, etc. These various rates, called the short rates, represent the interest rate for a specified time interval (such as one year). These rates for different points in time (today, one year from now, two years from now, etc.) are denoted as r₁, r_2, r₃, etc., where the subscript can be interpreted as the maturity year for a bond that was issued one year prior.

The PV of $1,000 to be paid in two years would be discounted at the current one year rate and also at the one year short rate one year from now, or PV = $1,000/(1+r₁)(1+r₂).

Spot rate (y_i) represents the yield today on a zero coupon bond with the specified maturity.

It is the collection of all of these spot rates at different maturities that gives us the pure yield curve described above (though technically we should be careful and refer to it as the term structure to avoid confusion with the on-the-run yield curve).

$1,000/(1+r₁)(1+r₂) = $1,000/(1+y₂)²

Question 2

Spot vs. Forward Rates

Answer 2

Let’s continue to assume that the future short rates are known with certainty and we wanted to invest $1,000 for three years. We could do this in two different ways:

1. Invest in a three year zero coupon bond at the current three year spot rate. This will give us a value of: $1,000„(1 + y₃)…³
2. Invest in a two-year zero coupon bond at the two-year spot rate and then plan to reinvest the proceeds at the one-year rate two years from now (the future short rate). This will give us proceeds equal to $10,00(„1 ‚ +y₂)…² in two years which will then be reinvested at the future short rate to obtain a total payment of: $1,000„(1 + y₂)…² (1+r₃)

We can use the zero coupon spot rates y_i to solve for the future short rates r_i. (1+y₃)³ = (1+y₂)²(1+r₃)

Since the future short rates cannot really be known with certainty, to be careful we will refer to the future short rate calculated in this fashion as the forward rate, and rewrite this as: (1+y₃)³ = (1+y₂)²(1+f₃)

Question 3

Interest Rate Uncertainty and Forward Rates

Answer 3

In the case where investors wanted to invest long term E„(r₂)… > f₂ but in the case where investors wanted to invest short term E„(r₂)… < f₂. What this shows you is that the relationship between the forward rate and the expected spot rate depends on the relative preferences for long term and short term bonds and investors’ willingness to assume interest rate risk.

Question 4

Term Structure

Answer 4

This relationship between the various spot rates is known as the Term Structure. If spot rates are higher for longer time periods, then we say the term structure slopes upward, if the rates are all the same we say it is flat, and if the rates are higher for earlier time periods we say it is downward sloping.

Question 5

Explain Expectation Hypothesis

Answer 5

The forward rate equals the market consensus expectation of the future short interest rate.; that is , f2= E(r2) and liquidity premiums are zero.

This means that we can express the relationship between the spot rates as: (1+r₁) [1 + E(r₂)] = (1+y₂)²

Therefore, if the two year rate (y₂) is higher than the one year rate (r₁=y₁), meaning if the term structure is upward sloping, then it must be because we expect the short rate to rise. Similarly, if the term structure is downward sloping, it must be because we expect the short rate to fall.

Question 6

Explain Liquidity Preference

Answer 6

Short-term investors will be unwilling to hold long-term bonds unless f₂>E(r₂), whereas long-term investors will be unwilling to hold shor bonds unless E(r₂)>f₂. => Both groups of investors require a premium to hold bonds with maturities different from their investment horizons. Advocates of the liquidity preference theory believe that short-term investors dominate the market so that the forward rate will generally exceed the expected short rate. The excess of f₂ over E(r₂), the liquidity premium, is predicted to be positive.

Question 7

Market Segmentation Theory

Answer 7

There need be no relationship between short-, median, and long-term interest rates. Under the theory, a major investor invests in bonds of a certain maturity and does not readily switch from one to another. The yields in these different markets are determined by supply and demand in each of these markets.

Question 8

What are the two reasons that the forward rate could be high/increasing?

Answer 8

1. Investors expect rising interest rates, meaning that E(r_n) is high/increasing
2. Investors require a large premium for holding longer-term bonds.

Question 9

Forward Rates as Forward Contracts