Forward Rates

Investment Theory

Lorenzo Naranjo

Fall 2024

Computing Forward Rates

Synthesizing Forward Rates

• The term-structure of interest rates allow investors to lock future interest rates using forward-rates that can be synthesized from zero-coupon rates.
• Denote by Z(n) the price of a zero-coupon bond with face value equal to $1 and expiring at year n.
• Consider now two zero-coupon bonds with expirations m < n, that we will denote by M and N, respectively.
• If we sell \frac{Z(n)}{Z(m)} units of bond M and invest the proceeds in bond N.
  • The cost of buying \frac{Z(n)}{Z(m)} units of M equals the units you buy times the cost per unit, i.e., \frac{Z(n)}{Z(m)} \times Z(m) = Z(n).
  • Therefore, by selling \frac{Z(n)}{Z(m)} units of M you get the amount needed to buy 1 unit of bond N.

Cash Flows from the Strategy

• The table below shows that this strategy creates a forward-starting zero-coupon bond.

  Year	0	m	n
  Sell \frac{Z(n)}{Z(m)} units of M	Z(n)	-\frac{Z(n)}{Z(m)}	0
  Buy 1 unit of N	-Z(n)	0	1
  Total	0	-\frac{Z(n)}{Z(m)}	1

• Instead of investing a certain amount today at a known interest rate, and therefore guaranteeing a known amount at maturity, this newly created security guarantees today that if you invest a certain amount in year m, you will earn a known interest rate in year n.

The Forward Rate

• The implicit annualized interest rate f(m, n) that applies to this newly created security is called the forward-rate from year m to n, which must satisfy \frac{1}{(1 + f(m, n))^{(n - m)}} = \frac{Z(n)}{Z(m)}, so that f(m, n) = \left( \frac{Z(m)}{Z(n)} \right)^{\!\frac{1}{n - m}} - 1. \tag{1}

Example 1 (Computing Forward Rates) Given

Bond	Maturity (years)	Price ($)
Z_{1}	1	920
Z_{2}	2	840
Z_{3}	3	760
Z_{4}	4	710

\begin{aligned} f(2, 4) & = \left(\frac{840}{710}\right)^{\!1/2} - 1 = 8.77\%, \\ f(1, 4) & = \left(\frac{920}{710}\right)^{\!1/3} - 1 = 9.02\%, \\ f(3, 4) & = \frac{760}{710} - 1 = 7.04\%. \end{aligned}

Forward and Zero-Coupon Rates

• Sometimes, the forward rate is expressed as a function of the zero-coupon rates implicit in the zero-coupon bond prices.
• If we denote by r_{m} and r_{n} the zero-coupon rates corresponding to maturities m and n, respectively, 1 can be re-written as f(m, n) = \left( \frac{(1 + r(n))^{n}}{(1 + r(m))^{m}} \right)^{\!\frac{1}{n - m}} - 1. \tag{2}

Intuition

• Note that if we could fix the forward rate f(m, n) beforehand, then it must be the case that investing from 0 to m at r(m), and then rolling the deposit from m to n at f(m, n), should be equivalent to investing all the way from 0 to n at r(n).
  • If not, there would be a simple arbitrage opportunity where we could invest using the strategy that delivers the most and borrow using the other alternative.
• Therefore, it must be the case that: (1 + r(m))^{m} (1 + f(m, n))^{n - m} = (1 + r(n))^{n}, which also delivers (2).

Example 2 (One-Year Forward Rates) The table below shows the zero-coupon rate per year compounded annually for different maturities. The forward rate that applies from the previous to the current year is computed in the last column.

Year	Zero Rate (%)	Forward Rate (%)
1	4.0
2	5.0	6.01
3	5.6	6.81
4	6.0	7.21
5	6.3	7.51

In the table, the forward rate that applies from year 1 to 2 is computed as f(1, 2) = \frac{1.05^{2}}{1.04} - 1 = 6.01\%, whereas the forward rate that applies from year 4 to 5 is given by f(4, 5) = \frac{1.063^{5}}{1.06^{4}} - 1 = 7.51\%.

Theories of the Term-Structure of Interest Rates

The Term Structure of Interest Rates

• We call the collection of YTM of zero-coupon bonds the term structure of zero-coupon bond yields or the term structure of zero rates.
• The yield curve usually refers to a similar concept which is the par yields of coupon bonds.
• It is possible to derive the term structure of zero rates from par yields.
• Formally, the term structure of interest rates is a function y(n) that determines the zero rate for a specific maturity n.
• In these notes, we will express these rates per year with annual compounding.

Different Shapes

• The term structure of zero rates can take different shapes as a function of the time-to-maturity:
  • upward sloping (most typical),
  • downward sloping,
  • flat,
  • and hump shaped.

The Expectations Hypothesis

• There have been many theories proposed to explain the shape of the term structure of zero rates, and among them the most popular one is the expectations hypothesis (EH).
• In a world populated by risk-neutral investors, the expected (holding period) return of investing in any asset should be the risk-free zero-rate that applies to that period.

The Holding Period Return

• Denote by B the price of a bond paying annual coupons C and expiring at time T.
• If the expectations hypothesis holds, the expected HPR of buying this bond at time 0 for B_{0} and selling this bond next year for B_{1} must be r_{0}(1), the zero rate at time 0 that applies to cash flows in year 1, \operatorname{E}(R_{1}) = \frac{\operatorname{E}(B_{1}) + C}{B_{0}} - 1 = r_{0}(1).
• Note that B_{1} is unknown today since we do not know how interest rates will evolve over time.

Example 3 (Computing an Expected Bond Price) Suppose you have the following information on zero-coupon rates.

Maturity (years)	Rate (%)
1	5.0%
2	5.5%
3	6.0%
4	6.3%
5	6.5%

Consider a 4% annual-paying coupon bond over a notional of $1,000 expiring in 5 years. The price of this bond today can be computed by discounting its cash flows at the appropriate zero-rate. B_{0} = \frac{40}{1.05} + \frac{40}{1.055^{2}} + \frac{40}{1.06^{3}} + \frac{40}{1.063^{4}} + \frac{1040}{1.065^{5}} = 898.02

If the EH holds, the price of this bond next year, B_{1}, should be such that \frac{B_{1} + C}{B_{0}} = 1 + r_{0}(1), implying B_{1} = 898.02 \times 1.05 - 40 = \$902.92.

Expected Future Discount Rates

• Our version of the EH is consistent with assuming that expected future discount rates can be computed using forward rates, i.e., \operatorname{E}(Z_{t}(n)) = \operatorname{E}\left(\frac{1}{(1 + r_{t}(n))^{n}}\right) = \frac{1}{(1 + f(t, t+n))^{n}}. \tag{3}
• We can use (3) to derive an expression that relates expected future zero-rates and forward rates.
• The first thing to notice is that Jensen’s inequality implies that \operatorname{E}\left(\frac{1}{(1 + r_{t}(n))^{n}}\right) > \frac{1}{(1 + \operatorname{E}(r_{t}(n)))^{n}}, which combined with 3 delivers \operatorname{E}(r_{t}(n)) > f(t, t + n).

An Approximation

• Therefore, our version of the EH implies that forward rates are a downward biased estimate of future zero-rates.
• The difference between the two, however, is in general very small.
• To see why, consider the function h(x) = \frac{1}{(1 + x)^n}.
• The second order Taylor expansion of h(x) around x = \mu is: \frac{1}{(1 + x)^n} \approx \frac{1}{(1 + \mu)^n} - n \frac{(x - \mu)}{(1 + \mu)^{n + 1}} + \frac{1}{2} n (n + 1) \frac{(x - \mu)^{2}}{(1 + \mu)^{n + 2}}. \tag{4}

The Approximation is Correct

• Consider now a random variable X such that \operatorname{E}(X) = \mu and \operatorname{V}(X) = \sigma^{2}.
• Taking expectations on both sides of (4) shows that \operatorname{E}\left(\frac{1}{(1 + X)^n}\right) \approx \frac{1}{(1 + \mu)^n} + \frac{1}{2} n (n + 1) \frac{\sigma^{2}}{(1 + \mu)^{n + 2}}. \tag{5}
• Denoting f = f(t, t + n), \mu = \operatorname{E}(r_{t}(n)) and \sigma^{2} = \operatorname{V}(r_{t}(n)), we can combine (3) and (5) to obtain \frac{1 + \mu}{1 + f} \approx \left(1 + \frac{1}{2} n (n + 1) \left(\frac{\sigma}{1 + \mu}\right)^{\!2} \right)^{\!1/n} \approx 1.
• Thus, if the EH holds, forward rates are good forecasts of expected future zero-rates, i.e., \operatorname{E}(r_{t}(n)) \approx f(t, t+n).

Implications of EH for the Shape of the Term Structure

• To understand the implications of the EH for the shape of the term structure of zero rates, consider the evolution of the one-year rate r_{n}(1) over time.
• If the EH holds, (2) implies that \small 1 + \operatorname{E}(r_{n}(1)) \approx \frac{(1 + r_{0}(n+1))^{n+1}}{(1 + r_{0}(n))^{n}} = (1 + r_{0}(1)) \left(\frac{1 + r_{0}(n+1)}{1 + r_{0}(1)}\right) \left(\frac{1 + r_{0}(n+1)}{r_{0}(n)}\right)^{n}
  • If the term structure is upward sloping, r_{0}(1) < r_{0}(n) < r_{0}(n+1), one-year zero-rates are expected to increase until the term structure is flat, and vice-versa.
  • Therefore, the EH implies that the term structure of zero rates should be flat on average.

The Liquidity Preference Theory

• Empirically, we observe that the term-structure of interest rates is most of the time upward sloping.
• Thus, the EH does not seem to explain the term-structure of interest rates well in practice.
• One potential pitfall of the EH is that it basically assumes that all zero coupon bonds are perfect substitutes, independent of their maturity.

Risk-Aversion

• When investors are risk averse, they care not only about the expected short rate but also about its volatility.
• Investors in long-term bonds want to be compensated for committing their funds for a long time since they face price-risk uncertainty if they need to sell before maturity.
• Conversely, issuers of bonds are willing to pay a higher interest rate on long-term bonds because they can lock in an interest rate for many years.
• Thus, the liquidity preference theory implies that: r_{0}(2) > \frac{r_{0}(1) + \operatorname{E}(r_{1}(2))}{2}.

Segmented Markets Theory

• This theory is also known as the preferred habitat theory.
• According to this theory, some investors only trade short-term bonds implying that short-term interest rates are solely determined by supply and demand among these investors.
• Other investors, like insurance companies, only trade long-term bonds, implying that long-term interest rates are determined by supply and demand among these investors.
• This view may explain why 30-year rates are typically lower than 20-year rates.