Spot And Forward Rates, Part 2
What you'll learn
- LOS 23.cDescribe the assumptions concerning the evolution of spot rates in relation to forward rates implicit in active bond portfolio management.
- LOS 23.dDescribe the strategy of rolling down the yield curve.
Relationships Between Spot and Forward Rates
For an upward sloping spot curve, the forward rate rises as j increases. (For a downward sloping yield curve, the forward rate declines as j increases.) For an upward sloping spot curve, the forward curve will be above the spot curve as shown in Figure 23.1.
Conversely, when the spot curve is downward sloping, the forward curve will be below it.
Figure 23.1 shows spot and forward curves as of July 2013. Because the spot yield curve is upward sloping, the forward curves lie above the spot curve.
Figure 23.1: Spot Curve and Forward Curves
From the forward rate model:
which can be expanded to:
In other words, the spot rate for a long maturity security will equal the geometric mean of the one period spot rate and a series of one year forward rates.
Forward Price Evolution
If the future spot rates actually evolve as forecasted by the forward curve, the forward price will remain unchanged. Therefore, a change in the forward price indicates that the future spot rate(s) did not conform to the forward curve. When spot rates turn out to be lower (higher) than implied by the forward curve, the forward price will increase (decrease).
A trader expecting lower future spot rates (than implied by the current forward rates) would purchase the forward contract to profit from its appreciation.
For a bond investor, the return on a bond over a one year horizon is always equal to the one year risk free rate if the spot rates evolve as predicted by today’s forward curve. If the spot curve one year from today is not the same as that predicted by today’s forward curve, the return over the one year period will differ, with the return depending on the bond’s maturity.
An active portfolio manager will try to outperform the overall bond market by predicting how the future spot rates will differ from those predicted by the current forward curve.
Jane Dash, CFA, has collected benchmark spot rates as shown here:
The expected spot rates at the end of one year are as follows:
Calculate the one year holding period return of a:
- 1 year zero coupon bond.
- 2 year zero coupon bond.
- 3 year zero coupon bond.
Solution
First, note that the expected spot rates provided just happen to be the forward rates implied by the current spot rate curve.
Recall that:
Hence:
- The price of a one year zero coupon bond given the one year spot rate of 3% is 1 / (1.03) or 0.9709.
After one year, the bond is at maturity and pays $1 regardless of the spot rates.
Hence the holding period
- The price of a two year zero coupon bond given the two year spot rate of 4%:
After one year, the bond will have one year remaining to maturity, and based on a one year expected spot rate of 5.01%, the bond’s price will be1 / (1.0501) = $0.9523
Hence, the holding period
- The price of three year zero coupon bond given the three year spot rate of 5%:
After one year, the bond will have two years remaining to maturity. Based on a two year expected spot rate of 6.01%, the bond’s price will be
Hence, the holding period
Hence, regardless of the maturity of the bond, the holding period return will be the one year spot rate if the spot rates evolve consistent with the forward curve (as it existed when the trade was initiated).
If an investor believes that future spot rates will be lower than corresponding forward rates, then she will purchase bonds (at a presumably attractive price) because the market appears to be discounting future cash flows at “too high” of a discount rate.
“Rolling Down the Yield Curve”
The most straightforward strategy for a bond investor is maturity matching, purchasing bonds that have a maturity equal to the investor’s investment horizon.
However, with an upward sloping interest rate term structure, investors seeking superior returns may pursue a strategy called riding the yield curve (also known as rolling down the yield curve). Under this strategy, an investor will purchase bonds with maturities longer than his investment horizon. In an upward sloping yield curve, shorter maturity bonds have lower yields than longer maturity bonds.
As the bond approaches maturity (i.e., rolls down the yield curve), it is valued using successively lower yields and, therefore, at successively higher prices.
If the yield curve remains unchanged over the investment horizon, riding the yield curve strategy will produce higher returns than a simple maturity matching strategy, increasing the total return of a bond portfolio. The greater the difference between the forward rate and the spot rate, and the longer the maturity of the bond, the higher the total return.
Consider Figure 23.2, which shows a hypothetical upward sloping yield curve and the price of a 3% annual pay coupon bond (as a percentage of par).
Figure 23.2: Price of a 3%, Annual Pay Bond
A bond investor with an investment horizon of five years could purchase a bond maturing in five years and earn the 3% coupon but no capital gains (the bond can be currently purchased at par and will be redeemed at par at maturity). However, assuming no change in the yield curve over the investment horizon, the investor could instead purchase a 30 year bond for $63.67, hold it for five years, and sell it for $71.81, earning an additional return beyond the 3% coupon over the same period.
In the aftermath of the financial crisis of 2007−08, central banks kept short term rates low, giving yield curves a steep upward slope. Many active managers took advantage by borrowing at short term rates and buying long maturity bonds. The risk of such a leveraged strategy is the possibility of an increase in spot rates.