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Suppose that yields to maturity on 1-year, 2-year, and 3-year zero-coupon government bonds are 1.00%, 2.00%, and 2.50%, respectively. To preserve a bit of realism, these yields are quoted on a semiannual bond basis, meaning annual percentage rates for two periods per year. The periodicity assumption is totally arbitrary on zeros, but semiannual compounding is the norm in practice. I call these rates the 0 × 1, 0 x 2, and 0×3. These bonds presumably trade in the cash market, so the first number is the starting date and the second is the ending date. The difference is the time frame, or tenor, of the bond. So, 2.50% is the 0×3 yield (usually said “zero by three”) – the yield on a 3-year zero-coupon bond starting today.

We can use this simple yield curve to infer the 1×2 and 2×3 forward rates. These are the implied yields on 1-year bonds starting one and two years into the future (i.e., forward in time). An implied forward rate (IFR) is the answer to this question: At what rate must one be able to reinvest the proceeds of an investment in a shorter-term bond to equal the proceeds of an investment in a longer-term bond? Intuitively, the 1×2 IFR has to be about 3.00% and the 2×3 about 3.50%. We need “about” here because we're neglecting compounding for the time being – that will be added later in the chapter.

The idea is that if an investor can buy a 1-year bond to yield 1.00% and reinvest for the second year at 3.00%, the total return matches that of 2.00% per year on the 2-year bond. Likewise, if an investor can buy the 2-year at 2.00% and reinvest for the third year at 3.50%, the same total return is obtained as on buying the 3-year bond at 2.50%. Another IFR that we can deduce from this yield curve is the 1×3. That is the annual rate on a 2-year bond starting in year 1 and ending in year 3. It has to be about 3.25%. Suppose the investor buys the 1-year earning 1.00% for the first year and then reinvests for the next two years at 3.25% per year. That strategy produces the same return over the three years as buying the 3-year at an annual yield of 2.50%.

These calculations, of course, can be formalized into official bond math equations. Let Rate0×A be the shorter-term rate for the 0 x A time period, Rate0×B the longer-term rate for the 0 x B period, and RateA×B the implied

forward rate between years A and B. The two time periods are AYears and BYears. Equation 5.1 captures the idea of the same total return (neglecting compounding).


The longer-term rate is a weighted average of the shorter-term rate and the implied forward rate, whereby the weights are the shares of the overall time frame.

A direct equation for the A × B IFR comes from rearranging equation 5.1.


The 1 × 2, 2 × 3, and 1×3 IFRs can be estimated by substituting into equation 5.2.

We'll see that this formula provides an excellent approximation for the more accurate result once we include compounding and the periodicity of the quoted rates.

A few words on notation are needed at this point. I really like the notation for the forward rates that I'm using, that is, the 1×2, 2×3,1×3, and so on. You can visualize the time frames easily, I think. But, alas, what I like in theory is not always used in practice. A friend who works for a major bank tells me that he often sees these forward rates written as the lyly, 2y1y, and 1y2y, respectively. The first part is the forward time period and the second is the tenor of the underlying security. An advantage to this notation is that you can mix months and years. For instance, what is the 3m5y forward rate? It's the rate on a 5-year bond (or interest rate swap), 3 months into the future. My notation would be a bit awkward; it would have to be the 0.25 x 5.25. Anyway, I use my favored notation and avoid the awkward time frames.

The averaging implicit in equations 5.1 and 5.2 suggests an analogy to textbook microeconomic theory. Remember marginal cost and average cost? Average cost is total cost divided by the quantity (usually of widgets) produced. The average cost curve is U-shaped with increasing returns to scale at first, then later with diminishing returns. Marginal cost is the incremental cost for increasing production by one unit. The marginal cost curve crosses the average cost curve at its lowest point.

The analogy is that the forward curve is like marginal cost and the yield curve is like average cost. The buyer of a 2-year bond at 2.00% obtains an incremental, or marginal, return of 3.00% for the second year after earning 1.00% for the first year. The buyer of the 3-year at 2.50% gets a marginal return of 3.50% for the third year after earning 2.00% for the first two years (or a 1-year rate of 1.00% followed by 3.00%). This leads naturally to using the implied forward rate in maturity choice decisions. The issue will be how one's own expectation for a future rate compares to that priced into the forward curve.

Consider an investor who has a known, certain 2-year horizon and can buy any of these three zero-coupon government bonds. The obvious strategy is to buy the 2-year zero at 2.00% and lock in the rate of return (barring default on the government bond and neglecting inflation). But our investor might consider “riding the yield curve” and buy the 3-year at 2.50%. The hope is that the yield at the time of sale is “low” (and the sale price is “high”). How low? Lower than 3.50%, the 2×3 IFR. The companion fear is that the yield is above 3.50%. Our investor might also consider buying the 1-year at 1.00% and hope that rates rise for the second year. How high? Higher than 3.00%, the 1×2 implied forward rate. The risk is that the reinvestment rate is less than 3.00%. The forward curve is the benchmark for your hopes and fears.

The key point is that the investor's maturity choice decision depends, in part at least, on the held view on future cash market rates vis-a-vis the implied forward rates. Other factors undoubtedly matter as well – the risk of underperforming the obvious maturity-matching strategy, the cost of reinvesting cash, the cost and risk of having to sell the bond at the horizon date. In any case, implied forward rates provide information useful in making the decision.

While our investor is choosing an investment strategy, what is the market thinking about future government bond yields? Does the market (somehow defined) expect the 1-year yield to rise from 1.00% to 3.00% and then to 3.50%? Said differently, is the implied forward curve that we can calculate from currently observed zero-coupon market rates a reasonable forecast for future rates? The classic textbook theories of the term structure of interest rates offer different answers that question.

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