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Implied spot and forward rates need to be useful in making financial decisions to qualify as information and not just create more data. Fortunately, they really can be quite informative. An obvious application for bootstrapping the implied spot curve is to get the zero-coupon rates needed to derive the implied forward curve. That is, the implied spot curve can be just an intermediate step to get to get the information we need to help make a maturity choice decision.

Summary of Implied Forward Rate Calculations

FIGURE 5.2 Summary of Implied Forward Rate Calculations

Figure 5.2 summarizes the various paths to the implied forward rate formulas. You first have to know if you are in the money market or the bond market. The time frame for the rates is a good clue – days or months versus years. If you are in the money market, you next need to know if the rates are quoted on an add-on or on a discount basis. Then you can use equation 5.6 or 5.8 to get the comparably quoted IFR. If you are in the bond market, you need to know if the observed yields are on zero-coupon bonds or, more likely, on coupon bonds. If the latter, you first have to bootstrap the implied spot rates and then calculate the IFR using equation 5.2 or 5.4, depending on your need for precision.

Suppose you have a 4-year investment horizon and can buy any of the four government bonds in the preceding example. You are deciding whether to buy the 1-year zero and reinvest at higher yields in each future year, you hope, or to buy the 4%, 4-year bond. How high would rates have to be to prefer the rollover strategy? The answer, of course, is higher than the IFRs. But you cannot just plug the yields to maturity into equation 5.2 or 5.4. Respecting bond math protocol, you need to use the implied spot rates.

To expect a higher total return after four years by buying and rolling over

1-year bonds, you project the 1-year yield to track a path, on average, above 2.55%, 5.08%, and 6.40%.

Now suppose that you are a commercial banker working with a middle market business customer on a 12-month loan to build up working capital. The loan rate will be tied to 6-month or 12-month LIBOR. If the customer chooses 12-month LIBOR (currently 4.00%), the cost of funds is set. But if the customer chooses 6-month LIBOR (currently 3.50%), there is interest rate risk at the reset date in six months. Your customer asks for your recommendation. You know that the maturity-choice decision depends on the rate view (i.e., where 6-month LIBOR is expected to be vis-a-vis the implied forward rate). You also know that the 6 x 12 IFR is not 4.50% because of the periodicity problem. You use equation 5.6 to get a breakeven level for LIBOR of 4.42% (assuming 180 and 360 days).

The customer's decision turns on whether 6-month LIBOR is expected to be above or below 4.42% in six months. That might or might not be the rate that market participants in general are expecting. That does not matter – what matters is the decision maker's own rate view. The commercial banker here can help the process by providing some historical data, the bank economist's view on monetary policy and economic conditions, and the like. The advantage of the IFR is that as a breakeven rate it provides a framework for the above-or-below decision.

Another application of the implied spot curve is bond valuation. In the example, the 4-year actively traded benchmark bond has a coupon rate of 4% and is priced at 99.3125 to yield 4.1902% on an annually compounded street convention basis. Now suppose that we want to calculate the fair value on another 4-year bond, this one having a 9% coupon rate. We neglect taxation and assume that this bond has the same liquidity and default risk as the four benchmark securities. The key point is that in a world of no arbitrage, this bond will not be priced to yield 4.1902%. Instead, its (no-arbitrage) price is the present value of the cash flows discounted at the implied spot rates.

Based on that price, the yield-to-maturity statistic is 4.1274%.

The usual candidates to explain why the yields on any two bonds differ are time to maturity (the shape of the yield curve), credit risk (the probability of default and the assumed recovery rate), liquidity risk, and taxation. This example reveals another reason – coupon structure. We have three 4-year bonds for the same risk class: one yielding 4.2525% (the implied 0% coupon bond), another yielding 4.1902% (the benchmark 4% coupon bond), and this one yielding 4.1274% (the 9% coupon bond). The yield differences are entirely due to coupon structure. The more “weight” that is placed on the first few cash flows (i.e., the higher the coupon rate), the lower the yield to maturity. Of course, that conclusion depends on the shape of the curve.

Implied zero-coupon rates can be used in valuation problems beyond just bonds having different coupon rates. They have applications in corporate finance as well. Let's suppose now that the four benchmark bonds are corporate securities for a specific bond rating, say, single A. Better yet, we could assume that they are unsecured liabilities of the same A-rated issuer. Even though there might be no market for zero-coupon corporate bonds, we still can carry out the implied spot rate bootstrapping calculations. The idea is that if such corporate zeros did exist, they would have to trade at those spot yields if there were to be no arbitrage opportunities (and assuming no transactions costs to exploit those opportunities).

Now suppose that we need to value some project that has credit risk deemed to be equivalent to these corporate bonds. Clearly, this is a valuation problem begging for discounted cash flow (DCF) analysis. Assume first that our analyst does the obvious – this is a 4-year project, so he or she uses the 4-year yield to maturity of 4.1902% to do the discounting. I have always wondered why DCF corporate finance problems, at least in the textbooks I've seen, invariably use a single discount rate or cost of capital for all time periods. Coming from a bond market perspective, I see the usually upward slope to the yield curve instructing us to discount year-1 cash flow at a lower rate than year-2 cash flow.

Better analysis, in my opinion, is to use the implied spot rates (i.e., 3.0264%, 2.7903%, 3.5476%, and 4.2525%) instead of the yield on the maturity-matching coupon bond. These implied spot rates correspond to the timing of the specific cash flows, which are no different than the face values on four zero-coupon bonds. Depending on the amount and timing of the projected cash flows, this DCF calculation could produce a higher valuation for the project. All we have done to get the more appropriate discount rates is reconfigure data from the underlying yield curve and add the assumption of no arbitrage.

To be fair, the source of error in most corporate finance valuation problems is not the lack of precision in the discount rates in the denominators. Instead, the difficulty is uncertainty regarding the expected cash flows in the numerators. Often those numbers are just best-guess projections of future sales and operating costs. Then our “laser-sharp” implied spot rates represent bond math overkill. In those circumstances, we should use appropriate technology and discount all cash flows at some reasonable rate. However, there are situations when the numerators are scheduled amounts on scheduled dates – for instance, with financial contracts such as lease agreements or interest rate swaps. Then the implied spots could produce a better valuation.

For hedge funds and proprietary trading desks at financial institutions, the implied spot curve is used to identify arbitrage opportunities. The two strategies we first saw in Chapter 2 are: (1) coupon stripping – buy the coupon bond and sell the cash flows separately as zero-coupon coupon bonds (e.g., TIGRS, CATS, LIONS back in the early 1980s and Treasury STRIPS since 1985), and (2) bond reconstitution – buy the zero-coupon bonds in sufficient quantity to build and sell a coupon bond. The implied spot curve identifies when one of these arbitrage strategies might be profitable.

By design, the implied spot curve is the sequence of zero-coupon rates such that those trading two strategies break even. If actual zero-coupon bonds are trading at yields above those implied rates, a bond reconstitution strategy might work. Actual zeros would have “high yields” and therefore “low prices” relative to breakeven. The arbitrageur buys the zeros, builds the coupon bond, and sells it for a profit. To be sure, the profit would have to cover transactions costs. Those could be included directly, but then we would not have an implied spot curve; instead we would have an implied spot cone. An arbitrage opportunity arises when actual zeros are trading outside the cone. If actual zero-coupon bonds can be sold at yields below the implied spot curve (or below the cone), the arbitrageur undertakes the coupon-stripping strategy, as Merrill Lynch did in the 1980s when it created TIGRS.

A hugely important application for implied spot and forward rates is in pricing interest rate derivatives. We see this in detail in Chapter 8 on interest rate swaps. The idea is that the forward curve is a sequence of “hedge-able” future rates. They indicate the rates that can be locked in using derivatives. Once again the caveats are that there are no arbitrage opportunities and transactions costs are small enough to be neglected. Those are fine and acceptable assumptions in normal financial market conditions and active trading in derivatives. However, during a financial crisis when liquidity dries up, some arbitrage opportunities can lie there unexploited. Sad but true.

One more bond math application of the implied forward curve is worth some attention before moving on to duration and convexity. A horizon yield on a coupon bond, as we know from Chapter 3, depends critically on coupon reinvestment rates. Remember that one of the assumptions in using the yield to maturity as a measure of an investor's total return is that all coupons are reinvested at that same yield. Obviously, future rates are random. But suppose we can hedge that interest rate risk using costless and riskless derivatives and can lock in rates along the implied forward curve.

Consider again the 4-year, 4% annual payment government bond priced at 99.3125 to yield 4.1902%. We calculated the IFRs above: The 1 x 2 is 2.5547%, the 2 x 3 is 5.0790%, and the 3 x 4 is 6.3961%. What will be the 4-year horizon yield (i.e., the total pretax return after four years assuming no default) if coupons are in fact reinvested at those specific rates? Are you guessing 4.1902%? Higher? Lower?

The total return turns out to be 117.314087 (percent of par value).

In the first term in brackets, the initial coupon is rolled over at the 1×2, then the 2×3 and 3×4 IFRs. The second coupon is reinvested at the 2×3 and 3×4, while the third is rolled over only once at the 3×4 IFR. The last coupon is received at maturity along with the principal.

The 4-year horizon yield, or holding-period rate of return, is the solution for HPR, the annual rate connecting the purchase price to the total return.

Notice that 4.2525% is the 0×4 implied spot rate. Derivatives (in particular, costless and riskless derivatives) lock in for the investor the 4-year spot rate, not the 4.1902% yield to maturity on the 4-year bond.

That is not a coincidence. Suppose that the bond buyer has only a 3- year investment horizon. What is the 3-year horizon yield assuming coupon reinvestment at the 1×2 and 2×3 IFRs and the ability to lock in using derivatives the sale price of the bond at the end of the third year at the 3×4 IFR? The total return after three years is 110.261642 (percent of par value).

When the bond is sold, the proceeds are just the final coupon and principal discounted at the 3×4 IFR. The 3-year horizon yield is 3.5476%, which – no surprise now – is the 0×3 implied spot rate.

These examples demonstrate the interconnectedness between the underlying yield curve on traded coupon bonds, the implied spot, and implied forward rates. The connection is the assumption of no arbitrage.

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