SECURITIES
IV. THE ARBITRAGE-FREE VALUATION APPROACH
The fundamental flaw of the traditional approach is that it views each security as the same package of cash flows. For example, consider a 10-year U.S. Treasury issue with an 8% coupon rate. The cash flows per $100 of par value would be 19 payments of $4 every six months and
$104 twenty 6-month periods from now. The traditional practice would discount each cash flow using the same discount rate.
The proper way to view the 10-year 8% coupon Treasury issue is as a package of zero-coupon bonds whose maturity value is equal to the amount of the cash flow and whose maturity date is equal to each cash flow’s payment date. Thus, the 10-year 8% coupon Treasury issue should be viewed as 20 zero-coupon bonds. The reason this is the proper way to value a security is that it does not allow arbitrage profit by taking apart or ‘‘stripping’’ a security and selling off the stripped securities at a higher aggregate value than it would cost to purchase the security in the market. We’ll illustrate this later. We refer to this approach to valuation as thearbitrage-free valuation approach.6
6In its simple form, arbitrage is the simultaneous buying and selling of an asset at two different prices in two different markets. The arbitrageur profits without risk by buying cheap in one market and simultaneously selling at the higher price in the other market. Such opportunities for arbitrage are rare.
Less obvious arbitrage opportunities exist in situations where a package of assets can produce a payoff (expected return) identical to an asset that is priced differently. This arbitrage relies on a fundamental principle of finance called the ‘‘law of one price’’ which states that a given asset must have the same price
Chapter 5 Introduction to the Valuation of Debt Securities 111
EXHIBIT 4 Comparison of Traditional Approach and Arbitrage-Free Approach in Valuing a Treasury Security
Each period is six months
Discount (base interest) rate Cash flows for∗ Period Traditional approach Arbitrage-free approach 12% 8% 0%
1 10-year Treasury rate 1-period Treasury spot rate $6 $4 $0 2 10-year Treasury rate 2-period Treasury spot rate 6 4 0 3 10-year Treasury rate 3-period Treasury spot rate 6 4 0 4 10-year Treasury rate 4-period Treasury spot rate 6 4 0 5 10-year Treasury rate 5-period Treasury spot rate 6 4 0 6 10-year Treasury rate 6-period Treasury spot rate 6 4 0 7 10-year Treasury rate 7-period Treasury spot rate 6 4 0 8 10-year Treasury rate 8-period Treasury spot rate 6 4 0 9 10-year Treasury rate 9-period Treasury spot rate 6 4 0 10 10-year Treasury rate 10-period Treasury spot rate 6 4 0 11 10-year Treasury rate 11-period Treasury spot rate 6 4 0 12 10-year Treasury rate 12-period Treasury spot rate 6 4 0 13 10-year Treasury rate 13-period Treasury spot rate 6 4 0 14 10-year Treasury rate 14-period Treasury spot rate 6 4 0 15 10-year Treasury rate 15-period Treasury spot rate 6 4 0 16 10-year Treasury rate 16-period Treasury spot rate 6 4 0 17 10-year Treasury rate 17-period Treasury spot rate 6 4 0 18 10-year Treasury rate 18-period Treasury spot rate 6 4 0 19 10-year Treasury rate 19-period Treasury spot rate 6 4 0 20 10-year Treasury rate 20-period Treasury spot rate 106 104 100
∗Per $100 of par value.
By viewing any financial asset as a package of zero-coupon bonds, a consistent valuation framework can be developed. Viewing a financial asset as a package of zero-coupon bonds means that any two bonds would be viewed as different packages of zero-coupon bonds and valued accordingly.
The difference between the traditional valuation approach and the arbitrage-free approach is illustrated in Exhibit 4, which shows how the three bonds whose cash flows are depicted in Exhibit 3 should be valued. With the traditional approach, the discount rate for all three bonds is the yield on a 10-year U.S. Treasury security. With the arbitrage-free approach, the discount rate for a cash flow is the theoretical rate that the U.S. Treasury would have to pay if it issued a zero-coupon bond with a maturity date equal to the maturity date of the cash flow.
Therefore, to implement the arbitrage-free approach, it is necessary to determine the theoretical rate that the U.S. Treasury would have to pay on a zero-coupon Treasury security for each maturity. As explained in the previous chapter, the name given to the zero-coupon Treasury rate is theTreasury spot rate. In Chapter 6, we will explain how the Treasury spot rate can be calculated. The spot rate for a Treasury security is the interest rate that should be used to discount a default-free cash flow with the same maturity. We call the value of a bond based on spot rates thearbitrage-free value.
regardless of the means by which one goes about creating that asset. The law of one price implies that if the payoff of an asset can be synthetically created by a package of assets, the price of the package and the price of the asset whose payoff it replicates must be equal.
EXHIBIT 5 Determination of the Arbitrage-Free Value of an 8% 10-year Treasury Period Years Cash flow ($) Spot rate (%)∗ Present value ($)∗∗
1 0.5 4 3.0000 3.9409
2 1.0 4 3.3000 3.8712
3 1.5 4 3.5053 3.7968
4 2.0 4 3.9164 3.7014
5 2.5 4 4.4376 3.5843
6 3.0 4 4.7520 3.4743
7 3.5 4 4.9622 3.3694
8 4.0 4 5.0650 3.2747
9 4.5 4 5.1701 3.1791
10 5.0 4 5.2772 3.0829
11 5.5 4 5.3864 2.9861
12 6.0 4 5.4976 2.8889
13 6.5 4 5.6108 2.7916
14 7.0 4 5.6643 2.7055
15 7.5 4 5.7193 2.6205
16 8.0 4 5.7755 2.5365
17 8.5 4 5.8331 2.4536
18 9.0 4 5.9584 2.3581
19 9.5 4 6.0863 2.2631
20 10.0 104 6.2169 56.3830
Total $115.2621
∗The spot rate is an annual discount rate. The convention to obtain a semiannual discount rate is to take one-half the annual discount rate. So, for period 6 (i.e., 3 years), the spot rate is 4.7520%. The semiannual discount rate is 2.376%.
∗∗The present value for the cash flow is equal to:
Cash flow (1+Spot rate/2)period
A. Valuation Using Treasury Spot Rates
For the purposes of our discussion, we will take the Treasury spot rate for each maturity as given. To illustrate how Treasury spot rates are used to compute the arbitrage-free value of a Treasury security, we will use the hypothetical Treasury spot rates shown in the fourth column of Exhibit 5 to value an 8% 10-year Treasury security. The present value of each period’s cash flow is shown in the last column. The sum of the present values is the arbitrage-free value for the Treasury security. For the 8% 10-year Treasury, it is $115.2619.
As a second illustration, suppose that a 4.8% coupon 10-year Treasury bond is being valued based on the Treasury spot rates shown in Exhibit 5. The arbitrage-free value of this bond is $90.8428 as shown in Exhibit 6.
In the next chapter, we discuss yield measures. The yield to maturity is a measure that would be computed for this bond. We won’t show how it is computed in this chapter, but simply state the result. The yield for the 4.8% coupon 10-year Treasury bond is 6.033%.
Notice that the spot rates are used to obtain the price and the price is then used to compute a conventional yield measure.It is important to understand that there are an infinite number of spot rate curves that can generate the same price of $90.8428 and therefore the same yield.(We return to this point in the next chapter.)
Chapter 5 Introduction to the Valuation of Debt Securities 113
EXHIBIT 6 Determination of the Arbitrage-Free Value of a 4.8% 10-year Treasury Period Years Cash flow ($) Spot rate (%)∗ Present value ($)∗∗
1 0.5 2.4 3.0000 2.3645
2 1.0 2.4 3.3000 2.3227
3 1.5 2.4 3.5053 2.2781
4 2.0 2.4 3.9164 2.2209
5 2.5 2.4 4.4376 2.1506
6 3.0 2.4 4.7520 2.0846
7 3.5 2.4 4.9622 2.0216
8 4.0 2.4 5.0650 1.9648
9 4.5 2.4 5.1701 1.9075
10 5.0 2.4 5.2772 1.8497
11 5.5 2.4 5.3864 1.7916
12 6.0 2.4 5.4976 1.7334
13 6.5 2.4 5.6108 1.6750
14 7.0 2.4 5.6643 1.6233
15 7.5 2.4 5.7193 1.5723
16 8.0 2.4 5.7755 1.5219
17 8.5 2.4 5.8331 1.4722
18 9.0 2.4 5.9584 1.4149
19 9.5 2.4 6.0863 1.3578
20 10.0 102.4 6.2169 55.5156
Total 90.8430
∗The spot rate is an annual discount rate. The convention to obtain a semiannual discount rate is to take one-half the annual discount rate. So, for period 6 (i.e., 3 years), the spot rate is 4.7520%. The semiannual discount rate is 2.376%.
∗∗The present value for the cash flow is equal to:
Cash flow (1+Spot rate/2)period
B. Reason for Using Treasury Spot Rates
Thus far, we simply asserted that the value of a Treasury security should be based on discounting each cash flow using the corresponding Treasury spot rate. But what if market participants value a security using the yield for the on-the-run Treasury with a maturity equal to the maturity of the Treasury security being valued? (In other words, what if participants use the yield on coupon-bearing securities rather than the yield on zero-coupon securities?) Let’s see why a Treasury security will have to trade close to its arbitrage-free value.
1. Stripping and the Arbitrage-Free Valuation The key in the process is the existence of the Treasury strips market. As explained in Chapter 3, a dealer has the ability to take apart the cash flows of a Treasury coupon security (i.e., strip the security) and create zero-coupon securities. These zero-coupon securities, which we called Treasury strips, can be sold to investors. At what interest rate or yield can these Treasury strips be sold to investors? They can be sold at the Treasury spot rates. If the market price of a Treasury security is less than its value using the arbitrage-free valuation approach, then a dealer can buy the Treasury security, strip it, and sell off the Treasury strips so as to generate greater proceeds than the cost of purchasing the Treasury security. The resulting profit is an arbitrage profit. Since, as we will see, the value
determined by using the Treasury spot rates does not allow for the generation of an arbitrage profit, this is the reason why the approach is referred to as an ‘‘arbitrage-free’’ approach.
To illustrate this, suppose that the yield for the on-the-run 10-year Treasury issue is 6%.
(We will see in Chapter 6 that the Treasury spot rate curve in Exhibit 5 was generated from a yield curve where the on-the-run 10-year Treasury issue was 6%.) Suppose that the 8%
coupon 10-year Treasury issue is valued using the traditional approach based on 6%. Exhibit 7 shows the value based on discounting all the cash flows at 6% is $114.8775.
Consider what would happen if the market priced the security at $114.8775. The value based on the Treasury spot rates (Exhibit 5) is $115.2621. What can the dealer do? The dealer can buy the 8% 10-year issue for $114.8775, strip it, and sell the Treasury strips at the spot rates shown in Exhibit 5. By doing so, the proceeds that will be received by the dealer are $115.2621. This results in an arbitrage profit of $0.3846 (=$115.2621−$114.8775).7 Dealers recognizing this arbitrage opportunity will bid up the price of the 8% 10-year Treasury issue in order to acquire it and strip it. At what point will the arbitrage profit disappear? When the security is priced at $115.2621, the value that we said is the arbitrage-free value.
To understand in more detail where this arbitrage profit is coming from, look at Exhibit 8.The third column shows how much each cash flow can be sold for by the dealer if it is stripped. The values in the third column are simply the present values in Exhibit 5 based on discounting the cash flows at the Treasury spot rates. The fourth column shows how much the dealer is effectively purchasing the cash flow if each cash flow is discounted at 6%. This is the last column in Exhibit 7. The sum of the arbitrage profit from each cash flow stripped is the total arbitrage profit.
2. Reconstitution and Arbitrage-Free Valuation We have just demonstrated how coupon stripping of a Treasury issue will force its market value to be close to the value determined by arbitrage-free valuation when the market price is less than the arbitrage-free value. What happens when a Treasury issue’s market price is greater than the arbitrage-free value? Obviously, a dealer will not want to strip the Treasury issue since the proceeds generated from stripping will be less than the cost of purchasing the issue.
When such situations occur, the dealer will follow a procedure calledreconstitution.8 Basically, the dealer can purchase a package of Treasury strips so as to create a synthetic (i.e., artificial) Treasury coupon security that is worth more than the same maturity and same coupon Treasury issue.
To illustrate this, consider the 4.8% 10-year Treasury issue whose arbitrage-free value was computed in Exhibit 6. The arbitrage-free value is $90.8430. Exhibit 9 shows the price assuming the traditional approach where all the cash flows are discounted at a 6% interest rate.
The price is $91.0735. What the dealer can do is purchase the Treasury strip for each 6-month period at the prices shown in Exhibit 6 and sell short the 4.8% 10-year Treasury coupon issue whose cash flows are being replicated. By doing so, the dealer has the cash flow of a 4.8% coupon 10-year Treasury security at a cost of $90.8430, thereby generating an arbitrage profit of $0.2305 ($91.0735−$90.8430). The cash flows from the package of Treasury strips
7This may seem like a small amount, but remember that this is for a single $100 par value bond. Multiply this by thousands of bonds and you can see a dealer’s profit potential.
8The definition ofreconstituteis to provide with a new structure, often by assembling various parts into a whole.Reconstitutionthen, as used here, means to assemble the parts (the Treasury strips) in such a way that a new whole (a Treasury coupon bond) is created. That is, it is the opposite ofstripping a coupon bond.
Chapter 5 Introduction to the Valuation of Debt Securities 115
EXHIBIT 7 Price of an 8% 10-year Treasury Valued at a 6% Discount Rate Period Years Cash flow ($) Spot rate (%)∗ Present value ($)∗∗
1 0.5 4 6.0000 3.8835
2 1.0 4 6.0000 3.7704
3 1.5 4 6.0000 3.6606
4 2.0 4 6.0000 3.5539
5 2.5 4 6.0000 3.4504
6 3.0 4 6.0000 3.3499
7 3.5 4 6.0000 3.2524
8 4.0 4 6.0000 3.1576
9 4.5 4 6.0000 3.0657
10 5.0 4 6.0000 2.9764
11 5.5 4 6.0000 2.8897
12 6.0 4 6.0000 2.8055
13 6.5 4 6.0000 2.7238
14 7.0 4 6.0000 2.6445
15 7.5 4 6.0000 2.5674
16 8.0 4 6.0000 2.4927
17 8.5 4 6.0000 2.4201
18 9.0 4 6.0000 2.3496
19 9.5 4 6.0000 2.2811
20 10.0 104 6.0000 57.5823
Total 114.8775
∗The discount rate is an annual discount rate. The convention to obtain a semiannual discount rate is to take one-half the annual discount rate. So, since the discount rate for each period is 6%, the semiannual discount rate is 3%.
∗∗The present value for the cash flow is equal to:
Cash flow (1.03)period
purchased is used to make the payments for the Treasury coupon security shorted. Actually, in practice, this can be done in a more efficient manner using a procedure for reconstitution provided for by the Department of the Treasury.
What forces the market price to the arbitrage-free value of $90.8430? As dealers sell short the Treasury coupon issue (4.8% 10-year issue), the price of the issue decreases. When the price is driven down to $90.8430, the arbitrage profit no longer exists.
This process of stripping and reconstitution assures that the price of a Treasury issue will not depart materially from its arbitrage-free value. In other countries, as governments permit the stripping and reconstitution of their issues, the value of non-U.S. government issues have also moved toward their arbitrage-free value.
C. Credit Spreads and the Valuation of Non-Treasury Securities
The Treasury spot rates can be used to value any default-free security. For a non-Treasury security, the theoretical value is not as easy to determine. The value of a non-Treasury security is found by discounting the cash flows by the Treasury spot rates plus a yield spread to reflect the additional risks.
The spot rate used to discount the cash flow of a non-Treasury security can be the Treasury spot rate plus a constant credit spread. For example, suppose the 6-month Treasury
EXHIBIT 8 Arbitrage Profit from Stripping the 8% 10-Year Treasury
Period Years Sell for Buy for Arbitrage profit
1 0.5 3.9409 3.8835 0.0574
2 1.0 3.8712 3.7704 0.1008
3 1.5 3.7968 3.6606 0.1363
4 2.0 3.7014 3.5539 0.1475
5 2.5 3.5843 3.4504 0.1339
6 3.0 3.4743 3.3499 0.1244
7 3.5 3.3694 3.2524 0.1170
8 4.0 3.2747 3.1576 0.1170
9 4.5 3.1791 3.0657 0.1134
10 5.0 3.0829 2.9764 0.1065
11 5.5 2.9861 2.8897 0.0964
12 6.0 2.8889 2.8055 0.0834
13 6.5 2.7916 2.7238 0.0678
14 7.0 2.7055 2.6445 0.0611
15 7.5 2.6205 2.5674 0.0531
16 8.0 2.5365 2.4927 0.0439
17 8.5 2.4536 2.4201 0.0336
18 9.0 2.3581 2.3496 0.0086
19 9.5 2.2631 2.2811 −0.0181
20 10.0 56.3830 57.5823 −1.1993
115.2621 114.8775 0.3846
EXHIBIT 9 Price of a 4.8% 10-Year Treasury Valued at a 6% Discount Rate Period Years Cash flow ($) Spot rate (%) Present value ($)
1 0.5 2.4 6.0000 2.3301
2 1.0 2.4 6.0000 2.2622
3 1.5 2.4 6.0000 2.1963
4 2.0 2.4 6.0000 2.1324
5 2.5 2.4 6.0000 2.0703
6 3.0 2.4 6.0000 2.0100
7 3.5 2.4 6.0000 1.9514
8 4.0 2.4 6.0000 1.8946
9 4.5 2.4 6.0000 1.8394
10 5.0 2.4 6.0000 1.7858
11 5.5 2.4 6.0000 1.7338
12 6.0 2.4 6.0000 1.6833
13 6.5 2.4 6.0000 1.6343
14 7.0 2.4 6.0000 1.5867
15 7.5 2.4 6.0000 1.5405
16 8.0 2.4 6.0000 1.4956
17 8.5 2.4 6.0000 1.4520
18 9.0 2.4 6.0000 1.4097
19 9.5 2.4 6.0000 1.3687
20 10.0 102.4 6.0000 56.6964
Total 91.0735
Chapter 5 Introduction to the Valuation of Debt Securities 117
spot rate is 3% and the 10-year Treasury spot rate is 6%. Also suppose that a suitable credit spread is 90 basis points. Then a 3.9% spot rate is used to discount a 6-month cash flow of a non-Treasury bond and a 6.9% discount rate to discount a 10-year cash flow. (Remember that when each semiannual cash flow is discounted, the discount rate used is one-half the spot rate−1.95% for the 6-month spot rate and 3.45% for the 10-year spot rate.)
The drawback of this approach is that there is no reason to expect the credit spread to be the same regardless of when the cash flow is received. We actually observed this in the previous chapter when we saw how credit spreads increase with maturity. Consequently, it might be expected that credit spreads increase with the maturity of the bond. That is, there is aterm structure of credit spreads.
Dealer firms typically estimate a term structure for credit spreads for each credit rating and market sector. Generally, the credit spread increases with maturity. This is a typical shape for the term structure of credit spreads. In addition, the shape of the term structure is not the same for all credit ratings. Typically, the lower the credit rating, the steeper the term structure of credit spreads.
When the credit spreads for a given credit rating and market sector are added to the Treasury spot rates, the resulting term structure is used to value bonds with that credit rating in that market sector. This term structure is referred to as thebenchmark spot rate curveor benchmark zero-coupon rate curve.
For example, Exhibit 10 reproduces the Treasury spot rate curve in Exhibit 5. Also shown in the exhibit is a hypothetical credit spread for a non-Treasury security. The resulting benchmark spot rate curve is in the next-to-the-last column. It is this spot rate curve that is used to value the securities that have the same credit rating and are in the same market sector.
This is done in Exhibit 10 for a hypothetical 8% 10-year issue. The arbitrage-free value is
$108.4616. Notice that the theoretical value is less than that for an otherwise comparable Treasury security. The arbitrage-free value for an 8% 10-year Treasury is $115.2621 (see Exhibit 5).