## Monday, November 07, 2011

### Arbitrage-free Valuation Approach

CFA Level 1 - Fixed Income Investments

Under a traditional approach to valuing a bond, it is typical to view the security as a single package of cash flows, discounting the entire issue with one discount rate. Under the arbitrage-free valuation approach, the issue is viewed, instead, as various zero-coupon bonds that should be valued individually and added together to determine value. The reason this is the correct way to value a bond is that it does not allow a risk-free profit to be generated by "stripping" the security and selling the parts at a higher price than purchasing the security in the market.
As an example, a five-year bond that pays semi-annual interest would have 11 separate cash flows, and be valued using the appropriate yield on the curve that matches its maturity. So the markets implement this approach by determining the theoretical rate the U.S. Treasury would have to pay on a zero-coupon treasury for each maturity. The investor then determines the value of all the different payments using the theoretical rate and adds them together. This zero-coupon rate is the treasury spot rate. The value of the bond based on the spot rates is the arbitrage-free value.

Determining Whether a Bond is Under or Over Valued
What you need to be able to do is value a bond like we have done before using the more traditional method of applying one discount rate to the security. The twist here, however, is that instead of using one rate, you will use whatever rate the spot curve has that coordinates with the proper maturity. You will then add the values up as you did previously to get the value of the bond. You will then be given a market price to compare to the value that you derived from your work. If the market price is above your figure, then the bond is undervalued and you should buy the issue. If the market price is below your price, then the bond is overvalued and you should sell the issue.

How Does a Dealer Generate Arbitrage Profits?
A dealer has the ability to strip a security or to take apart the cash flows that make up the bond. These Treasury strips can be sold to investors. So if the market price of a Treasury security is less than the value using the arbitrage-free valuation, a dealer will buy the security, strip the bond and then sell the Treasury strips at a higher amount than the purchase price for the whole bond.

On the other hand, if the market price is more than the value using the arbitrage-free valuation, the dealer will buy the strips, make the bond "whole" and sell it at a higher price than that of the purchased strips.

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