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Unformatted text preview: Hedging Interest Rate Risk 1 I n t e r e s t R a t e R i s k The most important price in the economy is the interest rate, so it stands to reason that it would be the most important source of risk. Interest rate changes can take on an in f nity of values. But these changes are mostly of degree, not kind. Hence interest rate risk can usually be hedged with a few securities. This is especially true when the present value is a di f erentiable function of interest rates. Consider a 10-year plus a day coupon bond with a coupon rate of 9% . Its face value is $100 . Let today’s one day interest rate be some small number. We do not say what the price of the bond is today. Indeed, that is what we are trying to f gure out. We care about replicating the value of the bond tomorrow (by which time it will become precisely a 10 year bond). Suppose that within a day the annual interest rate will jump to either 7 . 7% , 7 . 8% , 7 . 9% , 8 . 0% , 8 . 1% , 8 . 2% , or 8 . 3% , and stay there for 10 years. Please see the accompanying spread sheet ”hedging interest rate risk”. What will the price of the coupon bond be in each of the seven states? The price of the coupon bond tomorrow will be given by its present value: P V ( r ) = 9 X t =1 9 (1 + r ) t + 109 (1 + r ) 10 The prices at di f erent interest rates can be calculated using Excel. The results are 8 . 3% 104 . 6342 8 . 2% 105 . 3200 8 . 1% 106 . 0119 8 . 0% 106 . 7101 7 . 9% 107 . 4145 7 . 8% 108 . 1252 7 . 7% 108 . 8424 To fully replicate these payo f s we would need to f nd seven benchmark securities. But the crucial thing to notice is that though there are many possible states (actually in f nitely many if we realistically included every possible interest rate change) the states have a structure to them. As the interest rates get higher, the bond loses value. Not only that, but the loss in value for every 10 basis point increase (i.e. each .1% increase, or equivalently, each increase of .001) in interest rates leads to an additional loss of about . 7 , as can be seen by inspection. For example, the change in value from r = 8% to r = 8 . 1% is . 6982 = 106 . 7101 − 106 . 0119 . This f gure is not exact for other changes, but it is close, as we shall see. 1 2 Hedging by a Replicating Portfolio Suppose for some reason we were able to purchase the above coupon bond for a very low price P at time 0, say because other people mistakenly thought it might default between today and tomorrow (though they all agree there is no subsequent chance of default). How could we lock in our pro f t? If we simply bought the bond for P , we might be right about it not defaulting, but we might end up losing money anyway because the interest rates went up. What should we do?...
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This note was uploaded on 12/08/2009 for the course ECON 251 at Yale.