Interest Rate Risk II Duration and Interest Rate Sensitivity. February 22nd, 2010 Average Life This is simply the average time that we wait to receive our principal. If it is a 5 year bullet, the answer is simple the average life is just 5 years. If the loan amortizes each year equally over the five years then the average life of the loan is obviously less than the first example of 5 years since you don’t wait the full length of time to get all of your principal back. Average Life Example Average Life is simply the time weighted average of the principal repayments. Principal Repayment multiplied by time to receive it divided by Total Principal. See Excel Spread Sheet for examples. Duration Two bonds or two loans can have the exact same maturity and similar interest rates but have one very different characteristic from a risk management/interest rate sensitivity perspective: Duration. Duration is a more complete measure of an asset or liability’s sensitivity to interest rate movements since it takes into account the timing of the underlying cash flows (principal payment, amortization, and interest payments) Bond 1 Bond 1 trades at par and has a 5 year bullet maturity and pays interest annually. By “bullet” we mean all the principal is due at maturity so there is no principal payment made in the intervening years. Par refers to the face value of the bond (let’s assume $1,000). Annually means that the interest is due at the end of each year on the anniversary of the issuance date. Bond 2 Bond 2 also trades at par and has a maturity of 5 years but amortizes over the period in 5 equal payments annual payments. Interest is also paid annually. Therefore, if we assume this bond also has a face value of $1,000, then $200 of principal is paid at the end of each year along with an interest payment. Comparison Obviously, you get your money back faster in the case of Bond 2 compared to Bond 1. However another way of looking at it is that if Bank A owned Bond 1 and Bank B owned Bond 2. We could actually look at Bond B as being 5 different Bonds. Or looking at it differently Bond 1: $1,000 face/par value 5 year bullet with an annual interest payment. Bond 2: $200 face/par value 5 year bullet with an annual interest payment. $200 face/par value 4 year bullet with annual interest payment. $200 3 year, 2 year, 1 year bullets with annual interest payments. Match funding To match fund Bank A who holds Bond 1, you would just need to go out and borrow 5 year money with interest payable annually and principal due at maturity. For Bank B/Bond2 you could either borrow with same terms and conditions as Bond 2 (5 year money with annual principal amortization and interest payment) or you could make 5 different borrowings matching the 5 different bonds that we broke Bond 2 down into. Portfolio Basis Banks have to do this on a portfolio basis. Therefore, if they are able to track all interest and principal payments (bullets and amortizations) then they can come up with an average duration for their portfolio on both the asset and liability side. They can then adjust their funding and investments accordingly to attain the exact balance of funding they want (short or long funding or match funding). Given the sophistication of information systems these days, this isn’t so difficult. Calculating Duration Duration is calculated as the present value of the weighted average of the different cash flows divided by the present value of the bond itself. See Excel File “Duration Problem” for illustration. We assume here that market rate equals coupon rate. Interest Rates? In the example of Bonds 1 and 2, I gave actual numbers for principal along with the maturities and frequency of interest payments. The bonds also had the same interest rate. If this were true what would such interest rates indicate about the risk of the two bonds? First let’s assume, yield curve is upward sloping which is the more typical interest environment. Risk/Tenor Premium If Bond 1 had the same risk rating as Bond 2, Bond 1 would normally require a higher risk premium due to the longer average tenor (5 year or 4.16 years vs 2.66 years as duration indicates). This is because the yield curve slopes upward and credit risk is higher in that more can go wrong with a company over a longer period of time. Remember that Bond 2 is really 5 different bonds and the average of the different pieces is being priced at 10%. This means that the 5 year piece is costing more than 10% given the upward sloping yield curve. For simplicity sake, let’s assume the 1 year maturity is priced at 8%, 2 year at 9%, 3 year at 10% , 4 year at 11% and 5 year at 12%. Thus yielding an average of 10% (actual result would not prove to be the case due to time value of money). Broken out this way it is clear that Bond 2 is much higher risk since for the same 5 year maturity the market requires 12% return vs. 10% for Bond 1. Using duration of the two bonds 4.17 years vs 2.66 years with both having a 10% coupon it is much easier to see that the shorter duration bond in an upward sloping yield curve environment is the riskier bond. Interest Rate Sensitivity Not only can we use Duration to see if we are matched, long or short-funded, we can actually come up with a measure of just how sensitive our portfolio is to a movement in interest rates. We call this the elasticity of interest rates or the first order derivative (we remember our calculus class? ) Calculus First Order/Elasticity are probably big and scary words to many. However, we are just measuring interest sensitivity. A 1 basis point (.01 percent or .0001 movement) in interest rates results in an X percent movement in principal whether it is for a bond trader or for an asset/liability manager. Duration is actually quite useful If you are a bond trader, it is difficult to have to get out your calculator every time you are faced with a bond purchase decision where there are intervening payments. Hence, if you can price bonds based on their risk attributes (in this case public ratings) and then look in the market what similar bonds with similar durations trade at, then you can easily compare the rates on the bonds.