Term Structure and Forward Interest Rates Interest Rate Notation The following presents notation to allow us to handle interest rates of bonds of various maturities at different points of time For example we might consider a one year bond versus a ten year bond Or we might consider a one year bond that is being sold today versus a one year bond that will be sold 1 2 or 3 years from now In general let ix y be the interest rate on a bond with x years to maturity i e an x year bond that will be sold at year y Here are some examples i1 t is the interest rate on a one year bond that is for sale today year t i1 t 2 is the interest rate on a one year bond that will be sold two years from now in year t 2 i5 t is the interest rate on a five year bond that is for sale today i10 t 1 is the interest rate on a ten year bond that will be sold one year from now If the date is in the future then today we don t know for sure what that interest rate will be In that case we put a superscript e on the interest rate to denote that it is what we expect to happen For example today in year t we don t know for sure what interest the rate will be on one year bond two years from now We write ie1 t 2 for what we expect that interest rate to be Note that this notation differs somewhat from that in the Mishkin textbook He does not use the semi colon in the subscript Implied Forward Rates With this notation for bonds of different maturities at different times we can consider strategies that involve purchasing bonds in the future Suppose you have B0 dollars to invest for two years You might follow one of these two strategies Strategy 1 Buy a two year bond today and hold it for two years Strategy 2 Buy a one year bond today and reinvest the proceeds in another one year bond in a year s time Which is the better strategy Under Strategy 1 you buy a two year bond today so its interest rate is i2 t After two years you will have 1 Bt 2 1 i2 t 2Bt Strategy 2 is a little more complicated After the first year you will have 2 Bt 1 1 i1 t Bt You will then invest Bt 1 at ie1 t 1 the interest rate that you expect at year t to prevail on one year bonds in year t 1 After the second year you expect to have 3 Bt 2 1 ie1 t 1 Bt 1 1 ie1 t 1 1 i1 t Bt You will choose the strategy to follow based on whether Bt 2 is larger in equation 1 or equation 3 Let s consider a simple example Suppose you have 1 000 to invest for two years Current interest rates are 6 for one year bonds and 8 on two year bonds and you expect one year rates to go up to 9 by next year These numbers mean B0 1 000 i1 t 0 06 i2 t 0 08 and ie1 t 1 0 09 Under Strategy 1 you buy the two year bond and earn 8 interest for two years Your ending balance will be 4 Bt 2 1 0 08 2 1 000 1 166 40 Under Strategy 2 you buy the one year bond today and earn 6 interest for the first year Then you re invest your money in another one year bond which you expect to earn 9 interest Your ending balance will be 5 Bt 2 1 0 09 1 0 06 1 000 1 155 40 As you can see you make an extra 11 under Strategy 1 so you will buy the two year bond However everyone else that wants to invest for two years will see the same thing you do so everyone will try to buy the two year bond This will push the price of that bond up which will push its interest rate down until the returns from the two strategies are exactly equal Mathematically this means 6 1 i2 t 2 1 ie1 t 1 1 i1 t Accordingly if you know two of these interest rates you can determine what the third must be For example if the market interest rates on one year and two year bonds are 5 and 6 respectively then what does the market expect the one year interest rate to be starting one year from now Just plug the numbers into equation 6 7 1 0 06 2 1 ie1 t 1 1 0 05 so 8 1 ie1 t 1 1 06 2 1 05 1 0701 which means the market expects the one year interest rate to be about 7 01 a year from now The expected interest rate ie1 t 1 in equation 6 is called an implied forward rate It is a forward rate because it deals with a bond that will exist at a future date This future rate is implied by the rates that we observe now Equation 6 is called a no arbitrage condition arbitrage The simultaneous purchase and sale of equivalent assets in different markets so as to profit from price discrepancies For example if Metallica tickets am I showing my age are selling for 45 over the internet and your friend is about to buy one from a neighbor for 55 you can step in and offer your friend the ticket for 50 buy it over the internet for 45 and pocket the 5 difference That 5 is an arbitrage profit The same kind of profit opportunities would exist if equation 6 didn t hold For example suppose the one year rate is 9 the two year rate is 10 and you expect one year rates to go up to 13 in a year s time In that case the two year bond would be a bad investment You would borrow money at the two year rate use the proceeds to buy a one year bond reinvest those proceeds after a year and pay back the borrowed money after the second year with money to spare Mathematically for each 1 000 you borrowed your profit would be 9 1 13 1 09 1 000 1 10 2 1 000 21 70 If these were the prevailing and expected interest rates how much money would you borrow A lot The more you borrow the more profit you reap So far we have been looking at a two year horizon The same principles apply generally Consider a ten year horizon One strategy would be to buy a ten year bond today Another would be to buy a nine year bond today and a one year bond in nine year s time The returns of both strategies should be equal by the no arbitrage condition Accordingly 10 1 ie1 t 9 1 i9 t 9 1 …