The term structure of interest rates Definitions and illustrations

Objective Understand the relationship between interest rates and expectations of future interest rates

Outline Spot vs. forward interest rates Implications for bond valuations Term structures of interest rates Theories on term structures

Spot vs. forward interest rates Spot rate Simple annual rate of interest, or the YTM on a pure discount bond maturing at time t Forward rate Implied future interest rate, as estimated today Forward rates reveal expectations of future interest rates

On spot rates Spot rates can be: quoted (ex: banks on GIC) calculated from bond data

Calculation of spot rates: Exemplification 1 A pure discount bond (PDB A) matures in one year, has a face value of $80, and sells for $75. P=$75=$80/(1+r) YTM(A) = 6.67%. 6.67% is the one-year spot rate, s(0,1)

Calculation of spot rates: Exemplification 2 A pure discount bond (bond B) matures in two years, has a face value of $80, and sells for $69. P=$69=$80/(1+r) 2 YTM(B) = 7.67% 7.67%. is the two-year spot rate, s(0,2)

Calculation of spot rates: Exemplification 3 A third pure discount bond (bond C) matures in three year, has a face value of $1080, and sells for $810. P=$810 =$1,080/(1+r) 3 YTM(C) = 10.06% 10.06% is the three-year spot rate, s(0,3)

Note A portfolio of the three pure discount bonds ( A, B, and C) will produce the same cash inflows as a three-year, 8% coupon bond. The 8% coupon bond would sell for $75+$69+$810=$954 and would yield approximately 9.7% Observation $80/(1.097) + $80/(1.097) 2 + $1,080/(1.097) 3 = = $80/(1.0667) + $80/(1.0767) 2 + $1,080/(1.1006) 3

Implication: Exemplification 4 A three-year, 8% coupon bond with a face value of $1,000 yields 9.7%. The one-year spot rate is 6.67%, the two-year spot rate is 7.67%. What is the three-year spot rate?. P= $80/(1.0667) + $80/(1.0767) 2 + $1,080/(1+s 3 ) 3 but P = $80/(1.097) + $80/(1.097) 2 + $1,080/(1.097) 3 hence $80/(1.097)+ $80/(1.097) 2 +$1,080/(1.097) 3 = $80/(1.0667)+$80/(1.0767) 2 + $1,080/(1+s 3 ) 3 s 3 = 10.06%

Calculation of forward rates: Exemplification 1 Assume that today (2001) you contact your broker and agree that one year from today (2002) you will buy a PDB maturing in one year (2003) for a price of $ $ = 1000/(1+r) YTM = 8.67%. Comment You have locked in a 8.67% return for the year This is the one-year forward rate, f(1,2)

Calculation of forward rates: Exemplification 2 Assume that today (2001) you contact your broker and agree that one year from today (2002) you will buy a PDB maturing in 2004 for a price of $ $ = 1000/(1+r) 2 YTM = 5.73%. Comment You locked in a 5.73% annual return for This is the two- year forward rate, f(1,3).

Relationship between spot and forward rates Exemplification Assume that you have a two-year investment horizon and you are facing the following choices:A You can buy the PDB that matures in two years and yields 7.67%, B You can buy the PDB that matures in one year and yields 6.67% and lock in the price of the PDB to be issued next year, maturing one year later, and yielding 8.67%.

Relationship between spot and forward rates Exemplification (cont’d) What is your return for each strategy? A Buy the two-year PDB bond. Hold it until maturity. Return =7.67%B Buy the one-year PDB maturing next year, and lock in the price of the second one-year PDB maturing two years from now. Return = [( )( ) -1] 1/2 = 7.67%

No arbitrage Imagine that the today's price of the one-year PDB to be issued next year were $900 instead of $ It follows that f(1,2)) would be (1000/900) -1 = 11.11% instead of 8.67%. By choosing the second strategy your return for the two years would be 8.82%/year instead of 7.67%/year. You would be better off by "rolling over" the two one-year PDB instead of buying the two-year PDB and holding it until maturity. Everyone would “roll over” until arbitrage would not be possible anymore.

Relationship between spot and forward rates: Two -year horizon [1+s(0,2)] 2 = [1+s(0,1)][1+f(1,2)]

Relationship between spot and forward rates: Three -year horizon [ 1+s(0,3)] 3 = [1+s(0,1)][1+f(1,2)][1+f(2,3)] = [1+s(0,2)] 2 [1+f(2,3)] Note [1+f(1,2)][1+f(2,3)] = [1+f(1,3)] 2 hence [1+s(0,3)] 3 = [1+s(0,1)][1+f(1,3)] 2

Generalization [1 + s(0,k)] k [1 + f(k,t)] (t-k) = [1 + s(0,t)] t

The term structure of interest rates Relationship between bond yields and various bond maturity dates

The term structure of interest rates Upward sloping Downward sloping Flat

Upward sloping term structure % 4% 6%

Downward sloping term structure % 15% 20% 5%

Flat term structure

Theories on term structures Unbiased Expectations Liquidity Preference Market Segmentation

Unbiased Expectations Investors believe f = E(s) Does not explain why term structures are mostly upward sloping

Liquidity Preference Investors believe f = E(s) + liquidity premium

Market Segmentation There are: short-term borrowers and short-term lenders medium-term borrowers and medium-term lenders long-term borrowers and long-term lenders The term structure depends on the relative demand/supply in each market segment