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CH26supp

# CH26supp - Spot and forward interest rate Hedging Futures...

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Spot and forward interest rate Hedging Futures Ch. 26 supplementary note JHT Kim November 19, 2009 1/34

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Spot and forward interest rate Hedging Futures Spot interest rate Spot and forward interest rate Spot interest rate The n - year spot interest rate is the rate on an investment made for n years, starting from today. For example, a two-year spot rate of 10% means that an investment will earn 10% every year for two years So, the n -year spot rate = n -year zero-coupon yield Example: Consider two zero-coupon bonds A and B from the same issuer. - Bond A matures in one year ( t = 1) and trades at \$925.926 - Bond B matures in two years ( t = 2) and trades at \$826.446 The one-year and two-year spot rates are therefore: Year spot rate 1 8% 2 10% 2/34
Spot and forward interest rate Hedging Futures Spot interest rate Spot and forward interest rate Forward interest rate The term structure refers to the relationship of spot rates with different maturities Note from the above table that 10% is the rate you get ONLY when your investment is locked in for two years. The interest rate for the second year (i.e., the rate you get over [1, 2] for the investment made at t = 1) is not shown in the table This rate is called the forward rate Formally, the forward rate is the interest rate in the future implied by the spot rates The forward rates can be computed from the spot rates As an example, let us compute the forward rate for the second year from the above table 3/34

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Spot and forward interest rate Hedging Futures Spot interest rate Spot and forward interest rate Forward interest rate From the table, investing a unit dollar in bond B for two years will give \$(1 . 10) 2 = \$1 . 21 at t = 2 To get the rate for the second year we observe \$1 . 21 = \$(1 + Rate yr 1 ) × (1 + Rate yr 2 ) \$1 × (1 . 10) 2 | {z } 2 - yr spot rate = \$1 × 1 . 08 |{z} 1 - yr spot price × (1 + Rate yr 2 ) Rate yr 2 = 12 . 04% So the forward rate for the second year, 12.04% in this example, is the rate you will earn during [1 , 2] if you invest a unit dollar at time 1 4/34
Spot and forward interest rate Hedging Futures Spot interest rate Spot and forward interest rate The forward rate is a hypothetical future interest rate implied by (or squeezed by) the spot rates year sport rate forward rate 1 8% N/A 2 10% 12.04% By convention, we denote the n -year spot rate by r n and the year n forward rate by f n in the example, r 1 = 8%, r 2 = 10%, and f 2 = 12 . 04% Also note that in this example, (1 + r 2 ) 2 = (1 + r 1 ) × (1 + f 2 ) 5/34

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Spot and forward interest rate Hedging Futures Spot interest rate Spot and forward interest rate year sport rate forward rate 1 r 1 N/A 2 r 2 f 2 . . . . . . . . . n r n f n General relationship between the spot rate and the forward rate: (1 + r n ) n = (1 + r n - 1 ) n - 1 × (1 + f n ) Forward rate f n is the implied return during [ n - 1 , n ] f n can therefore be thought as the “expected” interest rate during [ n - 1 , n ], guessed by market investors at t = 0 But we never know the actual (or true) interest rate during [ n - 1 , n ] until the time comes to n - 1 We often draw the graph of the spot rates over time 6/34
Spot and forward interest rate Hedging Futures Spot interest rate

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CH26supp - Spot and forward interest rate Hedging Futures...

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