# Bootstrapping Interest Rate Curves

Bootstrapping Interest Rate Curves
1
##### Starting Info
Suppose we have the following Treasury yields (based roughly on Bloomberg.com, Nov. 22, 2004)
Maturity (yrs) Coupon Price 32nds Yield
1.00 2.250 \$99.66 99 21/32 2.61
2.00 2.500 \$99.15 99  5/32 2.96
3.00 3.000 \$99.53 99 17/32 3.19
4.00 3.500 \$100.58 ##### 3.37
5.00 3.500 \$99.77 99 25/32 3.58
2
##### Spot Curve
The above curve reflects the yield for current securities with certain maturities.
The spot curve (or zero curves) tells us what the spot or interest rate is for a zero coupon bond of a particular maturity.
In effect, it is the discount rate applied to single cash flow in time for any of the coupon bonds above.
We can “bootstrap” out a zero curve from the data above.
We know the 0.25 and 0.50 spot rates since they are discount securities.
Maturity (yrs) Coupon Price 32nds Yield Spot Rate
0.25 n/a \$99.46 2.17 2.17
0.50 n/a \$98.82 2.38 2.38
1.00 2.250 \$99.66 99 21/32 2.61
A
##### 1 Year Spot
The one-year spot rate is easily found by equalizing the cash flows.
y is the yield to maturity, z1 and z2 are the two zero rates (6mo and 1yr):
C1/(1+y/2) + (100+C2)/(1+y/2)^2 = C1/(1+z1/2) + (100+C2)/(1+z2/2)^2
1.1105 + 98.5364 = 1.1118 + 101.1250/(1+Z2/2)^2
Solving for z2, the 1yr zero rate:
99.6469 = 1.1118 + 101.1250/(1+Z2/2)^2
98.5351 = 101.1250/(1+Z2/2)^2
(1+Z2/2)^2 = 1.02628
1+Z2/2 = 1.01306 (Square root)
Z2/2 = 0.01306
Z2 = 2.6113 Percent
In this case, the 1-year spot rate matches the yield; that isn’t always the case.
Maturity (yrs) Coupon Price 32nds Yield Spot Rate
0.25 n/a \$99.46 2.17 2.17
0.50 n/a \$98.82 2.38 2.38
1.00 2.250 \$99.66 99 21/32 2.61 2.6113
B
##### Rest of Spots
Recurse through the rest of maturities, one by one, to get their spot rates.
Maturity (yrs) Coupon Price 32nds Yield Spot Rate Sum of Prior Coupons’ PVs Final Payment
0.25 n/a \$99.46 2.17 2.17
0.50 n/a \$98.82 2.38 2.38
1.00 2.250 \$99.66 99 21/32 2.61 2.6113
1.50 2.250 \$99.28 99  9/32 2.76 2.7440 2.2080 #####
2.00 2.500 \$99.15 99  5/32 2.96 2.9448 3.6532 #####
2.50 2.875 \$99.63 99 20/32 3.05 3.0359 5.5571 #####
3.00 3.000 \$99.53 99 17/32 3.19 3.1784 7.1898 #####
3.50 3.125 \$99.64 99 20/32 3.26 3.2497 8.9109 #####
4.00 3.500 \$100.58 ##### 3.37 3.3651 ##### #####
4.50 3.375 \$99.64 99 20/32 3.49 3.4889 ##### #####
5.00 3.500 \$99.77 99 25/32 3.58 3.5835 ##### #####
Plotting the regular yield curve (in blue) versus the spot curve (in yellow):
We can see that as maturity extends, the two curves cross.
##### Forward Curve
A forward curve is simply a graph of the x-month forward rate at different points in the future.
Unlike the other two curves, the x-axis represents the “starting point” in the future for the forward contract, not
it’s maturity.
In this example, we’ll determine the 6-month forward curve from the above information.
A
##### 6mo Forward in 6mos
The 6mo forward rate in 6 months can be thought of as what we could borrow/lend at for 6 months, 6 months from now.
Confusing enough?
By the law of no arbitrage, investing our money now for 1 year or now for 6months, with the next 6mo rate locked in,
must result in the same present value.
y is the yield to maturity, z1 is the 6mo spot rate, and f1 is the 6mo forward rate 6months from now.
C1/(1+y/2) + (100+C2)/(1+y/2)^2 = C1/(1+z1/2) + (100+C2)/(1+z1/2)(1+f1/2)
1.1105 + 98.5364 = 1.1118 + 101.1250/((1+z1/2)(1+f1/2))
Solving for f1:
99.6469 = 1.1118 + 101.1250/((1+z1/2)(1+f1/2))
98.5351 = 101.1250/((1+2.38/2)(1+f1/2))
98.5351 = 99.9358 /(1+f1/2)
(1+f1/2) = 1.01421
f1/2 = 0.01421
f1 = 2.8429
Maturity (yrs) Coupon Price 32nds Yield Spot Rate 6mo Fwd Rate
0.00 2.38
0.50 n/a \$98.82 2.38 2.38 2.8429
1.00 2.250 \$99.66 99 21/32 2.61 2.6113
B
##### Rest of Forwards
Recurse through the rest of maturities, one by one, to get the forward rates.
Maturity (yrs) Coupon Price 32nds Yield Spot Rate 6mo Fwd Rate
0.00 2.17 2.38
0.50 n/a \$98.82 2.38 2.38 2.8429
1.00 2.250 \$99.66 99 21/32 2.61 2.6113 3.0209
1.50 2.250 \$99.28 99  9/32 2.76 2.7440 3.5997
2.00 2.500 \$99.15 99  5/32 2.96 2.9448 3.4176
2.50 2.875 \$99.63 99 20/32 3.05 3.0359 3.9537
3.00 3.000 \$99.53 99 17/32 3.19 3.1784 3.6984
3.50 3.125 \$99.64 99 20/32 3.26 3.2497 4.2473
4.00 3.500 \$100.58 ##### 3.37 3.3651 4.5875
4.50 3.375 \$99.64 99 20/32 3.49 3.4889 4.5123
5.00 3.500 \$99.77 99 25/32 3.58 3.5835
To summarize:
##### Yield Curve
Yields are bond-specific; given a bond’s market price and coupons, the yield
is the average rate that all cash flows are discounted to make present and
future values the same.
##### Spot Curve
The spot curve diagrams what pure discount rate the market applies to any
cash flow at each maturity point.  It is not bond specific.
Also called the zero curve.
##### Forward Curve
This is a plot of what the market charges to borrow money for a 6 month
the period starting at certain future dates.
Note that forward curves could be made for any borrowing term
(i.e. 1 year forwards, 3 month forwards, etc.)
##### Disclaimer
This is a very rudimentary example.  In practice, bootstrapping is a much more difficult process, mainly due to the difficulty
of getting a clean, accurate original yield curve.  There are not actively traded Treasury securities at every maturity point.
The Treasury no longer issues 30 year Bonds, making the long end of the curve tricky.  Etc., etc.
This worksheet is meant more as an explanation for the concept of bootstrapping, the process of generating a
spot curve from a yield curve, and a forward curve from a spot curve.

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