By Zachary Emig, MBA Class of 2005, Ross School of Business
Bootstrapping Interest Rate Curves
1 Starting Info
By Zachary Emig
Suppose we have the following Treasury yields (based roughly on Bloomberg.com, Nov. 22, 2004) Maturity (yrs) Coupon Price 32nds Yield Treasury Yields 0.25 na $99.46 2.17 4 0.50 na $98.82 2.38 3.5 1.00 2.250 $99.66 99 21/32 2.61 1.50 2.250 $99.28 99 9/32 2.76 3 2.00 2.500 $99.15 99 5/32 2.96 2.5 2.50 2.875 $99.63 99 20/32 3.05 2 3.00 3.000 $99.53 99 17/32 3.19 1.5 3.50 3.125 $99.64 99 20/32 3.26 1 4.00 3.500 $100.58 100 19/32 3.37 0.5 4.50 3.375 $99.64 99 20/32 3.49 0 5.00 3.500 $99.77 99 25/32 3.58 0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
2 Spot Curve
The above curve reflects the yield for current securities with certain maturities. The spot curve (or zero curve) 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 a 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) 0.25 0.50 1.00 Coupon na na 2.250 Price $99.46 $98.82 $99.66 32nds Yield 99 21/32 2.17 2.38 2.61 Spot Rate 2.17 2.38
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 Solving for z2, the 1yr zero rate: 99.6469 98.5351 (1+Z2/2)^2 1+Z2/2 Z2/2 Z2 = = = = = = = 1.1118 + 101.1250/(1+Z2/2)^2 1.1118 + 101.1250/(1+Z2/2)^2 101.1250/(1+Z2/2)^2 1.0262836 1.0130566 (Square root) 0.0130566 2.6113 Percent
In this case, the 1 year spot rate matches the yield; that isn't always the case. Maturity (yrs) 0.25 0.50 1.00 Coupon na na 2.250 Price $99.46 $98.82 $99.66 32nds Yield 2.17 2.38 2.61 Spot Rate 2.17 2.38 2.6113
99 21/32
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By Zachary Emig, MBA Class of 2005, Ross School of Business
B Rest of Spots Recurse through the rest of maturities, one by one, to get their spot rates. Maturity (yrs) 0.25 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 Coupon na na 2.250 2.250 2.500 2.875 3.000 3.125 3.500 3.375 3.500 Price $99.46 $98.82 $99.66 $99.28 $99.15 $99.63 $99.53 $99.64 $100.58 $99.64 $99.77 32nds Yield 2.17 2.38 2.61 2.76 2.96 3.05 3.19 3.26 3.37 3.49 3.58 Sum of Prior Final Spot Rate Coupons' PVs Payment 2.17 2.38 2.6113 2.7440 2.2080 101.1250 2.9448 3.6532 101.2500 3.0359 5.5571 101.4375 3.1784 7.1898 101.5000 3.2497 8.9109 101.5625 3.3651 11.5435 101.7500 3.4889 12.6078 101.6875 3.5835 14.5725 101.7500
99 99 99 99 99 99 100 99 99
21/32 9/32 5/32 20/32 17/32 20/32 19/32 20/32 25/32
Plotting the regular yield curve (in blue) versus the spot curve (in yellow):
Yield Spot Rate Yield Spot Rate
3.8 3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 2 0.25 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 i
3.65
3.6
3.55
3.5
3.45 3.4 3.35 3.3 3.25 3.50 4.00
i
4.50
5.00
We can see that as maturity extends, the two curves cross.
3 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 its 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 though 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 Solving for f1: 99.6469 98.5351 98.5351 (1+f1/2) f1/2 f1 = = = = = = 1.1118 + 101.1250/((1+z1/2)(1+f1/2)) 101.1250/((1+2.38/2)(1+f1/2)) 99.935764 /(1+f1/2) 1.0142145 0.0142145 2.8429 = 1.1118 + 101.1250/((1+z1/2)(1+f1/2))
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By Zachary Emig, MBA Class of 2005, Ross School of Business
Maturity (yrs) 0.00 0.50 1.00
Coupon na 2.250
Price $98.82 $99.66
32nds Yield 2.38 2.61
Spot Rate
99 21/32
6mo Fwd Rate 2.38 2.38 2.8429 2.6113
B Rest of Forwards Recurse through the rest of maturities, one by one, to get the forward rates. Maturity (yrs) 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 Coupon na 2.250 2.250 2.500 2.875 3.000 3.125 3.500 3.375 3.500 Price $98.82 $99.66 $99.28 $99.15 $99.63 $99.53 $99.64 $100.58 $99.64 $99.77
Yield
32nds Yield 2.38 2.61 2.76 2.96 3.05 3.19 3.26 3.37 3.49 3.58
99 99 99 99 99 99 100 99 99
21/32 9/32 5/32 20/32 17/32 20/32 19/32 20/32 25/32
Spot Rate 6mo Fwd Rate 2.17 2.38 2.38 2.8429 2.6113 3.0209 2.7440 3.5997 2.9448 3.4176 3.0359 3.9537 3.1784 3.6984 3.2497 4.2473 3.3651 4.5875 3.4889 4.5123 3.5835
Spot Rate
6mo Fwd Rate
5 4.5 4 3.5 3 2.5 2 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00
3 Summarizing the Curves
To summarize: Yield Curve Spot Curve Forward 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 at to make present and future values the same. 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. This is a plot of what the market charges to borrow money for a 6 month 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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By Zachary Emig, MBA Class of 2005, Ross School of Business
Pricing Interest Rate Swaps
1 Starting Info
We have the following rate curves (from the Bootstrapping worksheet). I've added LIBOR spot and 6mo fwd rates. LIBOR Spot Maturity (yrs) Coupon Price 32nds Yield Spot Rate 6mo Fwd Rate Rates 0.00 2.38 0.50 na $98.82 2.38 2.38 2.8429 2.8800 1.00 2.250 $99.66 99 21/32 2.61 2.6113 3.0209 3.1113 1.50 2.250 $99.28 99 9/32 2.76 2.7440 3.5997 3.2640 2.00 2.500 $99.15 99 5/32 2.96 2.9448 3.4176 3.6348 2.50 2.875 $99.63 99 20/32 3.05 3.0359 3.9537 3.6459 3.00 3.000 $99.53 99 17/32 3.19 3.1784 3.6984 3.7884 3.50 3.125 $99.64 99 20/32 3.26 3.2497 4.2473 3.7997 4.00 3.500 $100.58 100 19/32 3.37 3.3651 4.5875 3.9451 4.50 3.375 $99.64 99 20/32 3.49 3.4889 4.5123 4.0789 5.00 3.500 $99.77 99 25/32 3.58 3.5835 4.1735
i
i
I created the LIBOR forward rates simply because most IR Swaps use LIBOR for the floating leg.
Yield Spot Rate 6mo Fwd Rate LIBOR Spot Rates LIBOR 6mo Fwd Rate
5.5 5
4.5
4 3.5 3 2.5 2 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00
2 Swap Info
Assume we want to buy (go long) a swap, i.e. pay a fixed rate, receive floating (LIBOR 6mo). Here are the contract details I'm looking for: Notional Term (Years) Settlement Floating Rate $100,000,000 4 Every 6mos LIBOR 6mo
3 Pricing
So we are expecting, based on LIBOR forward rates to receive the following 8 cash inflows. We can discount each using the [LIBOR] spot rates. Time (Years) 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 6mo LIBOR LIBOR Discount PV of Cash Fwd Rate Cash Flow In Spot Rate Factor In 2.8800 0 0.0000 0 0 3.3429 $2,880,000 2.8800 0.9858 $2,839,117 3.5697 $3,342,897 3.1113 0.9696 $3,241,266 4.7513 $3,569,689 3.2640 0.9526 $3,400,471 3.6905 $4,751,328 3.6348 0.9305 $4,421,065 4.5024 $3,690,508 3.6459 0.9136 $3,371,765 3.8673 $4,502,399 3.7884 0.8935 $4,022,969 4.9659 $3,867,317 3.7997 0.8766 $3,389,962 $4,965,916 3.9451 0.8553 $4,247,494 Total PV of Floating Payments $28,934,109
Naturally, in a no-arbitrage world, the PV of the fixed payments we make out must also be $28,934,109 . Do the math; the fixed leg of the swap is simply an annuity: $28,934,109 = Pmt * Sum ( Discountfactors )
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By Zachary Emig, MBA Class of 2005, Ross School of Business
$28,934,109 = Pmt * 7.3775 $3,921,922 = Pmt Thus, the fixed interest rate is $3,921,922 / 3.9219 $100,000,000
This would probably be quoted at a spread over the equivalent (4yr Treasury): 55.2 bp
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By Zachary Emig, MBA Class of 2005, Ross School of Business
By Zachary Emig
LIBOR 6mo Fwd Rate 2.8800 3.3429 3.5697 4.7513 3.6905 4.5024 3.8673 4.9659 5.1522 5.0273
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