bootstrapping math

8/8/2019 Bootstrapping Math

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By Zachary Emig, MBA Class of 2005, Ross School of Business

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Bootstrapping Interest Rate Curves By Zachary Emig

1 Starting InfoSuppose we have the following Treasury yields (based roughly on Bloomberg.com, Nov. 22, 2004)

Maturity (yrs) Coupon Price 32nds Yield0.25 na $99.46 2.170.50 na $98.82 2.381.00 2.250 $99.66 99.66 2.611.50 2.250 $99.28 99.28 2.762.00 2.500 $99.15 99.15 2.962.50 2.875 $99.63 99.63 3.053.00 3.000 $99.53 99.53 3.193.50 3.125 $99.64 99.64 3.264.00 3.500 $100.58 100.58 3.374.50 3.375 $99.64 99.64 3.495.00 3.500 $99.77 99.77 3.58

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) Coupon Price 32nds Yield Spot Rate0.25 na $99.46 2.17 2.170.50 na $98.82 2.38 2.381.00 2.250 $99.66 99.66 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)^298.5351 = 101.1250/(1+Z2/2)^2

(1+Z2/2)^2 = 1.031+Z2/2 = 1.01 (Square root)Z2/2 = 0.01Z2 = 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 na $99.46 2.17 2.170.50 na $98.82 2.38 2.381.00 2.250 $99.66 99.66 2.61 2.6113

0.25 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.000

0.5

1

1.5

2

2.53

3.5

4 Column G


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B Rest of SpotsRecurse through the rest of maturities, one by one, to get their spot rates.

Maturity (yrs) Coupon Price 32nds Yield Spot Rate0.25 na $99.46 2.17 2.170.50 na $98.82 2.38 2.38

1.00 2.250 $99.66 99.66 2.61 2.61131.50 2.250 $99.28 99.28 2.76 2.7440 2.2080 101.12502.00 2.500 $99.15 99.15 2.96 2.9448 3.6532 101.25002.50 2.875 $99.63 99.63 3.05 3.0359 5.5571 101.43753.00 3.000 $99.53 99.53 3.19 3.1784 7.1898 101.50003.50 3.125 $99.64 99.64 3.26 3.2497 8.9109 101.56254.00 3.500 $100.58 100.58 3.37 3.3651 11.5435 101.75004.50 3.375 $99.64 99.64 3.49 3.4889 12.6078 101.68755.00 3.500 $99.77 99.77 3.58 3.5835 14.5725 101.7500

Plotting the regular yield curve (in blue) versus the spot curve (in yellow):

We can see that as maturity extends, the two curves cross.

3 Forward CurveA 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, notits 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 = 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 = 99.94 /(1+f1/2)

(1+f1/2) = 1.01f1/2 = 0.01f1 = 2.8429

Sum of PriorCoupons' PVs

FinalPayment

0.25 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.002

2.2

2.4

2.6

2.8

3

3.2

3.43.6

3.8

Yield Spot R ate

3.50 4.00 4.50 5.003.25

3.3

3.35

3.4

3.45

3.5

3.553.6

3.65

Yield Spot Rate


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Maturity (yrs) Coupon Price 32nds Yield Spot Rate 6mo Fwd Rate0.00 2.380.50 na $98.82 2.38 2.38 2.84291.00 2.250 $99.66 99.66 2.61 2.6113

B Rest of ForwardsRecurse through the rest of maturities, one by one, to get the forward rates.

Maturity (yrs) Coupon Price 32nds Yield Spot Rate 6mo Fwd Rate0.00 2.17 2.380.50 na $98.82 2.38 2.38 2.84291.00 2.250 $99.66 99.66 2.61 2.6113 3.02091.50 2.250 $99.28 99.28 2.76 2.7440 3.59972.00 2.500 $99.15 99.15 2.96 2.9448 3.41762.50 2.875 $99.63 99.63 3.05 3.0359 3.95373.00 3.000 $99.53 99.53 3.19 3.1784 3.69843.50 3.125 $99.64 99.64 3.26 3.2497 4.24734.00 3.500 $100.58 100.58 3.37 3.3651 4.58754.50 3.375 $99.64 99.64 3.49 3.4889 4.51235.00 3.500 $99.77 99.77 3.58 3.5835

3 Summarizing the Curves To summarize:

Yield Curve Yields are bond-specific; given a bond's market price and coupons, the yieldis the average rate that all cash flows are discounted at to make present andfuture values the same.

Spot Curve The spot curve diagrams what pure discount rate the market applies to anycash 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 monthperiod 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 difficultyof 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 aspot curve from a yield curve, and a forward curve from a spot curve.

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.002

2.5

3

3.5

4

4.5

5 Yield Spot Rate 6mo Fwd Rate


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Pricing Interest Rate Swaps

1 Starting InfoWe have the following rate curves (from the Bootstrapping worksheet). I've added LIBOR spot and 6mo fwd rates.

Maturity (yrs) Coupon Price 32nds Yield Spot Rate 6mo Fwd Rate0.00 2.380.50 na $98.82 2.38 2.38 2.8429 2.88001.00 2.250 $99.66 99.66 2.61 2.6113 3.0209 3.11131.50 2.250 $99.28 99.28 2.76 2.7440 3.5997 3.26402.00 2.500 $99.15 99.15 2.96 2.9448 3.4176 3.63482.50 2.875 $99.63 99.63 3.05 3.0359 3.9537 3.64593.00 3.000 $99.53 99.53 3.19 3.1784 3.6984 3.78843.50 3.125 $99.64 99.64 3.26 3.2497 4.2473 3.79974.00 3.500 $100.58 100.58 3.37 3.3651 4.5875 3.94514.50 3.375 $99.64 99.64 3.49 3.4889 4.5123 4.07895.00 3.500 $99.77 99.77 3.58 3.5835 4.1735

I created the LIBOR forward rates simply because most IR Swaps use LIBOR for the floating leg.

2 Swap Info

Here are the contract details I'm looking for:

Notional ### Term (Years) 4Settlement Every 6mosFloating Rate LIBOR 6mo

3PricingSo 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 2.8800 0 0.0000 0 00.50 3.3429 $2,880,000 2.8800 0.9858 $2,839,1171.00 3.5697 $3,342,897 3.1113 0.9696 $3,241,2661.50 4.7513 $3,569,689 3.2640 0.9526 $3,400,4712.00 3.6905 $4,751,328 3.6348 0.9305 $4,421,0652.50 4.5024 $3,690,508 3.6459 0.9136 $3,371,7653.00 3.8673 $4,502,399 3.7884 0.8935 $4,022,9693.50 4.9659 $3,867,317 3.7997 0.8766 $3,389,9624.00 $4,965,916 3.9451 0.8553 $4,247,494

Total PV of Floating Payments ###

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:

LIBOR SpotRates

Assume we want to buy (go long) a swap, i.e. pay a fixed rate, receive f loating (LIBOR 6mo).

6mo LIBORFwd Rate

Cash FlowIn

LIBORSpot Rate

DiscountFactor

PV of CashIn

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.002

2.5

3

3.5

4

4.5

5

5.5

Yield Spot Rate 6mo Fwd Rate

LIBOR Spot Rates LIBOR 6mo Fwd Rate


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$28,934,109 = Pmt * Sum ( Discountfactors )$28,934,109 = Pmt * 7.3775

$3,921,922 = Pmt

Thus, the fixed interest rate is $3,921,922 / ###3.9219

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 Bus iness

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By Zachary Emig

2.88003.34293.56974.75133.69054.50243.86734.96595.15225.0273

LIBOR 6moFwd Rate


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