Building a Zero Coupon Yield Curve Clive Bastow, CFA, CAIA ABSTRACT Create and use a zero- coupon yield curve from quoted LIBOR, Eurodollar Futures, PAR Swap and OIS rates. www.elpitcafinancial.com
Risk- free rate... 3 Day Count Convention... 3 Creating the Zero- curve out to 1 year... 4 LIBOR Rates... 4 Example... 4 Eurodollar Futures... 5 Convexity Adjustment... 6 Extending the Zero- curve beyond 1 year... 8 Par Swap Rates... 8 Putting it all together... 11 Overnight Indexed Swap (OIS) Rates... 12 Forward Rates... 14 Example... 14 2
Risk- free rate Traditionally financial institutions used LIBOR rates as a proxy for risk- free rates. LIBOR is the rate AA- rated financial institutions can borrow short- term funds from other financial institutions. However, LIBOR is not totally free of credit risk as there is a small chance that the borrower will default during the borrowing period. LIBOR is quoted for maturities up to 1 year. We can use the quoted LIBOR rates to construct our initial zero- curve for maturities up 1 year. Day Count Convention Number of days between dates Number of days in reference period The day count conventions commonly used are: 1. Actual/Actual 2. 30/360 3. Actual/360 4. Actual/365 USD Libor is quoted Actual/360 basis. The interest for the USD LIBOR period is given as: Interest earned in reference period LIBOR Rate Actual number of days in period 360 The discount factor for USD LIBOR period is given as: 1 + LIBOR Rate 1 Actual number of days in period 360 In the crisis of 2007 LIBOR quotes included higher levels of credit risk and many financial institutions have moved to using the OIS (overnight index swap) rate as a proxy for risk- free rate. We will start by constructing a LIBOR based zero- coupon curve and then see how this changes once we consider Eurodollar, Par Swap and OIS data. 3
Creating the Zero- curve out to 1 year LIBOR Rates As there are no intermediate payments on LIBOR quotes, we can use them as starting point for obtaining our zero- rates (or spot rates). The PV of USD 1 received on the stated maturity for the LIBOR quotes give us our DF (discount factor) for that date. Example On 27 January 2017, USD 1- Month LIBOR is quoted as 0.77833% The actual number of days between 27 January 2017 and 27 February 2017 is 31. The interest per USD 1 notional for the 31- day period is USD 0.000670299 paid at the end of the term. The Discount Factor (DF) is 0.99933022. This represents the present value of USD 1 received on 27 February 2017 given the LIBOR interest rate. We can repeat the process for all LIBOR quotes out to 1 year to generate the following table. Term End Date LIBOR Rate Days DF Zero Overnight 28/01/17 0.69 1 0.999980834 0.006995766 1 Week 03/02/17 0.72011 7 0.999859998 0.007300604 1 Month 27/02/17 0.77833 31 0.99933022 0.007888758 2 Months 27/03/17 0.84222 59 0.998621598 0.008533287 3 Months 27/04/17 1.039 90 0.99740923 0.010520648 6 Months 27/07/17 1.35878 181 0.993214711 0.013729675 12 Months 27/01/18 1.724 365 0.982820838 0.017328436 The continuously compounded zero- rate (or spot rate) for each date above can be calculated using the following formula: ln 1 DF 365 Days One problem with using LIBOR rates is that credit risk becomes increasingly embedded within the quoted rate for longer durations. 4
Eurodollar Futures Eurodollar futures are contracts listed on the Chicago Mercantile Exchange (CME). A Eurodollar is a dollar deposited in a US or Foreign bank outside the United States. The Eurodollar interest rate is the rate of interest earned on Eurodollars deposited by one financial institution with another. The 3- month Eurodollar future is a contract on the interest that will be paid on a notional of USD 1 million for a future three- month period. Eurodollar futures are listed out to 10 years on the CME, although liquidity (and therefore informational value) drops for maturities beyond 3-4 years. Eurodollar futures are quoted as 1- R where R is the actual 3- month Eurodollar rate with quarterly compounding and actual/360 day count basis. Because the contracts are listed on an exchange and margining occurs daily the credit risk observed in the LIBOR rate is largely removed. Profits and losses are settled daily with the exchange acting as central counterparty. The following table shows the settlement prices for Eurodollar futures on 27 January 2017. Contract Settle Volume Settlement Mar 2017 98.91 106,997 13- Mar- 17 Apr 2017 98.855 2,025 13- Apr- 17 May 2017 98.8 3,603 15- May- 17 Jun 2017 98.745 134,341 19- Jun- 17 Jul 2017 98.715 308 17- Jul- 17 Sep 2017 98.62 120,896 18- Sep- 17 Dec 2017 98.475 152,856 18- Dec- 17 Mar 2018 98.355 142,299 19- Mar- 18 Jun 2018 98.23 113,717 18- Jun- 18 Sep 2018 98.12 119,641 17- Sep- 18 Dec 2018 98 158,920 17- Dec- 18 Mar 2019 97.915 115,654 18- Mar- 19 Jun 2019 97.835 78,067 17- Jun- 19 Sep 2019 97.75 68,111 16- Sep- 19 Dec 2019 97.66 110,406 16- Dec- 19 Mar 2020 97.61 49,451 16- Mar- 20 Before we use these quotes to extend our zero- curve we need to make a convexity adjustment to the implied Eurodollar rates, owing to the daily 5
settlement. The final settlement is on the Settlement date (listed in the table above) whereas for a forward the settlement would be 3 months later. Convexity Adjustment The convexity adjustment required is given by the formula: Where: Forward rate = Futures rate 1 2 σ! T! T! T1 is the time to maturity of the futures contract. T2 is the time to maturity of the rate underlying the futures contract. σ is the standard deviation of the change in short- term interest rates in 1 year (we will assume 1.2 bps for calculation purposes here). The first future matures exactly half way between our 1- month and 2- month LIBOR dates. We will use the average of the two rates calculated for LIBOR 1- month and 2- month to apply to the time between now and the maturity of the first Eurodollar future. Applying the convexity adjustment per the above table results in the following table of zero rates out to a maturity of 2.5 years. Contract Continuous Convexity 3m- Fwd Rate Zero Date Zero 1.5m Libor 13 Mar 17 0.008103 Mar 17 0.010885 0.00000 0.010882 19 Jun 17 0.010007 Jun 17 0.012530 0.00002 0.012512 18 Sep 17 0.010981 Sep 17 0.013776 0.00004 0.013734 18 Dec 17 0.011752 Dec 17 0.015221 0.00007 0.015146 19 Mar 18 0.012495 Mar 18 0.016416 0.00012 0.016299 18 Jun 18 0.013177 Jun 18 0.017661 0.00017 0.017493 17 Sep 18 0.013834 Sep 18 0.018756 0.00023 0.018527 17 Dec 18 0.014454 Dec 18 0.019950 0.00030 0.019652 18 Mar 19 0.015060 Mar 19 0.020796 0.00038 0.020419 17 Jun 19 0.015530 If we compare the zero rates generated from Eurodollar futures to those from the LIBOR rates we can observe the impact of credit risk on the LIBOR quotes directly. As can be seen on the graph below, once we get beyond 2- months the LIBOR rates (shown on the blue line) start to price in materially higher credit risk, and therefore does not produce true risk- free rates. Thus, we prefer to use the Eurodollar rates in creating our zero- curve out to 1 year. 6
We could continue to use Eurodollar futures out to March 2020 as there is adequate liquidity. 7
Extending the Zero- curve beyond 1 year Par Swap Rates PAR interest rate swap rates can now be used to extend the LIBOR yield curve beyond 1 year. On 27 January 2017, USD 2- year par swap was quoted at 1.516%. Swap rates are quoted on a semi- annual basis with a 30/360 day- count. In a semi- annual swap, half the coupon rate is paid every 6- months. Term in Years Coupon 0.5 0.758% 1 0.758% 1.5 0.758% 2 0.758% USD Swap rates were quoted on 27 January 2017 as follows: Tenor Par swap rate 1 1.261 2 1.516 3 1.725 4 1.887 5 2.018 6 2.125 7 2.214 8 2.287 9 2.348 10 2.401 The 1- year swap rate again demonstrates the credit risk embedded in the longer dated LIBOR rates. The swap rate represents the fixed leg payment (paid semi- annually) on a 1- year swap with the counterparty paying 3- month LIBOR settled every quarter. The swap therefore has a lower amount of credit risk then 12- month LIBOR where the entire repayment is at the end of the 12- month period. Because the associated credit risk is lower for the swap, the rate is lower. The 1- year swap rate is in- line with the rates implied by the Eurodollar futures. 8
The first step in extending the Zero- curve is to calculate what the 1- year DF should be given our knowledge of 6- month discount factor. We can use the following equation to calculate the coupon recognizing that the present value of the future swap payments must sum to 1.. 5 x Sc x DF6m + 0.5 x Sc x DF12m + 1 x DF12m = 1 Where: Sc is the par swap coupon DF6m is the 6- month discount factor DF12m is the 12- month discount factor From our Eurodollars work we can interpolate the 6m rate as 0.9947778. From this we calculate that the 1- year discount factor is 0.99372793. We can confirm this as follows: Term Cashflow Discount Factor PV 0.5 0.006305 0.9947778 0.00627207 1 1.006305 0.9875017 0.99372793 Swap NPV 1.00 We now have the 1- year discount factor. The next step is to interpolate a1.5- year swap rate and use this to calculate the DF for the 1.5 year point. The following formula will be used for interpolation: 1 yr Swap rate 2yr Swap rate!.! yielding a 1.5 year swap rate of 1.382633719. Again present value of the cashflows associated with the 1.5- year swap should sum to 1. We need to solve the following equation which will give us the 1.5 year discount factor:. 5 x Sc x DF6m + 0.5 x Sc x DF12m + 0.5 x Sc x DF18m + 1 x DF18m = 1 Plugging in the known variables and solving for DF18m gives us the 1.5 year discount factor of 0.97952455. We can then use this information and the 2- year swap rate to find the discount factor for the 2- year point. We can reiterate this process for each 6- month period using the swap rates and interpolated swap rates to complete our tale of discount factors out to 10- years. 9
The results of this process can be seen in the table below: Year Swap Coupon Discount Factor 1.00 1.2610000 0.98750173 1.50 1.3826337 0.97952455 2.00 1.5160000 0.97019544 2.50 1.6171271 0.96044151 3.00 1.7250000 0.94961229 3.50 1.8041826 0.93883024 4.00 1.8870000 0.92727354 4.50 1.9514010 0.91585546 5.00 2.0180000 0.90386373 5.50 2.0708090 0.89211098 6.00 2.1250000 0.87993829 6.50 2.1690436 0.86803581 7.00 2.2140000 0.85582667 7.50 2.2502040 0.84397354 8.00 2.2870000 0.83190926 8.50 2.3172993 0.82017931 9.00 2.3480000 0.80830743 9.50 2.3743521 0.79669782 10.00 2.4010000 0.78499228 10
Putting it all together Combining this data with the data we already have gives us DF factors from overnight out to 10 years from using LIBOR rates until the first Eurodollar contract, Eurodollars out to 2.5 years, and then swap rates. Date Discount Factor Zero Rate Date Discount Factor Zero Rate 28/01/17 0.999980834 0.006995657 27/07/20 0.938830238 0.01804152 03/02/17 0.999859998 0.007300615 27/01/21 0.927273538 0.018863749 27/02/17 0.99933022 0.007888762 27/07/21 0.915855465 0.019538551 13/03/17 0.99900153 0.008102747 27/01/22 0.903863728 0.020204264 19/06/17 0.996087018 0.010007273 27/07/22 0.892110982 0.020762396 18/09/17 0.992984674 0.010981273 27/01/23 0.879938291 0.02130752 18/12/17 0.989590384 0.011752073 27/07/23 0.868035808 0.021777253 19/03/18 0.985860595 0.01249451 27/01/24 0.855826666 0.022232358 18/06/18 0.981862515 0.013177425 27/07/24 0.843973538 0.022613755 17/09/18 0.977589722 0.013834113 27/01/25 0.831909264 0.022988242 17/12/18 0.973084494 0.01445398 27/07/25 0.820179311 0.023317688 18/03/19 0.968328493 0.015060413 27/01/26 0.80830743 0.02363148 17/06/19 0.964274233 0.015245162 27/07/26 0.796697816 0.023920743 27/01/20 0.94961229 0.017233831 27/01/27 0.784992279 0.024194882 We can now use the curve to calculate the forward price of USD 1 invested today at any time up to 10 years. The formula for the forward price (F) is given as: F = Se!" Where S is the present value, r is the present value and t is the time in years. Example To calculate the forward value of USD 1 invested on 27- January- 2017 maturing on 27- July- 2025 we use the formula with S = USD 1, r = 2.23317688% and t = (3013 days / 365 days per year). The forward value is therefore USD 1.21924558. We can confirm this by dividing the present value of USD 1 by the discount factor of 0.820179311. The concept of continuously compounded rates is extremely important for the valuation of derivatives. Once we have built our zero coupon curve we can use an interpolation method to obtain the zero rates for any given date. The market standard is to use the cubic spline method for interpolation. 11
Zero curve 0.03 0.025 0.02 0.015 0.01 0.005 0 As we have demonstrated, the zero- coupon curve constructed from LIBOR rates represents near- risk- free rates although these contain an increasing degree of credit risk the longer the maturity, inherent in the AA rating of the financial institution. We have improved on this by using Eurodollar futures to remove much of the credit risk element inherent in the LIBOR rates, creating the short- end of the curve out to 2.5- years. Since the financial crisis, there has been widespread use of collateralization through the adoption of Credit Support Annex s (CSA) attached to ISDA master agreements coupled with central clearing of derivatives to further mitigate counterparty credit risk. The standard practice in the market now is to determine discount rates from overnight indexed swap (OIS) rates when valuing all fully collateralized derivatives transactions. Overnight Indexed Swap (OIS) Rates An overnight indexed swap (OIS) is a swap where a fixed rate for a period (e.g., 1 month, 3 months, or 1 year) is exchanged for the geometric average of the overnight rates during the period. For OIS swaps of up to 1 year there is only a single leg resulting in a single payment at maturity (similar to the LIBOR rates) so the short- end of the zero curve is relatively simple to adjust to OIS rates. 12
Where the zero- curve is required for dates beyond 1 year the assumption that the LIBOR- OIS spread is constant for all maturities beyond the longest OIS quote can be made. (The LIBOR- OIS spread has become a widely used indicator for bank credit and liquidity risk.) Alternatively, basis swaps where 3- month LIBOR is exchanged for the average federal funds rate plus a spread can be used. These swaps have maturities out to 30 years in the US. The OIS rates for 27 January 2017 were as follows: Term OIS Rate Libor Rate Difference 3m 0.69 1.039 0.349 6m 0.76 1.35878 0.59878 1y 0.90 1.261 0.361 As can be seen the OIS rates could be used to create a zero- curve for collateralized transactions out to 1 year. To extend this zero- curve out beyond 1- year we could either track basis swaps, or use the par swap spreads and use a fixed LIBOR- OIS spread for all maturities beyond 1 year of 36.1 basis points. The zero- curve is a very important concept in finance. Once we have built our curve we can calculate the present value, forward values, implied forward rates for any term and use it to value bonds, swaps, forwards, futures and derivatives. Understanding and building the zero- curve is the first step in understanding how to model and value derivatives. 13
Forward Rates Using our zero curve we can now calculate a forward rate for any period in the future. Example We need to calculate the forward rate from 27 January 2018 to 27 January 2019. We know that USD 1 invested today will grow to USD 1.01265646 by 27 January 2018 using the continuous zero- rate (or spot rate) for that date. We also know that USD 1 invested today will grow to USD 1.3072016 by 27 January 2018. We can get calculate these directly through our earlier formula: F = Se!" We can calculate the value of the 12- month forward rate, from 27 January 2018 to 27 January 2019 through the following formula. Where: S is USD 1 F = Se!!!! e!!!! r1 is the zero (spot) rate from the current period to 27 January 2018 T1 is the time from the current period to 27 January 2018 r2 is the forward rate from 27 January 2018 to 27 January 2019 T2 is the time from 27 January 2018 to 27 January 2019 We know the value of F already so we can rearrange the formula to solve to r2 as follows: r! = ln F Se!!!! T! 365 The value for r2 is 1.7680707%. This is the 1- year forward rate starting from 27 January 2018 to 27 January 2019. 14