CREATING A CORPORATE BOND SPOT YIELD CURVE FOR 

PENSION DISCOUNTING 

DEPARTMENT OF THE TREASURY 

OFFICE OF ECONOMIC POLICY 

WHITE PAPER 

FEBRUARY 7, 2005 

I. Introduction 

Plan sponsors, plan participants and financial markets should be able to depend on accurate 

measurement of pension liabilities. Accuracy requires that the discount rates used in 

calculation of the present value of a plan’s benefit obligations satisfy two criteria: the discount 

rates must reflect the timing of the future payments, and they should be based on current 

market-determined interest rates for similar obligations. The Administration proposes1

 to 

replace the current law method with a schedule of rates drawn from a spot yield curve of high 

grade corporate bonds.2

 Discounting future benefit cash flows using the rates from the spot 

yield curve is the most accurate way to measure a plan’s liability because, by matching the 

maturity of the discount rate with the timing of the obligation, it properly computes today’s 

cost of meeting that obligation. Spot yield curves are used routinely by financial market 

participants to value future cash flows, for example, those of loans, mortgages, bonds, and 

swaps. 

The Administration also proposes to require plans offering lump sum payments to use the same 

spot yield curve for both measuring pension liabilities and determining the minimum value of 

the lump sum payments. Changing the interest rate used to determine the lower bound for 

lump sums to the same market-determined spot yield curve as used in the liability calculations 

would ensure that the minimum lump sum is the same as the value of the pension annuity. 

Sponsors would still be free, as they are under current law, to be more generous if they wished. 

In such cases, it is important that the calculation of liabilities explicitly take into account the 

probability that future payments will be made in the form of a lump sum. 

The purpose of this paper is to describe the methodology that will be used to derive the spot 

yield curves in the Administration’s proposal. The paper begins by describing the proposed 

methodology for calculating spot yield curves; it then provides illustrative examples of the 

resulting curves. The results are presented in a series of charts, including a chart illustrating 

the evolution of a 90-business-day moving average spot yield curve over time. As the charts 

show, the 90-day average curves are smooth over the maturity range without excessive 

 

1

 See www.dol.gov/ebsa for a complete description of current law relating to valuing pension liabilities, the reason 

for change, and the Administration’s proposal. See also the General Explanations of the Administration’s FY 

2006 Revenue Proposals (Blue Book) at http://www.treas.gov/offices/tax-policy/library/bluebk05.pdf. For the 

underlying rationale for the use of the corporate bond yield curve in pension discounting, see the testimony of 

Mark J. Warshawsky, Assistant Secretary for Economic Policy, before the Special Committee on Aging, U.S. 

Senate, October 14, 2003. 

2

 A spot yield curve is simply a table or a graph that reports a set of yields at a particular point in time on single 

payment bonds (zero coupon bonds) of different maturities. A single payment bond (zero coupon bond) provides 

a single payment at a future time when the bond matures. 
2

fluctuations and evolve smoothly over time without distortion by very short-term market 

movements. They accurately reflect high grade corporate bond markets, while minimizing the 

idiosyncratic impact of individual issues and the impact of short-term market volatility. 

In order to produce spot yield curves that measure the market as accurately as possible, the 

estimation methodology should correct for influences on yields due to special characteristics of 

bonds that are not related to their underlying risk. In particular the presence of embedded 

options was found to have a significant effect on prices. The estimation methodology 

incorporates an appropriate correction for this factor. A technical appendix explains the details 

of the methodology. 

 

Treasury would compute the spot yield curves and would make the calculated curves available 

on a regular basis for use in pension discounting. Pension plans would apply the spot yields of 

each maturity to discount the future expected benefit cash flow at that maturity. 

II. Implementing the Proposal 

The spot yield curve is not directly observable, but must be estimated from data on high quality 

corporate bonds. This section presents the criteria that guide the choice of estimation 

methodology for spot yield curves and a description of the methodology. 

Criteria for Choice of Methodology

The methodology for estimating spot yield curves for use in pension discounting should meet 

the following criteria: 

o It should be as transparent in construction as possible, using accepted, logical, and 

accurate techniques and assumptions, enabling rates to be accepted by employers, plan 

beneficiaries, and the investment community. 

o It should accurately reflect high grade corporate bond markets while minimizing 

idiosyncratic impact of individual issues and the chance of manipulation by an issuer or 

source of data, by drawing upon data that are as broad and deep as possible at each 

maturity. 

o It should use readily and reliably available data that are updated frequently, and likely to 

remain available over a reasonably extended period. 

o The resulting spot yield curves should be smooth over the maturity range. Furthermore, 

the curves should evolve smoothly over time, so that they reflect changing conditions in 

established financial markets without inducing excessive short-term market volatility. 
3

Methodology 

Spot yield curves are estimated directly from data on corporate AA bonds. The process 

incorporates unbiased adjustments for bonds with embedded call options, and allows for 

unbiased projections of yields beyond a 30-year maturity.3

 

Estimation. The price of a bond is expressed as the sum of the cash flows discounted to the 

present, with an added factor adjusting for the price impact of any embedded call options. The 

discount function is estimated by a nonlinear least squares process, and the spot yield curve is 

then calculated. The yield curve is projected beyond maturities of 30 years using the estimated 

discount function. Par yield curves4

 are also presented for comparison with actual market 

yields. See Appendix 1 for details of the method. 

Data. The results in this paper are drawn from data on daily corporate AA bond price and par 

amount outstanding obtained from Merrill Lynch.5

 The data include only corporate bonds 

with fixed coupons and no embedded options other than calls, with maturities of 30 years or 

less, and par amounts outstanding of at least $250 million. To capture behavior at maturities 

less than a year, the bond price data are augmented with data from the Federal Reserve Board 

for AA commercial paper. 

Averaging over a 90-Day Period. As indicated above, the proposal includes a provision that 

the spot yields be averaged over a 90-business-day period. The averaging should reduce the 

impact of day-to-day market volatility in the liability measures and should help make 

contribution requirements more predictable while not compromising the accuracy of liability 

measures. The results are presented for both single-day yield curves and curves averaged over 

a 90-business-day period. 

Reflecting Lump Sums in the Liability Calculation. Because the minimum value of lump sum 

payments would be the same as the present value of the underlying pension annuity, the 

mechanics of the liability calculation for plans offering the minimum can proceed by 

discounting all future expected pension annuity cash flows to the present using the current spot 

yield curve. By contrast, in the case of a more generous sponsor, the calculation of liabilities 

must explicitly take the lump sum behavior into account. For example, a sponsor may offer an 

additional cash payment to early retirees taking lump sums, along with the lump sum generated 

as the present value of the pension annuity. In this example, the liability calculations should 

 

3 One other public methodology, familiar to pension analysts, is the Citigroup Pension Discount Curve, originally 

developed in response to the 1993 SEC guidance on discount rates under SFAS 87 and SFAS 106. The Citigroup 

approach uses Treasury par curve data to set the shape of the yield curve and calculates the AA corporate spot 

yields at each maturity by adding to the Treasury par curve an average of the difference between corporate and 

Treasury par yields within each of five maturity ranges. The Citigroup curves do not extend beyond a maturity of 

30 years and thus are not easily usable for discounting pension liabilities which can extend much beyond that 

maturity. The Citigroup Pension Discount Curve is published at each month end, in tabular form, at 6-month 

maturities between 0.5 years and 30 years. The month-end curves are available back to September 30, 1995. For 

further details, see Lawrence Bader, “Discounting Pension Liabilities under the New SEC Rules”, Pension Section 

News, June 1994; see also Bader and Ma, “The Salomon Brothers Pension Discount Curve and the Salomon 

Brothers Pension Liability Index, 1995 Update”, Salomon Brothers, January 1995. The Citigroup Pension 

Discount Curve and the two papers are available on the Society of Actuaries website, at http://www.soa.org. 4

 Par yield curves show the yields to maturity of coupon-paying bonds selling at par, at each maturity. 5

 Price data from several additional sources will be used in the preparation of published yield curves. 
include the value of the expected cash payments, discounted to the present using the spot yield 

curve. 

III. Results

The impact of the proposal is illustrated in the charts presented in this section. 

Results for a Single Date 

Charts 1 and 2 show the results for December 30, 2004. Chart 1 shows the calculated spot 

yield curve for that date. The curve starts below 3 percent and increases to between 6 and 7 

percent over the projection period out to 80 years. The curve is consistent with the typical 

shape of yield curves. 

CHART 1

0 10 20 30 40 50 60 70 80

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

SPOT YIELD CURVE

CORPORATE AA BONDS

12/30/04, Percent

Maturity, Years

4
Chart 2 shows the estimated par yield curve, the yields to maturity on coupon-paying bonds 

selling at par, for the same day, with the actual yields of the securities in the sample plotted as 

dots around it. The diamonds on the chart are the yields of two securities with maturities 

beyond 30 years and thus not included in the estimation process. As can be seen, the data are 

well approximated by the estimation process. 

CHART 2

0 20 40 60 80 100

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

PAR YIELD CURVE

CORPORATE AA BONDS

12/30/04, Percent

Maturity, Years

Averaging over a 90-Day Period 

Chart 3 shows the 90-business-day average spot yield curve for December 30, 2004. This 

curve is calculated by averaging the spot yields at each maturity from spot yield curves

constructed for the 90 business days up to and including December 30, 2004. The average 

curve rises smoothly from about 2 percent to between 6 and 7 percent at the longest maturities. 

5
To facilitate comparison with current law for plan years beginning in 2004 or 2005, the chart 

also shows the current and 4-year weighted average corporate bond rate, and the current and 

105 percent of the 4-year weighted average rate on 30-year Treasury securities.6

 The 90-

business-day average spot yield curve is presented as a table in Appendix 2. 

CHART 3

0 10 20 30 40 50 60 70 80

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

SPOT YIELD CURVE

CORPORATE AA BONDS

12/30/04, Percent

Average of 90

Business Days

Maturity, Years

Corporate:

100% Moving Average

100% Current

30-Year Treasury:

105% Moving Average

100% Current

Evolution over Time 

Chart 4 is a three-dimensional chart illustrating the stability of the 90-day average spot yield 

curves from January 2000 through December 2004. The average spot yield curves at each date 

are shown as front-to-back slices with interest rates on the vertical axis and maturities on the 

axis extending into the paper, away from the reader. The average curves are arrayed by date 

on the axis that runs left to right. 

6

 

6

 The 4-year Corporate Bond Weighted Average rate for December 2004 is 6.10 percent; 105 percent of the 30-

year Treasury Weighted Average rate is 5.35 percent. 
The evolution of the 90-day average spot yield curves illustrated in the chart accurately tracks 

(with a slight lag due to the moving average process) the general pattern of corporate bond 

markets over this period. The shortest maturity spot yields begin at between 5 and 6 percent in 

January 2000, rise somewhat through 2000, decline to around 1 percent over most of the 

following period through mid-year 2004, and increase to about 2 percent by yearend 2004. 

The long maturity yields begin above 8 percent in January 2000, dip to under 7.5 percent by 

2002, and recently are below 7 percent. As the chart indicates, the 90-day average spot yield 

curves evolve smoothly across these five years, yet still reflect important market movements. 

CHART 4 

7
Comparison of Liability Values over Time

Charts 5 and 6 compare pension liabilities using the yield curve to liabilities under current law, 

for a mature plan (short duration distribution of benefits) and a young plan (long duration 

distribution of benefits), at yearends 2001-2004. Chart 5 shows the ratio of the liability 

calculated using the yield curve to that calculated using the 4-year weighted average corporate 

rate in current law for both plans. 

As the chart shows, the mature plan’s liabilities calculated using the yield curve are uniformly 

larger than those using the rates in current law (the ratio is uniformly above 100 percent). This 

unsurprising result is due to the fact that the bulk of the mature plan’s cash outflows occur 

relatively early and the yield curve shorter-term rates are lower than the current law rate. 

The young plan’s liabilities under the yield curve are larger than those using current law at the 

beginning of the period but fall slightly below current law liabilities in 2004. This result is due 

to the fact that the bulk of the young plan’s liabilities occur at long horizons, with the result 

that the liabilities are primarily determined by the behavior of the long-term spot rates. The 

long-term rates in the yield curve are below the current law rate in 2001, resulting in higher 

liabilities compared to those under current law, then gradually rise above it, resulting in lower 

liabilities. 

CHART 5 

2001 2002 2003 2004

95

100

105

110

115

95

100

105

110

115

Mature Young

90-Day Yield Curve Average over Current Law

(4-Year Moving Average of Bond Index)

RATIO OF PENSION LIABILITIES

End of Year, Percent

8
The consecutive ratios are relatively stable. Compared to current law, the liabilities calculated 

using the yield curve evolve smoothly over time without much variation from year to year, 

notwithstanding the significant drop in short-term rates over the period. 

The comparisons in Chart 5 reflect the fact that the current law interest rate is based on an 

average of interest rates over the prior 48 months. As a result the current law calculation of 

liability reflects significantly higher out-of-date rates. It is helpful to focus just on the effects 

of an upward sloping yield curve (as opposed to the effects of smoothing). Chart 6 presents a 

comparison of liabilities calculated using the yield curve and using the corporate bond index 

rate from current law but averaged over the preceding 5 months – similar to the 90-businessday average for the yield curve. This comparison more clearly illustrates the impact from the 

use of the yield curve versus a single rate. 

CHART 6 

2001 2002 2003 2004

80

90

100

110

120

80

90

100

110

120

Mature Young

RATIO OF PENSION LIABILITIES

End of Year, Percent

90-Day Yield Curve Average

over 5-Month Bond Index Average

9
10

As Chart 6 shows, the mature plan’s liabilities calculated using the yield curve are consistently 

slightly higher than those using the 5-month average corporate bond index rate. Again these 

results reflect the fact that the single corporate index rate is higher than the shorter-term rates 

under the yield curve that are matched to the shorter-term cash flows that are the bulk of the 

mature plan’s cash flows. The young plan’s liabilities are uniformly lower using the yield 

curve than using the 5-month average corporate bond index rate, reflecting the fact that the 

corporate rate is lower than the long term yield curve rates appropriate to the bulk of the young 

plan’s cash flows. 

IV. Conclusion: Choosing Rates for Pension Discounting 

The 90-day average spot yield curves constructed as described in this paper can well serve as 

rates for pension discounting. They are constructed using accurate techniques and 

assumptions. They are based on readily and reliably available data from known sources. They 

accurately reflect high grade corporate bond markets, while minimizing the idiosyncratic 

impact of individual issues and the impact of short-term market volatility. Finally, as can be 

seen from the charts, they are smooth over the maturity range without excessive fluctuations 

and evolve smoothly over time without distortion by very short-term market movements. 

 
11

APPENDIX 1

COMPUTING THE SPOT YIELD CURVE7

The following notes sketch out technical details about the data and estimation methodology 

used to compute the spot yield curves in the proposal. 

The spot yield curves used in the proposal are estimated using data on daily corporate AA bond 

price and par amount outstanding obtained from Merrill Lynch and data on bond 

characteristics such as coupon and callability. The dataset includes only corporate bonds with 

semi-annual fixed coupons, maturities of 30 years or less, and amounts outstanding of at least 

$250 million. Bonds with special characteristics such as floating rate coupons are not included. 

Bonds with regular embedded call options but no embedded options other than calls are 

included. To capture behavior at maturities less than a year, the bond price data is augmented 

with data for AA commercial paper from the Federal Reserve Board. For a small number of 

days, the absence of a sufficient number of quotes that fulfill these criteria meant that that day 

was eliminated. Although the total number of quotes in each daily sample has varied over 

time, recently it has been a little above 200. The number of bonds with regular call options has 

varied from about 10 in 2001 to around 5 recently. 

To estimate the spot yield curve from the bond prices it is first postulated that the prices in the 

AA market are characterized by a discount function, which for each maturity gives today’s 

price for $1 received at that maturity through investment in AA quality securities. The spot 

yield curve is calculated from the discount function; in effect, allowing for maturity, the two 

curves are the inverse of each other, since discounting future payments by the spot rate from 

the curve is the same as applying the discount function to the future payments. The par yield 

curve, which shows the yield at each maturity of a bond paying coupons and trading at par, can 

also be derived from the discount function. 

Estimating the Discount Function 

A specific functional form of the discount function must first be chosen for estimation. In this 

methodology the discount function is expressed in terms of the instantaneous forward interest 

rate, and then the forward rate is modeled as a cubic spline. The knot points for the spline are 

chosen at maturities of 0, 1.5, 3, 7, 15, and 30 years, because these knots appear to work well 

in dividing the market into maturity segments. In addition, the spline is constrained to have a 

zero derivative at the 30-year knot to make it taper off smoothly, and to have a zero second 

derivative at the zero knot for increased stability at the short end. Finally, the forward rate 

spline is constrained at the 30-year knot to be equal to its mean value over the range from 15-

30 years. 

The bond price given by the present value of cash flows is then adjusted for the value of any 

call option embedded in the bond by expressing the price of the bond as the sum of the 

 

7

 This methodology was developed by James A. Girola, with participation by Lucy Huffman, both of the Office of 

Economic Policy, under the general supervision of Mark J. Warshawsky, Assistant Secretary for Economic 

Policy. 
12

 

discounted cash flows and the value of the call.8 The value of the call is modeled, in an 

unbiased way, as a coefficient multiplied by the length of time of call exposure, the time that 

the investor is exposed to the risk that the bond will be called.9

 For a bond with a nondiscrete 

call that has not yet reached its first call date (or for bonds with discrete calls that are between 

call dates), the call exposure time is the time from the next call date to maturity.10 For a bond 

with a nondiscrete call that is callable currently, the call exposure time is the time until 

maturity. 

The spline and the call exposure coefficient are jointly estimated by nonlinear least squares, 

that is, the sum of squared errors between predicted and actual sample prices (including 

accrued interest) is minimized.11 The data are weighted by the square root of outstanding par 

value to assure that the estimated yield curve reflects the movement of the market as a whole. 

The data are further weighted by the inverse of the square root of duration, to correct for 

heteroscedasticity. To facilitate the estimation process, the spline is written in terms of Bspline bases. To obtain a more refined estimate of the discount function, the estimation takes 

account of coupon and principal payment dates that fall on a weekend or holiday. 

The yield curve is projected beyond maturities of 30 years using the estimated splines. 12 

Because the forward rate spline at 30 years is constrained to be equal to its mean over years 15-

30, and its derivative is constrained to be zero at 30 years, the forward rate for maturities 

longer than 30 years is equal to the forward rate at 30 years. The projected spot rate gradually 

converges to a constant depending on this forward rate. The final step is to compute a 90-

business-day moving average of the spot yield curves. 

In general, the estimation proceeds very smoothly over the last several years, and the daily 

yield curves, as well as the 90-business day average curves, evolve along a stable path. In 

particular, the projection beyond 30 years produces results that are simple and unbiased, are 

smooth over the entire maturity range, and are consistent over time. Empirically, the stability 

8

 A number of models for the value of the embedded call option have been proposed in the academic literature, 

including the exact continuous Black-Scholes formulation, and lattice-computed discrete extensions. The 

resulting yield curves from the linear regression model appear to be more stable over time. Tao-Hsien Dolly 

King, “An Empirical Examination of Call Option Values Implicit in U.S. Corporate Bonds”, Journal of Financial 

and Quantitative Analysis, Vol. 37, December 2002, also estimates the value of embedded calls using a regression 

model. 

9

 The effects of several variables related to call option values were analyzed, including the exercise price, the call 

exposure time, and the time until the first call date for those bonds in the call protection period (the period before 

the first call date). The only variable consistently significant and of the correct sign was the call exposure time. 

Since all bonds with embedded calls in the data are rated AA, measures of volatility were not analyzed. 

10 A bond with a nondiscrete call is one that is callable at any time after the first call date at the terms specified in 

the call schedule. A bond with discrete call(s) is callable only on the date(s) specified in the call schedule. 

11 The impact of other factors was examined, including liquidity and tax effects, but only the call exposure time 

measure was found to be consistently effective over time. Elton et al, “Factors Affecting the Valuation of 

Corporate Bonds”, New York University, October, 2000, have examined the impact of different factors. 12 Some academic research has theorized on the behavior of spot rates for long maturities. Dybvig and Marshall, 

“Pricing Long Bonds: Pitfalls and Opportunities”, Financial Analysts Journal, January-February, 1996, suggest 

influences, including long range default risk, which would tend to increase long term spot rates, and interest rate 

risk of comparable Treasury bonds, which would result in a negative impact on spot rates related to convexity. 

They and market practitioners also cite the presence of investors with a strong preference for long-term bonds, 

which would tend to lower rates. Dybvig, Ingersoll and Ross, “Long Forward and Zero-Coupon Rates Can Never 

Fall”, Journal of Business, 1996, vol. 69, No. 1, argue that very long term spot rates should converge to a limit. 

The proposal’s projection beyond 30 years is an unbiased distillation of all these theoretical influences. 
13

appears to be aided by holding the knots fixed over time at reasonable values approximating 

maturity segmentation in the AA market, and by constraining the spline at 30 years as 

described. 
14

APPENDIX 2

SPOT YIELD CURVE, CORPORATE AA BONDS 

12/30/04, Percent 

Average of 90 Business Days 

Maturity, Spot Maturity, Spot Maturity, Spot Maturity, Spot

Years Rate Years Rate Years Rate Years Rate

0.5 2.33 20.5 5.99 40.5 6.52 60.5 6.69

1.0 2.59 21.0 6.01 41.0 6.52 61.0 6.70

1.5 2.84 21.5 6.04 41.5 6.53 61.5 6.70

2.0 3.07 22.0 6.06 42.0 6.54 62.0 6.70

2.5 3.27 22.5 6.08 42.5 6.54 62.5 6.71

3.0 3.44 23.0 6.10 43.0 6.55 63.0 6.71

3.5 3.61 23.5 6.13 43.5 6.55 63.5 6.71

4.0 3.76 24.0 6.15 44.0 6.56 64.0 6.71

4.5 3.89 24.5 6.16 44.5 6.57 64.5 6.72

5.0 4.03 25.0 6.18 45.0 6.57 65.0 6.72

5.5 4.15 25.5 6.20 45.5 6.58 65.5 6.72

6.0 4.27 26.0 6.22 46.0 6.58 66.0 6.72

6.5 4.38 26.5 6.23 46.5 6.59 66.5 6.73

7.0 4.48 27.0 6.25 47.0 6.59 67.0 6.73

7.5 4.58 27.5 6.26 47.5 6.60 67.5 6.73

8.0 4.68 28.0 6.28 48.0 6.60 68.0 6.73

8.5 4.77 28.5 6.29 48.5 6.61 68.5 6.74

9.0 4.86 29.0 6.30 49.0 6.61 69.0 6.74

9.5 4.94 29.5 6.32 49.5 6.61 69.5 6.74

10.0 5.02 30.0 6.33 50.0 6.62 70.0 6.74

10.5 5.09 30.5 6.34 50.5 6.62 70.5 6.75

11.0 5.16 31.0 6.35 51.0 6.63 71.0 6.75

11.5 5.23 31.5 6.36 51.5 6.63 71.5 6.75

12.0 5.29 32.0 6.37 52.0 6.64 72.0 6.75

12.5 5.35 32.5 6.38 52.5 6.64 72.5 6.75

13.0 5.41 33.0 6.39 53.0 6.64 73.0 6.76

13.5 5.46 33.5 6.40 53.5 6.65 73.5 6.76

14.0 5.51 34.0 6.41 54.0 6.65 74.0 6.76

14.5 5.56 34.5 6.42 54.5 6.66 74.5 6.76

15.0 5.60 35.0 6.43 55.0 6.66 75.0 6.76

15.5 5.65 35.5 6.44 55.5 6.66 75.5 6.77

16.0 5.69 36.0 6.45 56.0 6.67 76.0 6.77

16.5 5.73 36.5 6.46 56.5 6.67 76.5 6.77

17.0 5.77 37.0 6.47 57.0 6.67 77.0 6.77

17.5 5.80 37.5 6.47 57.5 6.68 77.5 6.77

18.0 5.84 38.0 6.48 58.0 6.68 78.0 6.78

18.5 5.87 38.5 6.49 58.5 6.68 78.5 6.78

19.0 5.90 39.0 6.50 59.0 6.69 79.0 6.78

19.5 5.93 39.5 6.50 59.5 6.69 79.5 6.78

20.0 5.96 40.0 6.51 60.0 6.69 80.0 6.78