Evaluation of the lookback method for discount rate v 01.12.17 [v2]

EVALUATION OF THE
LOOKBACK METHOD FOR
INFLATION, BOND AND
DISCOUNT RATE ESTIMATES
LAMBDA INTERNATIONAL CONSULTANTS
HOUSTON, TX

Objective: to evaluate the quality of inflation,
interest and discount rate projections, among
others, using a straight, symmetrical Lookback
Method
■ Inflation and interest rates are basic economic indicators, together they construct Real
Discount Rates
■ Real Discount Rates are used in many economic forecasts and projections; legal cases
do not have ‘equity risk premiums’ but rely on the difference between the yield on
government securities and inflation
■ Real Discount Rates are constructed from the netting of inflation rates from bond rates
in our analysis:
– US 1 year bond yields used from 1953-2016
– CPI-U from Treasury Department
– Formula for Discount Rate:
■ 1- (1+1 Yr Bond Yield Percent)/(1+Inflation Rate Percent)
■ Lookback periods established symmetrically with projection horizons; ie. 20 year
projection horizon would be based on averages from previous 20 year lookback period
■ 1, 3, 5, 10 and 20 year lookback and projection horizon periods were used; 30 year data
was available but insufficient for analysis (n=4) except for Inflation
(c) Lambda International Consultants, LLC 2

Methodology
■ Using the symmetrical lookback
method and data from 1953-2016,
(for the base calculations)
projections were established for
average inflation rates and US 1 yr.
bond rates to calculate projected
inflation, bond and Discount Rates
■ These were then back-tested against
real results for the horizon periods
1) Establish averages for each year for
bond yields and inflation from
1953-2016 using 1, 3, 5, 10 and
20 yr. lookback Periods
2) Calculate Projected Average
Discount Rates for each year and
each of the horizon periods
symmetric with the lookback
periods
3) Calculate the Real Average
Discount Rates for the years and
horizon periods
4) Calculate the degree of error in the
estimates vs the Real Average
Discount Rates that would have
been in effect

a. Symmetrical Methodology
CURRENT YEAR NOT COUNTED IN ‘Y’ PERIOD, BUT USED IN CALCULATION OF AVERAGES
Y=1,3,5,10, 20, (30 YEARS NOT USED BECAUSE OF LACK OF DATA)
Y YEARS BACK
Y YEARS
FORWARD
LOOKBACK PERIOD
Compare
estimates with
ex post facto
reality to arrive
at average error
rates,
distribution and
skewness of
errors
SYMMETRICAL
PERIODS
PROJECTION
HORIZON PERIOD
(1,3,5,10,20)

a. Symmetrical Methodology
Symmetrical estimate uses 10 year data to project 10 years forward

b. Stretching Methodology
Z= X * (1,3,5,10, 20) WHERE X IS BETWEEN 2 AND 3
Y=1,3,5,10, 20
Z YEARS BACK
Y YEARS
FORWARD
LOOKBACK PERIOD
Compare
estimates with
ex post facto
reality to arrive
at average error
rates,
distribution and
skewness of
errors
ASYMMETRICAL
PERIODS
PROJECTION
HORIZON PERIOD
(1,3,5,10,20)

b. Stretching Methodology
Stretched estimate uses 10 year data to project 5 years forward

c. Smoothing Methodology
W = CURRENT YEAR MINUS 1953
W YEARS BACK TO THE
BEGINNING OF DATA
SERIES (1953)
Y YEARS
FORWARD
LOOKBACK PERIOD
PROJECTION
HORIZON PERIOD
(1,3,5,10,20)
Compare
estimates with
ex post facto
reality to arrive
at average error
rates,
distribution and
skewness of
errors
ASYMMETRICAL
PERIODS

c. Smoothing Methodology (2)
Smoothed estimate is the accumulated un-weighed average for the entire
data series to the date of projection – 1953 to X year

Long term projections
are impacted by the
peak in the late
70’s/early 80’s in all
rates; shorter term is
impacted by 9/11 &
2009/10 crises
Long term projections still have to
include the possibility of another
inflationary spike caused by internal
or external factors; war, energy
crisis, etc.
(c) Lambda International
Consultants, LLC
10
The Challenge
OVERALL RATES
BREAK IN
OVERALL
TREND
OVERALL
TREND
CHANNEL

Using a Symmetrical Lookback period, computing Projected vs Real
Discount Rate estimates based on 1953-2016 results in a lag in
Estimated vs Real yet within a tight range of values for 10 and 20 yrs
Number of Observations
■ 1 yr= 63
■ 3 yr = 58
■ 5 yr = 54
■ 10 yr = 44
■ 20 yr = 24
Discount Rate =
1+ Bond Rate /
1+ Inflation Rate
DISCOUNT RATES
REAL DISCOUNT RATES PROJ. DISCOUNT RATES

Using a Symmetrical Lookback period, computing Projected vs Real
Discount Rate estimates based on 1953-2016 data yields R2’s in the
0.80 to 0.90 range
■ 1 yr= 63
■ 3 yr = 58
■ 5 yr = 54
■ 10 yr = 44
■ 20 yr = 24
Discount Rate =
1+ Bond Rate /
1+ Inflation Rate
DISCOUNT RATES

Symmetrical 20 Year Real Discount Rate Error Estimates based on
1953-2016 data remain constant within a +3 to -2 % range, but declines
in Real Discount Rates post 1990 drive an increase in absolute
Percentual Error
■ 20 yr = 24
DISCOUNT RATES
Not enough data for 30 year D’Rate analysis

The analysis may indicate a smile
curve phenomenon: short periods
have higher accuracy because of
serial correlation in economic data;
middle periods (+/- 5 yrs) lose
serial correlation without gaining
the benefit of large numbers; longer
projection periods (10+ yrs) gain
accuracy because of
counterbalancing errors and a
larger base of data with
Symmetrical Method…
Discount Rate Estimates
1 yr 3 yr 5 yr 10 yr 20 yr Average
Average error in D'rate estimate (R-P)/R 21.73% -36.09% 110.43% -56.79% -43.54% -0.85%
R2 (real vs proj) 0.76 0.38 0.10 0.07 0.92 0.45
St dev (real vs proj 1.93% 1.75% 1.58% 1.24% 0.74% 1.45%
Average proj D'rate 1.36% 1.53% 1.68% 1.91% 1.56% 1.61%
Average real D'rate 1.34% 1.41% 1.45% 1.52% 1.92% 1.53%
Coefficient of Variation (StDev/Avg) 144.04% 124.33% 108.70% 81.39% 38.25% 99.34%
Average error D'rate x average real D'rate 0.29% -0.51% 1.60% -0.86% -0.84% -0.06%
Average error D'rate/ Num. years projected 21.73% -12.03% 22.09% -5.68% -2.18% 4.79%
Observations 63.00 58.00 54.00 44.00 24.00
DISCOUNT RATES
SYMMETRICAL

..due to the ‘Law of Large Numbers’
the 20 year projections beat shorter
time horizons…
DISCOUNT RATES

… thus projection
channels created
by +/- Two
Standard
Deviations narrow
as the time horizon
lengthens
■ Number of Observations for
Analysis
■ 1 yr= 63
■ 3 yr = 58
■ 5 yr = 54
■ 10 yr = 44
■ 20 yr = 24
(c) Lambda International
Consultants, LLC
16
DISCOUNT RATES

Yet for the period under study, only 55% of estimates came within +/-
50% of the correct D’Rate result for Symmetrical Method Average of
Projection Periods
DISCOUNT RATES
55% WITHIN
+/- 50%

Stretching the lookback period vs the
horizon period improved +/- 50% results
vs Symmetrical periods for D’Rates
Lookback periods were stretched:
1 yr: 3 yr
3 yr: 5 yr
5 yr: 10 yr
10 yr: 20 yr
DISCOUNT RATES
StretchedSymmetical
55%
to
75%

Stretching the lookback period vs the horizon
period improved Average Absolute Error vs
Symmetrical periods for D’Rates
1 yr: 3 yr
3 yr: 5 yr
5 yr: 10 yr
10 yr: 20 yr
DISCOUNT RATES

Smoothed Lookback D’Rate projection is also
better than Standard Symmetrical at most
Projection Horizons for D’Rates
DISCOUNT RATES
For most Horizons Smoothed seems to perform better within +/- 50 % Errors
55%
to
63%
Smoothed
Symmetical

Over a long time frame, the Smoothed
Projection delivers results around a 40%
margin of error
38% Mean of
Rolling Average
DISCOUNT RATES

Smoothed Lookback projection is better at the 10 - 20
yr Projection Horizon while Stretched does better in
mid range horizons
DISCOUNT RATES
STRETCHED 1 yr 3 yr 5 yr 10 yr Average
Average error in D'rate smth est (R-P)/R 30.57% -30.49% 18.02% -70.27% -13.04%
R2 (real vs proj) 0.82 0.86 0.55 0.78 0.75
St dev (real vs proj 1.81% 1.56% 1.23% 0.91% 1.38%
Average proj D'rate 1.50% 1.68% 1.91% 1.56% 1.67%
Average real D'rate 1.37% 1.52% 1.81% 1.79% 1.62%
Coefficient of Variation (StDev/Avg) 132.21% 102.56% 68.22% 51.00% 88.50%
Average error D'rate x average real D'rate 0.42% -0.46% 0.33% -1.26% -0.24%
Average error D'rate/ Num. years projected 30.57% -10.16% 3.60% -7.03% 4.25%
Observations 59.00 54.00 44.00 24.00
SMOOTHED 1 yr 3 yr 5 yr 10 yr 20 yr Average
Average error in D'rate smth est (R-P)/R 112.05% 22.92% 160.67% -22.41% -27.60% 49.13%
R2 (real vs proj) 0.01 0.02 0.11 0.45 0.81 0.28
Average error D'rate x average real D'rate 1.50% 0.32% 2.34% -0.34% -0.53% 0.66%
Average error D'rate/ Num. years projected 112.05% 7.64% 32.13% -2.24% -1.38% 29.64%
Observations 63.00 58.00 54.00 44.00 24.00
SYMMETRICAL 1 yr 3 yr 5 yr 10 yr 20 yr Average
Average error in D'rate smth est (R-P)/R 21.73% -36.09% 110.43% -56.79% -43.54% -0.85%
R2 (real vs proj) 0.76 0.38 0.10 0.07 0.92 0.45
Average error D'rate x average real D'rate 0.29% -0.51% 1.60% -0.86% -0.84% -0.06%
Average error D'rate/ Num. years projected 21.73% -12.03% 22.09% -5.68% -2.18% 4.79%
Observations 63.00 58.00 54.00 44.00 24.00

Depending on the Time Horizon a
different method may be more
appropriate
1 YR 3 YR 5 YR 10
YR
20
YR
Average
error
Coeff of
Variation
r2 Avg Proj
– Avg
Real
D’Rate
SYMMETRICAL X -0.85% 99% .45 0.08%
STRETCHED X 13% 88% .75 0.05%
SMOOTHED X X X 49% 76% .28 0.04%

A ‘Counter-Stretch’ option using inverted
Lookback to Horizons (1yr:3yr; 3yr:5yr, etc.)
had poor results vs conventional methods
Counter Stretch Average of 3: 1 year Average of 5:3 year Average of 10:5 year Average of 20 :10 year Average
average error in D' rate -11.31% -73.84% 68.42% 92.02% 18.82%
R2 0.55 0.21 0.00 0.76 37.90%
st dev (real vs proj 1.87% 1.69% 1.42% 1.66% 1.66%
average proj d'rate 1.36% 1.53% 1.68% 1.52% 1.52%
average real d'rate 1.41% 1.45% 1.52% 1.46% 1.46%
coefficient of variation (std/avg) 132.55% 116.55% 93.37% 114.16% 114.16%
Average error Drate /average real -802.58% -5080.76% 4506.32% -459.01% -459.01%
Average error Drate/ No years -3.77% -14.77% 6.84% -3.90% -3.90%
Observations 58.00 54.00 44.00 24.00

In conclusion, D’rate projections
may be improved using
Smoothed or Stretched
methods over the Symmetrical
methods
A Hypothesis is that longer periods of Lookback Projection may improve
results in longer term projections
DISCOUNT RATES

Bond Rate Estimate Errors appear to account for far more of the D’Rate
Error than Inflation Rate Errors (using Symmetrical Method)
Calculation = (Real – Projected) / Real
OVERALL RATES

Average Bond Rate errors range from -75% (5yr) to -50% (20yrs) with
Symmetrical Analysis
BOND RATES

Bond Rate Estimate Errors appear to correlate for much of the D’Rate
Error with Symmetrical Analysis
BOND RATES

Average Error for estimating 20 yr Yields averaging 1 Year Bond Rates
across 20 year Lookback projection spans (1953-2016) yields Average
Error per calculation of 50% being over/under a +/- 50% margin of error
with Symmetrical Analysis
■ 20 yr = 24
BOND RATES

How can estimating Bond Yields be
improved to impact D’Rate estimates?
1. Use fixed 10 or 20 year bond rates in effect at time of
projection instead of Symmetrical Analysis
2. Stretched Method
3. Smoothed Method
BOND RATES

1. Using current 10 or 20 yr Bond Yields to
replace Lookback 10 Yr averaging does not
seem to be a game changer
BOND RATES
In this model, instead of using Symmetrical Analysis to determine future Bond
Rates, the current 10 or 20 yr Bond Rates for the Projection year were used

2. Stretched Estimate: the Stretched Estimate offers
improvement in the 10 yr projection horizon vs
Symmetric…
BOND RATES
Summation of 10 yr horizon periods
from 1973-1996 Lookback periods were stretched:
1 yr: 3 yr
3 yr: 5 yr
5 yr: 10 yr
10 yr: 20 yr

2. Stretched Estimate: … but very similar cumulative
error performance in the 10 yr Projections
BOND RATES
Summation of 10 yr horizon periods
from 1973-1996
1 yr: 3 yr
3 yr: 5 yr
5 yr: 10 yr
10 yr: 20 yr
Near 75%
within +/-
50% of real
number
Near 70%
within +/-
50% of real
number

3. Using a Smoothed estimate calculating on
the entire data series for B’rates improved 10
– 20 yr accuracy
BOND RATES
Smoothed estimate is the accumulated unweighed average for the entire
data series to the date of projection – 1953 to X year

3. Smoothed estimate calculating on the
entire data series for B’rates showed
distribution around 0% error
BOND RATES

3. Smoothed estimate (LT) calculating on the entire
data series for B’rates showed around 50-60% within
+/- 50% of the real value
BOND RATES

3. Smoothed Estimate… is somewhat poorer overall than the
Stretched or Symmetrical Estimates for the shorter horizons
BOND RATES
SYMMETRICAL 1 yr 3 yr 5 yr 10 yr 20 yr Average
Average error in B'rate estimate (symmetric) -6.81% -47.74% -72.27% -31.93% -50.14% -41.78%
R2 (real vs proj) 0.89 0.66 0.52 0.05 0.82 0.59
St dev (real vs proj 0.03 0.03 0.03 0.02 0.02 0.03
Average proj I'rate 4.98% 5.37% 5.68% 6.29% 6.90% 5.84%
Average real I'rate 4.97% 5.25% 5.45% 5.94% 5.63% 5.45%
Average error I'rate x average real I'rate -0.34% -2.51% -3.94% -1.90% -2.82% -2.30%
Average error I'rate/ Num. years projected -6.81% -15.91% -14.45% -3.19% -2.51% -8.58%
Observations 63.00 58.00 54.00 44.00 24.00
STRETCHED 1 yr 3 yr 5 yr 10 yr Average
Average error in B'rate estimate (stretched) -36.97% -74.14% -12.75% -13.12% -34.24%
R2 (real vs proj) 0.74 0.56 0.16 0.82 0.57
St dev (real vs proj 0.03 0.03 0.03 0.02 0.03
Average proj I'rate 5.37% 5.68% 6.29% 6.90% 6.06%
Average real I'rate 4.97% 5.25% 5.45% 5.94% 5.40%
Coefficient of Variation (StDev/Avg) 60.73% 53.00% 47.53% 38.84% 50.03%
Average error I'rate x average real I'rate -1.84% -3.89% -0.70% -0.78% -1.80%
Average error I'rate/ Num. years projected -12.32% -14.83% -1.27% -0.66% -7.27%
Observations 58.00 54.00 44.00 24.00
SMOOTHED 1 yr 3 yr 5 yr 10 yr 20 yr Average
Average error in B'rate estimate smoothed Avg -277.95% -220.68% -148.06% -24.17% -17.92% -137.76%
R2 (real vs proj) 0.04 0.00 0.00 0.21 0.79 0.21
St dev (real vs proj 0.02 0.02 0.02 0.02 0.01 0.02
Average proj I'rate 4.76% 4.86% 4.95% 5.14% 5.55% 5.05%
Average real I'rate 4.97% 5.25% 5.45% 5.94% 5.63% 5.45%
Average error I'rate x average real I'rate -13.81% -11.59% -8.07% -1.44% -1.01% -7.18%
Average error I'rate/ Num. years projected -277.95% -73.56% -29.61% -2.42% -0.90% -76.89%
Observations 63.00 58.00 54.00 44.00 24.00

Depending on the Time Horizon a
different method may be more
appropriate
1 YR 3 YR 5 YR 10
YR
20
YR
Average
error
Coeff of
Variation
r2 Avg Proj
– Avg
Real
B’Rate
SYMMETRICAL X X --42% 48% .59 -2.3%
STRETCHED X X -34% 50% .57 -1.8%
SMOOTHED X -138% 39% .21 -7.2%

In conclusion, bond rate
projections may be somewhat
improved using Smoothed or
Stretched methods over the
Symmetrical methods
Improvements were captured at the 10 year projection level
BOND RATES

Average Inflation Rate errors range from -10% (5yr) to -68%
(20yrs) with Symmetrical Analysis
INFLATION RATES

Standard Symmetrical Lookback Inflation
projection is better than Smoothed or
Stretched up to the 10 yr Projection Horizon
INFLATION RATES
For the 20 yr Horizon Smoothed seems to perform better within +/-
50 % Errors and averaging all periods, differences are minor

The Cleveland Fed also has inflation
projections since 1982 with good
accuracy
0%
10%
20%
30%
40%
50%
60%
70%
-125.00% -100.0% -75.0% -50.0% -25.0% 0.0% 25.0%
CLEVELAND FED'S INFLATION PROJECTION SCORECARD (1982-2016)
1 yr 3 yr 5 yr 10 yr 20 yr 30 yr Average

The Cleveland Fed has performed better
than Lookback Methods for accuracy
since 1983
CL FED 1 yr 3 yr 5 yr 10 yr 20 yr 30 yr
average error -13.71% -11.21% -13.98% -21.85% -36.18% -48.25%
R2 0.66 0.59 0.68 0.84 0.69 0.73
st dev 1.13% 1.05% 0.97% 0.82% 0.72% 0.79%
average real 2.80% 2.81% 2.83% 2.84% 2.79% 2.82%
coefficient of variation 40.46% 37.21% 34.07% 29.05% 25.73% 28.06%
Count 34.00 32.00 30.00 25.00 15.00 5.00
Symmetrical Lookback 1 yr 3 yr 5 yr 10 yr 20 yr 30 yr
Average error 30.03% 34.24% 38.07% 55.69% 118.41% 61.27%
R2 0.55 0.29 0.38 0.95 0.21 0.94
st dev 1.25% 1.45% 1.53% 1.71% 1.67% 0.91%
average real 2.80% 2.81% 2.83% 2.84% 2.79% 2.82%
Count 34.00 32.00 30.00 25.00 15.00 5.00
CLFED / Lookback 1 yr 3 yr 5 yr 10 yr 20 yr 30 yr
Average error -45.65% -32.74% -36.73% -39.23% -30.55% -78.74%
R2 1.19 2.06 1.80 0.89 3.22 0.77
st dev 90.48% 72.17% 63.22% 48.15% 42.98% 86.75%
average real 100.00% 100.00% 100.00% 100.00% 100.00% 100.00%
Count 1.00 1.00 1.00 1.00 1.00 1.00

In conclusion: there are various viable
alternatives to straight Symmetrical Lookback
Method that should be explored on a case by
case basis
■ Smoothed and Stretched methods perform better along
longer time spans
■ Cleveland Fed may be a valid checkpoint for Inflation Rate
projections
■ No one size fits all for projections; depending on whether
average accuracy or reduction of dispersion error is sought
■ All methods can work together and an average of results may
give the results required

Evaluation of the lookback method for discount rate v 01.12.17 [v2]

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Evaluation of the lookback method for discount rate v 01.12.17 [v2]