Download A Social Discount Rate for the US

International Research Journal of Finance and Economics
ISSN 1450-2887 Issue 25 (2009)
© EuroJournals Publishing, Inc. 2009
http://www.eurojournals.com/finance.htm
A Social Discount Rate for the US
Samih Antoine Azar
Faculty of Business Administration & Economics, Haigazian University
Mexique Street, Kantari, Beirut, Lebanon
E-mail: [email protected]
Tel: +9611349230; Fax: +9611349230
Abstract
The approach adopted in this paper in order to measure the US social discount rate
is market-driven and relies on trade-offs in financial markets. The underlying assumptions
are that public budgets displace the marginal private investment, and that US citizens
borrow to finance home mortgages. The average debt capacity of the US citizen, or the
representative investor, is 20%. It is argued that this produces an average return on net
wealth while what is needed is a marginal return on net wealth. Taking a debt service ratio
of 60% for the marginal investor yields a US social discount rate close to 3.7%, varying
between 3.68% and 3.73%. This rate is lower than the rates available in the literature but is
arguably more reasonable for very long-term public projects.
Keywords: Social discount rate, rate of time preference, public capital budget,
maximization of future consumption, Monte Carlo simulation, portfolio of one
risky and one risk-less asset, coefficient of relative risk aversion, elasticity of
marginal utility, home mortgages, debt service ratio.
1. Introduction
Lately there has been a renewed interest in measuring the social discount rates for Western nations.
This rate is critical in evaluating public budgets. A high rate will penalize long term projects, especially
projects that concern the next generations. A low rate will lead to an inefficient allocation of resources,
with the adoption of projects that are economically non-viable. A balance should be found. The
purpose of this letter is to propose an alternative, market-based, and original approach which is applied
to the US case.
In the literature, there are three main methods that are utilized to measure a social discount rate
for any country. The three are:
(1) Specifying a benchmark financial rate.
(2) Building upon the notion of a rate of time preference and using statistics about the longterm growth rate of a nation.
(3) Looking upon trade-offs in financial markets, which is the method selected in this letter.
This approach assumes perfect markets.
The first approach is the one followed by the US Office of Management and Budget (OMB), as
mentioned in its Circular No A-94, which provides “Guidelines and Discount Rates for Benefit-Cost
Analysis of Federal Programs,” and is available on the web site of the US OMB:
www.whitehouse.gov/omb/circulars/a94/a094.html. This site lists real Treasury interest rates from
International Research Journal of Finance and Economics - Issue 25 (2009)
204
1979 to 2008, rates that are used by the OMB to appraise public investments.1 The rate prescribed for
2008 and for 30-year projects is 2.8%. However the average rate over the years 1979 to 2008 is 4.4%.
Hence the OMB relies on the benchmark real Treasury interest rates to approximate the social discount
rate, assuming implicitly that the safe opportunity cost of capital should be the yield on long term Tsecurities. For an extensive discussion of this approach see Florio (2006) who calls such a discount rate
a Financial Discount Rate (FDR). Florio (2006) recommends a social discount rate between 3 % and
4%.
The second approach is based on a theory by Ramsey (1928), which is discussed in Barro and
Sala-i-Martin (1995: 60-67). See, for example, Pearce and Ulph (1995) who argue that the appropriate
social discount rate for the UK lies between 2% and 4%. For more recent applications see Evans and
Sezer (2002), Evans (2004b), Evans and Sezer (2004), and Percoco (2008). The underlying formula is
derived from Evans and Sezer (2002):
1 + SDR = (1 + g )γ (1 / π )(1 + ρ )
(1)
Where SDR stands for the social discount rate, g is the average growth rate of real per capita
consumption, γ is the absolute value of the elasticity of the marginal utility of consumption, π is the
average probability of survival of an individual, and ρ is the discount rate of the inter-temporal utility
of the consumer. Taking logs on both sides of equation (1), and generalizing the formula, the following
is obtained:
SDR ≈ γg + ρ + φ
(2)
Where φ is the probability of death. The parameters in equation (2) must all be estimated. For
the US g is estimated to be 2.2%, the same figure as the one reported in Evans and Sezer (2004), ρ is
at most 0.5%, if not zero, and φ is around 1%. This gives a value for the SDR equal to 3.7%. This
figure is lower than the one calculated in Evans and Sezer (2004) for the US which lies between 4.4%
and 4.6%, but it relies on estimates in Azar (2007a, and 2008). For example, Azar (2007a, 2008) finds
support for a γ equal to 1, while Evans and Sezer (2004) apply a γ equal to either 1.3 or 1.4 for the
US. Pearce and Ulph (1995) write that a γ equal to 1 is “defensible.” Evans and Sezer (2004) take
(ρ + φ ) to be equal to 1.5%, which is the same figure as the one I take. So the major issue is the
estimate of γ as Evans (2004a) insists upon. Evans and Sezer (2002), Evans (2004a,b), Evans and
Sezer (2004) and Percoco (2008) use other approaches to estimate γ . They compute γ from demand
functions for a preference-independent good, which is food, and/or from statistics on marginal and
average tax rates, while Azar (2008) computes γ from the first-order Euler equation of a CCAPM
model of stock returns and dividend payments.
The third approach to the measurement of the social discount rate considers the opportunity
cost of private investment instead of consumption, and assumes perfect markets. This approach is
market-driven, and relies on the first-order condition for the maximization of the expected utility of
wealth of an investor who has two assets to invest in: a risky asset and a risk-less asset. The
simulations in Azar (2007b) produce an estimate for the SDR between 5.62% and 5.71%, a range that
is above the ones reported in the literature surveyed above. For a detailed exposition of this approach
see the followings theoretical section, section 2. Section 3 gives the empirical results, while the last
section is a conclusion.
2. The Theory
The theoretical background is based on the maximization of expected future wealth in the presence of
two assets: a risky one and a risk-less one (Azar, 2007b). The maximand is:
1
This site was visited on February 19, 2008.
205
International Research Journal of Finance and Economics - Issue 25 (2009)
1−γ
+ α (~
r -rf ))) − 1
(3)
α
1− γ
Where W0 is initial wealth, ~
r is the risky outcome, rf is the risk-less return, E (.) is the
expectation operator, α is the share of wealth invested in the risky asset, and γ is the coefficient of
relative risk aversion (CRRA) in financial markets. The risky outcome is distributed as a normal
probability density function with mean 13.3% and standard deviation 20.1% (Ross et al., 2002: 233).
The CRRA is taken to be 4.5 (Azar, 2006). The first-order condition of equation (3) is:
E (W (1 + rf + α * (~
r − rf )))−γ W (~
r − rf ) = E (1 + rf + α * (~
r − rf ))−γ (~
r − rf ) = 0
(4)
α* ∈
(
arg
max E(U) where U =
0
0
(W0 (1 + rf
) (
)
Where W0 and W0−γ are simplified because they are given. Azar (2007b) simulates this
equation and obtains α * to be 0.5332 on average, a figure that ranges in a 95% interval between
0.5282 and 0.5381. Based on these findings the 95% confidence level of the social discount rate is
measured in Azar (2007b) to be between 5.62% and 5.71%. This social discount rate solves the
following equation:
α * E (r~ ) + (1 − α *)(rf ) − 0.032
(5)
Where E (~
r ) is taken to be 0.133, rf is taken to be a constant at 0.038 and the inflation rate is a
constant at 0.032 (Ross et al., 2002: 233). Azar (2007b) argues that this rate is the social discount rate
because it is independent of wealth by construction, and therefore applies to the representative investor.
Unfortunately the estimates of the social discount rate based upon equations (3), (4) and (5) are
relatively high, and might be considered unrealistic. This letter improves the model by assuming that
the representative investor can borrow as much as B dollars at the risk-free interest rate, can add this
borrowing to her net wealth, and will consume in the following period:
C t +1 = (W0 + B )(1 + rf + α (~
r − rf )) − (1 + rf ) B
(6)
If the expected utility of next-period consumption is maximized then one has:
α* ∈
arg
max E(U) where U =
((W0 + B )(1 + rf
α
And the first-order condition is:
⎛⎛⎛
B ⎞
B
⎜
⎟⎟(1 + rf + α * (~
E ⎜ ⎜⎜ ⎜⎜1 +
r − rf )) − (1 + rf )
W0
⎜ ⎝ ⎝ W0 ⎠
⎝
⎞
⎟
⎟
⎠
−γ
1−γ
r -rf )) − (1 + rf )B ) − 1
+ α(~
1− γ
(7)
⎞
(~r − rf )⎟⎟ = 0
⎟
⎠
(8)
Where W0 and W0−γ are simplified. This model, as it stands, is a money-making machine
because the investor can borrow at the risk-less rate and invest at a higher rate, unless α is zero.
However, I shall impose constraints on the model by limiting the amount borrowed, or more precisely,
by limiting the ratio of debt ( B ) to net wealth ( W0 ), i.e. B /W0 . It is implicitly assumed that the debt
contracted by the investor is mainly directed to finance home mortgages which are arguably profitable.
In the model of this paper the ratio of debt to net wealth is the same as the debt service ratio.
3. The Empirical Results
Candidates for the debt service ratio, or the ratio of debt to net wealth, (B / W0 ) , vary. Data in Girouard
et al. (2007: 23, Table 1) imply a ratio of 0.2, which is the average for the three years 1995, 2000, and
2005. Data from the US Federal Reserve Board on “Household Debt Service and Financial Obligations
Ratios” provide a range between 0.1932 (for home-owners) and 0.26 (for renters) for the third quarter
of 2007.2 Therefore the initial candidate for B /W0 is 0.2 (see Table 1). This estimate is an average
2
The web page is: http://www.federalreserve.gov/Releases/housedebt/ and it was visited on February 22, 2008.
International Research Journal of Finance and Economics - Issue 25 (2009)
206
over all borrowers. A better estimate is a marginal one because opportunity costs should be measured
relative to the marginal investor. Campbell and Viceira (2002: 187) report a standard deviation of 10%
for the growth in labor income. Taking three standard deviations away from zero for the marginal
laborer, the (B / W0 ) ratio becomes B / (0.7W0 ) , i.e. 0.2/0.7, which is around 0.3. In addition, if the
marginal investor is temporarily out of work she will receive 33% of her income from the
unemployment insurance fund (Hyman, 2008: 346). This means that the ratio B /W0 for the
unemployed is around 0.6, i.e. B / (0.33W0 ) or 0.2/0.33. Table 1 assumes four different plausible
estimates of the debt service ratio ( B /W0 ): 0.2, 0.3, 0.4, and 0.6.
The procedure implemented is as follows. The risky return is simulated from a normal
distribution of mean 13.3% and standard deviation 20.1% (Ross et al., 2002: 233). For each simulation
the condition in equation (8) is calculated for a given debt service ratio. The average of this equation
for the 1000 simulated risky outcomes is set to be equal to zero by varying the share of wealth in the
risky asset α. This is done through the Goal Seek command in Excel. Each run of 1000 simulations is
repeated a hundred times. Each run produces a given share of wealth in the risky asset α . In total there
are 100 estimates of α for a given debt service ratio ( B /W0 ). The simulations are repeated with
another value of the debt service ratio. The statistics and the results are provided in Table 1. The US
social discount rate is defined as the return on net wealth:
(rf + α * (E (r~ ) − rf )) − 0.032
(9)
~
Where E (r ) is taken to be 0.133, rf is taken to be 0.038, and the inflation rate is taken to be
0.032 (Ross et al., 2002: 233). This is the same equation as equation (5). For a given debt service ratio,
the return on gross wealth is:
⎛ B ⎞
⎛
B ⎞
⎟⎟ − 0.032
⎟⎟(1 + rf + α * (E (~
⎜⎜1 +
(10)
r ) − rf )) − (1 + rf )⎜⎜
W
W
0⎠
⎝ 0⎠
⎝
Table 1 provides all statistics on the results. The implied mean shares of wealth in the risky
asset ( α* ) vary between 0.44165 for a debt service ratio of 0.2 and 0.32651 for a debt service ratio of
0.6. The higher the debt service ratio the lower is this optimal mean share of wealth. The 95%
confidence intervals for these mean shares are all non-overlapping, meaning that all debt service ratios
imply different mean shares. The returns on gross wealth are between 5.53% and 5.79%, which
correspond approximately to the range in Azar (2007b) in an economy without debt. The higher the
debt service ratio, the lower are the social discount rates. The social discount rates vary between 4.8%
for a debt service ratio of 0.2 to 3.7% for a debt service ratio of 0.6. The estimate in Evans and Sezer
(2004) is between 4.4% and 4.6%, which corresponds to a debt service ratio of 0.3 in Table 1. My own
preferred estimate is 3.7% as I have argued in the introduction. This preferred estimate is exactly the
same as the one obtained with a debt service ratio of 0.6. It is also immaterially different from the
estimate of Evans (2004b) for France, which is 3.8%, and inside the range provided by Percoco (2008)
for Italy, which is between 3.69% and 3.83%. Therefore, to conclude, the US social discount rate is
estimated in this paper to be between 3.68% and 3.73%, which is the 95% confidence interval for a
debt service ratio of 0.6, corresponding to the marginal investor who is temporarily out of work.
207
Table 1:
International Research Journal of Finance and Economics - Issue 25 (2009)
Results of the random simulations, the implied real social discount rates, and their 95% confidence
r ) is distributed as a normal distribution with mean
intervals. The nominal random risky outcome (~
13.3% and standard deviation 20.1%. The risk-free rate ( rf ) is a nominal constant 3.8%. The
inflation rate is a constant 3.2%. The CRRA ( γ ) is a constant 4.5. The optimal share of wealth in
the risky asset ( α * ) is obtained by changing its value in the Excel Goal Seek command in order to
satisfy the first-order Euler equation of the following maximand:
1−γ
(
(
W0 + B )(1 + rf + α (~
r -rf )) − (1 + rf )B ) − 1
α * ∈ arg max E(U) where U =
α
1− γ
This first-order Euler equation is:
−γ
⎛⎛⎛
⎞
⎞ ~
⎞
B
B
⎜
~
E ⎜⎜ ⎜⎜1 + ⎟⎟(1 + rf + α * (r − rf )) − (1 + rf ) ⎟⎟ (r − rf )⎟ = 0
⎜
⎟
W0 ⎠
W0 ⎠
⎝⎝⎝
⎠
B / W0 = 0.2
# of simulations
# of runs
Sample size
Mean share of wealth in the risky asset
95% confidence interval for the mean share of wealth
in the risky asset
Implied mean real social discount rate on net wealth
95% confidence interval for the mean real social
discount rate on net wealth
Mean real return on gross wealth
95% confidence interval for the mean real return on
gross wealth
1000
100
100
0.44165
0.43820-
B / W0 = 0.3
1000
100
100
0.41603
0.41156-
B / W0 = 0.4
B / W0 = 0.6
1000
100
100
0.37435
0.37029-
1000
100
100
0.32651
0.32042-
4.80%
4.76%-4.83%
0.42050
4.55%
4.51%-4.59%
4.15%
4.12%-4.19%
0.32900
3.70%
3.68%-3.73%
5.64%
5.59%-5.68%
5.74%
5.68%-5.79%
5.57%
5.53%-5.63%
5.56%
5.53%-5.61%
0.44509
0.37840
4. Conclusion
In the literature there are three main approaches to measuring the US social discount rate. The first
takes, as a financial benchmark, real (inflation-adjusted) returns on T-securities. This approach
provides an average of 4.4% (US OMB). The second approach uses trade-offs in the inter-temporal
utility of consumption, together with an estimate of the probability of death, and the long run behavior
of consumption per capita. This approach produces a rate between 4.4% and 4.6% (Evans and Sezer,
2004) although this paper argues that the same approach yields a rate of 3.7% by changing the
underlying assumptions about the parameters, especially about the elasticity of the marginal utility of
consumption. The third approach is market-driven and looks at the trade-offs in financial markets, and
posits that public budgets displace private investment and not private consumption. The implied rate is
5.66% (Azar, 2007b). This third approach was refined in this paper by allowing borrowing by the
representative investor. This borrowing finances mainly home mortgages which are common and widespread in the US. Although markets are assumed to be perfect, the US social discount rate produced is
3.7%, varying in a 95% confidence interval between 3.68% and 3.73%. The estimate and the
confidence interval are based on the behavior of the marginal investor because the opportunity cost of
public budgets should be equal to the marginal and not the average return on net wealth.
International Research Journal of Finance and Economics - Issue 25 (2009)
208
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