Long-Run Investment Horizons and Implications for Mixed-Asset Portfolio Allocations

Transcription

1 Long-Run Investment Horizons and Implications for Mixed-Asset Portfolio Allocations Joseph L. Pagliari, Jr. Clinical Professor of Real Estate October 20, 2016 Institute for Private Capital he University of North Carolina Kenan-Flagler Business School

2 Long-Run Horizons & Mixed-Asset Portfolio Allocations 1 First Look: Annual Returns Serial Correlation & Long-Run Volatility Serial Correlation & Long-Run Correlation Second Look: Four-Year Returns aking Returns to the Limit

3 Some Background/Context Our Focus 2 raditional Investments: Stocks: lackluster returns (e.g., 7.4% average return for S&P 500 over the last ten years). Bonds: low interest rates (e.g., 2.6% ten-year and 3.5% thirty-year U.S. reasuries). Actuarial Assumptions: 5-8% (varies by plan and by country). \ Allocations: Non-raditional Investments: Alternative Investments: real estate, private equity, infrastructure, etc. Market Inefficiencies: belief that private markets due to their opaqueness provide investors the opportunity to exploit market inefficiencies. Diversification: quest to improve portfolio s risk/return characteristics. \ Allocations: Caveat Our Focus: Private-market assets often display inertia in returns (i.e., serial-correlation). he existence of this inertia alters the asset s long-run risk/return characteristics and, therefore, its role in a mixed-asset portfolio.

4 A First Look: Annual Returns Let s consider an MP-based analysis; we first need: Consider the (pre-fee) returns on major asset classes: Summary Statisitcs for the Annual Returns of Selected Asset Classes for the Years Ended x, 3 Asset Class: Geometric Mean Arithmetic Mean Median Standard Deviation Serial Correlation Sharpe Ratio S&P % 13.19% 15.90% 16.78% 0.45% U.S. Small Stocks 14.38% 16.36% 19.68% 20.91% % MSCI EAFE 7.67% 9.92% 9.80% 21.83% 14.27% U.S. L- Government Bonds 8.70% 9.46% 8.28% 13.09% % U.S. L- Corporate Bonds 8.78% 9.27% 9.09% 10.68% -7.22% Domestic High-Yield Corporate Bonds 9.76% 10.74% 10.06% 15.23% % U.S. 30-Day reasury Bills 5.05% 5.11% 5.03% 3.61% 88.31% NAREI-Equity 12.61% 14.07% 17.18% 17.27% 2.65% NCREIF Property 9.15% 9.44% 10.76% 7.90% 55.10% U.S. Inflation 3.74% 3.78% 3.01% 2.89% 77.35% Source: Morningstar We ll use the NCREIF return series as emblematic of private-market alternatives. NCREIF = U.S. institutionally held, privately held commercial real estate NCREIF return series begins in 1978 (one of the longest such series available) PE returns series (e.g., LBO, VC, etc.) neither as long nor as reliable Second highest

5 A First Look at Correlations 4 Before considering portfolio allocations, we also need the correlation matrix: Correlation Matrix for Selected Asset Classes Based Upon Annual Returns for the Period U.S. Small Stocks Correlation Coefficients Domestic U.S. L- Corp Hi-Yld Corp U.S. 30- Day -Bill MSCI U.S. L- NAREI- NCREIF U.S. Asset Class: S&P 500 EAFE Gvt Equity Property Inflation S&P U.S. Small Stocks MSCI EAFE U.S. L- Government Bonds U.S. L- Corporate Bonds Domestic High-Yield Corporate Bond U.S. 30-Day reasury Bills NAREI-Equity NCREIF Property U.S. Inflation Source: Morningstar Average Coefficient * * excluding inflation and itself. Proxy for the diversification benefit of various asset classes Lowest best

6 A First Look at the Efficient Frontier 5 Public Real Estate Private Real Estate

7 A First Look at Components of the Efficient Frontier 6 As the efficient frontier becomes highrisk/highreturn, the allocation to small stocks becomes untenable (we will revisit). 23.5% he average Real Estate allocation

8 Long-Run Horizons & Mixed-Asset Portfolio Allocations 7 First Look: Annual Returns Serial Correlation & Long-Run Volatility Serial Correlation & Long-Run Correlation Second Look: Four-Year Returns aking Returns to the Limit

9 A Second Look at Annual Returns 8 Let s look at serial-correlation by major asset classes: random walk Summary Statisitcs for the Annual Returns of Selected Asset Classes for the Years Ended Asset Class: Geometric Mean Arithmetic Mean Median Standard Deviation Serial Correlation Sharpe Ratio S&P % 13.19% 15.90% 16.78% 0.45% U.S. Small Stocks 14.38% 16.36% 19.68% 20.91% % MSCI EAFE 7.67% 9.92% 9.80% 21.83% 14.27% U.S. L- Government Bonds 8.70% 9.46% 8.28% 13.09% % U.S. L- Corporate Bonds 8.78% 9.27% 9.09% 10.68% -7.22% Domestic High-Yield Corporate Bonds 9.76% 10.74% 10.06% 15.23% % U.S. 30-Day reasury Bills 5.05% 5.11% 5.03% 3.61% 88.31% NAREI-Equity 12.61% 14.07% 17.18% 17.27% 2.65% NCREIF Property 9.15% 9.44% 10.76% 7.90% 55.10% U.S. Inflation 3.74% 3.78% 3.01% 2.89% 77.35% Source: Morningstar inertia high serial correlation is not prima facie evidence of an inefficient market

10 What Is Serial Correlation? Why Is It Important? 9 What Is Serial Correlation? A simple version: r = α + ϕr + ε t t 1 t where: r t = return in period t, α = constant, ϕ = measure of serial (or auto) correlation, and ε t = return in period t. his is an autoregressive model of order one (i.e., an AR(1) process). he Implication of Serial Correlation Upon Returns: If ϕ 0, then return series random walk (i.e., no discernible short-run pattern or white noise ) If ϕ < 0, then return series mean-reverting (i.e., there are reversals in the return pattern or blue noise ) If ϕ > 0, then return series inertia (i.e., red noise ) his is the case of (private) real estate and often alternative investments more generally

11 How Does Serial Correlation Effect Variance? 10 Again, we assume an AR(1) model of serial correlation: r = α + ϕr + ε t t 1 t By construction, we assume that variance is constant across all periods: = 1 =... = = t t+ t+ N What is the variance of two-period returns? = = = 2 tt, + 1 t t+ 1 tt, ϕ = ( ϕ) = + where: t,t+1 = auto-covariance of returns in periods t and t +1 = 2 ϕ

12 A wo-period Model of Correlated Variance 11 Recall the variance of two-period returns : = = = 2 tt, + 1 t t+ 1 tt, ( ϕ) = + We say: Variance (linearly) scales with time (i.e., 2-period variance = 2 2 ). he implication of serial correlation upon the volatility of returns: ype of Return Series If ϕ 0, then return series random walk (i.e., no discernible short-run pattern or white noise ) If ϕ < 0, then return series mean-reverting (i.e., blue noise ) If ϕ > 0, then return series inertia (i.e., red noise ) wo-period Variance 2 2 = 2 = = 2 < = 2 > 2 his is the heart of the issue! Variance grows faster than merely scaled with time (i.e., 2-period variance > 2 2 ).

13 A wo-period Model of Correlated Variance 12 Recall the variance of two-period returns : = = = 2 tt, + 1 t t+ 1 tt, ( ϕ) = + More broadly: Long-term variance differs by the level of auto-correlation. he implication of serial correlation upon the volatility of returns: ype of Return Series If ϕ 0, then return series random walk (i.e., no discernible short-run pattern or white noise ) If ϕ < 0, then return series mean-reverting (i.e., blue noise ) If ϕ > 0, then return series inertia (i.e., red noise ) wo-period Volatility 2 2 = 2 = = 2 < = 2 > 2

14 A Multi-Period Model of Correlated Variance 13 Our two-period model return variance can also be written as: = = = 2 tt, + 1 t t+ 1 tt, t tt, + 1 = 2 tt, + 1 t+ 1 he auto-covariance matrix (Ω) for long-term ( ) volatility is: Ω= 1,1 1,2 1,3 1,4 1, 2,1 2,2 2,3 2,4 2, 3,1 3,2 3,3 3,4 3, 4,1 4,2 4,3 4,4 4,,1,2,3,4,

15 A Multi-Period Model of Correlated Variance (continued) 14 Recall that variance is constant across all periods: By construction, the auto-covariance of S -period returns: t,t+s = 2 ϕ S So, the auto-covariance matrix (Ω) can be rewritten as: 1 = 1 =... = = t t+ t+ N ϕ ϕ ϕ ϕ ϕ 1 ϕ ϕ ϕ 2 2 ϕ ϕ 1 ϕ ϕ Ω= ϕ ϕ ϕ 1 ϕ ϕ ϕ ϕ ϕ 1

16 Special Case: Multi-Period Model of Variance 15 Special case: When ϕ = 0 ( random walk ), then auto-covariance matrix simplifies to: Ω= When ϕ =0, all the off-diagonal elements go to zero. 2 = As a result, long-run variance (linearly) scales with time.

17 A Multi-Period Model of Correlated Variance (continued) 16 2 More generally, long-run variance can be written as: t= 1 2 = + ( ) t = + ϕ t ( ) ϕ ϕ ϕ 1 ( 1 1 ϕ ) ( ϕ ) 2

18 A Multi-Period Variance Volatility (or Standard Deviation) 17 Let s convert long-run variance to the standard deviation: t= 1 ( ) t = + ϕ t ϕ ϕ = ( ) ϕ 1 1 ( 1 ϕ ) ( ϕ ) 2 In the special case of ϕ = 0, this simplifies to: ( ) 2 = = = Well-known result: the square root rule.

19 Illustration of Multi-Period Volatility as a f(ϕ ) 18 Let s look at an illustration of multi-period volatility as a f(ϕ): [arbitrarily assuming one-year volatility = 10%] After 25 periods, volatility has grown by more than five-fold (for ϕ =.75, volatility has grown more than twelve fold). After 25 periods, volatility has grown five-fold (for ϕ = 0).

20 Annualized (or Scaled) Volatility 19 It s often more intuitive to scale long-run standard deviation by : = t= 1 ( ) t + ϕ t ϕ ϕ = ϕ 1 In the special case of ϕ = 0, this simplifies to: 1 ( 1 ϕ ) ( ϕ ) 2 = = Another version of the wellknown square root rule. As before (p.17).

21 Annualized (or Scaled) Volatility Differently Stated 20 Begin again with the long-run standard deviation, but scaled by : = t= 1 ( ) t + 2 ϕ t ϕ ϕ = ϕ 1 In the limit, this expression simplifies to: 1 ( 1 ϕ ) ( ϕ ) 2 lim = 1+ ϕ 1 ϕ Again, the importance of estimating long-run volatility when asset classes display different levels of serial correlation.

22 Illustration of Scaled Multi-Period Volatility as a f(ϕ ) 21 Let s look at an illustration of annualized multi-period volatility as a f(ϕ): [arbitrarily assuming one-year volatility = 10%] After 25 periods, scaled volatility reduced to onehalf of original value (for ϕ =.75). After 25 periods, scaled volatility reduced to onefifth of original value (for ϕ = 0).

23 he Interplay of Long-Run Returns & Volatility 22 An illustration of the interplay of long-run (annualized) returns and volatility: [arbitrarily assuming one-year returns & volatility = 10%] 25% Illustration of Annualized Long-Run Average with Confidence Bands given the Absence of Serial Correlation 20% r + Z 15% Annualized Long-Run Average 10% with Confidence Bands: r, 5% 0% r ϕ t, t+1 =.0 ϕ t, t+1 =.0 r Z -5% Holding Period ( )

24 he Interplay of Long-Run Returns & Volatility (continued) 23 An illustration of the interplay of long-run (annualized) returns and volatility: [arbitrarily assuming one-year returns & volatility = 10%] 25% Illustration of Annualized Long-Run Average with Confidence Bands given Contrasting Serial Correlation Coefficients 20% ϕ t, t+1 =.5 15% Annualized Long-Run Average 10% with Confidence Bands: 5% ϕ t, t+1 =.0 ϕ t, t+1 =.0 0% ϕ t, t+1 =.5-5% Holding Period ()

25 Example: Interplay of Long-Run Returns & Volatility 24 Let s look at the interplay of long-run (annualized) returns and volatility:

26 Long- v. Short-Run Volatility as a f(ϕ ) 25 he ratio long-run (annualized) volatility to annual volatility as a f(ϕ): Ratio of Annualized Long-Horizon Volatility to Annual Volatility: Illustration of Scaled Long-Horizon Volatility to Annual Volatility with Various Serial Correlation Characteristics ϕ =.800 ϕ ~.946 ϕ ~.923 ϕ ~.882 ϕ ~.969 ϕ =.960 ϕ = x ϕ = 0 ϕ = NCREIF Serial Correlation (ϕ)

27 A Stylized Comparison of Long-Run Volatility 26 Public equities (e.g., S&P 500) and private real estate (e.g., NCREIF) have very different annual volatilities and serial correlation. Yet, the long run looks similar: 18% Illustration of Long-Run, Annualized Standard Deviation Common Stocks v. Private Real Estate 16% 14% Annualized Standard Deviation 12% 10% 8% 6% 4% 2% RE (ϕ.55) CS (ϕ 0) 0% Holding Period ( )

28 Long-Run Horizons & Mixed-Asset Portfolio Allocations 27 First Look: Annual Returns Serial Correlation & Long-Run Volatility Serial Correlation & Long-Run Correlation Second Look: Four-Year Returns aking Returns to the Limit

29 How Does Serial Correlation Effect Pair-wise Correlation? 28 he general formula for the periodic correlation between two asset classes (say, x and y) is given by: xy, By extension, the long-run ( > 1) pair-wise correlation is given by: ρ ρ x, y = = x xy, y x, y x y he previous section identified the denominator: long-run volatility ( x ). So, let s consider the numerator: long-run covariance ( x,y ).

30 A Multi-Period Model of Auto-Covariance 29 he long-term ( > 1) covariance matrix (Σ) of x and y is given by: x, y x, y x, y x, y x, y,1 1,1 2,1 3,1 4,1 x, y x, y x, y x, y x, y x, y x, y x, y x, y x, y Σ= x 4, y 1 x4, y 2 x4, y 3 x4, y 4 x4, y x, y x, y x, y x, y x, y

31 A Multi-Period Model of Auto-Covariance (continued) 30 For reasons similar to long-term variance, the long-term covariance matrix (Σ) of x and y can be rewritten as: ϕy ϕy ϕy ϕ y 2 2 ϕx 1 ϕy ϕy ϕy 2 3 ϕx ϕx 1 ϕy ϕ y Σ= ρxy, x y ϕx ϕx ϕx 1 ϕy ϕx ϕx ϕx ϕx 1

32 A Multi-Period Model of Auto-Covariance (continued) 31 For reasons similar to long-term variance, the long-term covariance matrix (Σ) of x and y can be rewritten as: ϕy ϕy ϕy ϕ y 2 2 ϕx 1 ϕy ϕy ϕy 2 3 ϕx ϕx 1 ϕy ϕ y Σ= ρxy, x y ϕx ϕx ϕx 1 ϕy ϕx ϕx ϕx ϕx 1 In the limit, long-term correlation simplifies to: lim ( ρxy) = ρ, xy, 1 ϕϕ x y 1 ϕ 1 ϕ 2 2 x y

33 he Ratio of Multi-Period Pair-wise Correlation to the Single Period 32 Perhaps the more relevant (and/or intuitive) measure is the ratio of the multiperiod pair-wise correlation to the single-period (or annual) pair-wise correlation; in the limit, this expression is given by: ρ x, y 1 ϕϕ x y lim = ρ 2 2 xy, 1 ϕx 1 ϕy Note: his ratio reaches its minimum when ϕ x = ϕ y regardless of their level. Recall: Lower (v. higher) correlation is more favorable with regard to MP diversification benefits. herefore, a summary of these effects: 1 ϕϕ x maximum, as ϕ, ϕ "bad" for MP 2 2 ϕ x ϕy minimum, as ϕ x ϕ y 1 1 y = x y "good" for MP

34 he Ratio of Multi-Period Pair-wise Correlation to the Single Period 33 A graphical representation of these relationships: When ϕ x ϕ y, long-term pair-wise correlation > annual pair-wise correlation. When ϕ x ϕ y, long-term pair-wise correlation annual pair-wise correlation.

35 Long-Run Horizons & Mixed-Asset Portfolio Allocations 34 First Look: Annual Returns Serial Correlation & Long-Run Volatility Serial Correlation & Long-Run Correlation Second Look: Four-Year Returns aking Returns to the Limit

36 Let s Look at Four-Year (Annualized) Returns 35 Let s apply these theoretical relationships to the data. We have 36 years worth of data. So, nine (independent) 4-year returns: Asset Class x Asset Class y Asset Class z Annual Four-Year Annual Four-Year Annual Four-Year Returns Returns Returns Returns Returns Returns r x,1 r y,1 r z,1 r x,2 r y,2 r z,2 R x,1 R y,1 r x,3 r y,3 r z,3 R z,1 r x,4 r y,4 r z,4 r x,5 r y,5 r z,5 r x,6 r y,6 r z,6 R x,2 R y,2 r x,7 r y,7 r z,7 R z,2 r x,8 r y,8 r z,8 r x,9 r y,9 r z,9 r x,10 r y,10 r z,10 R x,3 R y,3 r x,11 r y,11 r z,11 R z,3 r x,12 r y,12 r z,12 r x,33 r y,33 r z,33 r x,34 r y,34 r z,34 R x,9 R y,9 r x,35 r y,35 r z,35 R z,9 r x,36 r y,36 r z,36

37 A Second Look: 4-Year (Annualized) Returns 36 Consider the 4-year (annualized) returns on major asset classes ( x, ): Four-Year Returns of Selected Asset Classes for the Years Ended Summary Statistics Based on Nominal 4-Year (Annualized) Returns Geometric Mean Arithmetic Mean Standard Deviation Serial Correlation Sharpe Ratio S&P % 12.06% 7.93% 12.40% U.S. Small Stocks 14.38% 14.73% 9.43% % MSCI EAFE 7.67% 8.08% 10.25% 5.93% U.S. L- Government Bonds 8.70% 8.85% 6.11% % U.S. L- Corporate Bonds 8.78% 8.97% 6.80% % Domestic High-Yield Corporate Bond 9.76% 9.92% 6.27% % U.S. 30-Day reasury Bills 5.05% 5.10% 3.44% 87.34% NAREI-Equity 12.61% 12.90% 8.37% % NCREIF Property 9.14% 9.30% 6.00% -6.96% U.S. Inflation 3.74% 3.77% 2.76% 65.51% Sources: Morningstar and author's calculations Fourth lowest

38 Sources: Morningstar and author's calculations A Second Look: 4-Year (Annualized) Returns (continued) Let s compare the volatility of 1- and 4-year (annualized) returns: One- and Four-Year Returns of Selected Asset Classes for the Years Ended Summary Statistics Based on Nominal Returns Geometric Mean Arithmetic Mean Standard Deviation Serial Correlation Sharpe Ratio Panel A: Using One-Year Investment Horizons S&P % 13.19% 16.78% 0.45% U.S. Small Stocks 14.38% 16.36% 20.91% % MSCI EAFE 7.67% 9.92% 21.83% 14.27% U.S. L- Government Bonds 8.70% 9.46% 13.09% % U.S. L- Corporate Bonds 8.78% 9.27% 10.68% -7.22% Domestic High-Yield Corporate Bond 9.76% 10.74% 15.23% % U.S. 30-Day reasury Bills 5.05% 5.11% 3.61% 88.31% NAREI-Equity 12.61% 14.07% 17.27% 2.65% NCREIF Property 9.15% 9.44% 7.90% 55.10% U.S. Inflation 3.74% 3.78% 2.89% 77.35% As we d suspect, 4-year volatility drops by ½ when ϕ 0: = = Not so when ϕ > 0 Panel B: Using (Scaled) Four-Year Investment Horizons S&P % 12.06% 7.93% 12.40% U.S. Small Stocks 14.38% 14.73% 9.43% % MSCI EAFE 7.67% 8.08% 10.25% 5.93% U.S. L- Government Bonds 8.70% 8.85% 6.11% % U.S. L- Corporate Bonds 8.78% 8.97% 6.80% % Domestic High-Yield Corporate Bond 9.76% 9.92% 6.27% % U.S. 30-Day reasury Bills 5.05% 5.10% 3.44% 87.34% NAREI-Equity 12.61% 12.90% 8.37% % NCREIF Property 9.14% 9.30% 6.00% -6.96% U.S. Inflation 3.74% 3.77% 2.76% 65.51%

39 A Second Look: Correlations with 4-Year Returns 38 Before considering portfolio allocations, we need the correlation matrix: Correlation Matrix for Selected Asset Classes Based Upon Four-Year Returns for the Period U.S. Small Stocks Correlation Coefficients U.S. L- Corp Domestic Hi-Yld Corp U.S. 30 Day - Bill MSCI U.S. L- NAREI- NCREIF U.S. Asset Class: S&P 500 EAFE Gvt Equity Property Inflation S&P U.S. Small Stocks MSCI EAFE U.S. L- Government Bonds U.S. L- Corporate Bonds Domestic High-Yield Corporate Bond U.S. 30-Day reasury Bills NAREI-Equity NCREIF Property U.S. Inflation Average Coefficient * * excluding inflation and itself. Interestingly, these increased the most Source: Morningstar and author's calculations Proxy for the diversification benefit of various asset classes Lowest best

40 A Second Look: Efficient Frontier with 4-Year Returns 39 Efficient Frontier for Selected Asset Classes using (Annualized) Four-year Returns for the Period % U.S. Small Stk Annualized Return (Arithmetic Average) 12% 10% 8% 6% 4% 2% NCREIF Property U.S. 30 Day bill U.S. L Gvt Private Real Estate S&P 500 Domestic Hi-Yld Corp U.S. L Corp NAREI-Equity MSCI EAFE Public Real Estate 0% 0% 2% 4% 6% 8% 10% 12% Risk (Standard Deviation)

41 A Second Look: Components of the Efficient Frontier 40 As the efficient frontier becomes highrisk/highreturn, the allocation to small stocks becomes untenable (we will revisit). 11.6% he average Real Estate Allocation (no REIs)

42 A Second Look: Placing Allocation Constraints 41 What if the excessive allocations to small stocks (and/or any other asset class) are constrained? Changing Allocations to Real Estate as Maximum Weights Change for the Period Average (Public & Private) Real Estate Allocations Differences based on Using Maximum Weight to Using One-Year Returns Using Four-Year Returns One- v. Four-Year Returns: Any Asset Class Public Private otal Public Private otal Public Private otal 100% 8.1% 16.1% 23.5% 0.0% 11.6% 11.6% -8.1% -4.4% -11.9% 90% 8.1% 16.9% 24.5% 0.2% 12.1% 12.3% -7.8% -4.8% -12.2% 80% 8.5% 17.2% 25.6% 0.9% 12.5% 13.4% -7.6% -4.7% -12.3% 70% 9.9% 17.9% 27.8% 2.1% 12.9% 15.0% -7.8% -5.1% -12.9% 60% 12.5% 18.7% 31.2% 4.2% 13.2% 17.4% -8.3% -5.5% -13.8% 50% 15.6% 19.0% 34.6% 7.2% 13.6% 20.8% -8.3% -5.4% -13.7% Average 10.4% 17.6% 27.9% 2.4% 12.6% 15.1% -8.0% -5.0% -12.8% Standard Deviation 3.0% 1.1% 4.3% 2.8% 0.7% 3.5% 0.3% 0.4% 0.8% Coefficient of Variation 28.9% 6.4% 15.3% 114.8% 5.9% 23.2% -3.7% -8.1% -6.2% Both on a relative and absolute basis, the allocations to public real estate are more volatile than to private real estate. hough not directly observed here, the increasing allocations to public real estate reflect the tightening constraints on allocations to small-cap stocks his figure is still higher than the average (U.S.) pension plan s allocation to private real estate.

43 Long-Run Horizons & Mixed-Asset Portfolio Allocations 42 First Look: Annual Returns Serial Correlation & Long-Run Volatility Serial Correlation & Long-Run Correlation Second Look: Four-Year Returns aking Returns to the Limit

44 Finally, Let s Look at Infinite Holding Periods 43 Let s assume the sample of 36 years worth of data is representative of the entire population of returns. Our previous exercises resulted in examining volatility and correlation in, 1 1+ ϕ ρ ϕϕ the limit, as : lim = and lim x y = x y he other element we need for an MP-based analysis is the mean or average. With infinite holding-period returns, the average is represented by the geometric mean. Recall A few definitions with regard to the mean and volatility: t= 1 1 ϕ ρ 1 ϕ 1 ϕ 2 2 xy, x y ( ) 2 2 rt rt r t= 2 ( 1 ) 1; 1 t= r = + r = ; = 1 and t r r r 2 Clearly, we are in peril of pushing the data too far. Nevertheless, these infinite-horizon analyses reveal some real-world issues.

45 Finally: Infinite Holding-Period (Annualized) Returns 44 Consider the infinite holding period (annualized) returns on major asset classes: Annualized Summary Statistics for Selected Asset Classes Assuming Infinite Holding Periods and Based Upon Annual Returns for the Period Asset Class: Average Return Standard Deviation S&P 500 U.S. Small Stocks MSCI EAFE U.S. L Gvt U.S. L Corp Domestic Hi-Yld Corp U.S. 30 Day - Bill NAREI- NCREIF Equity Property S&P % 16.86% U.S. Small Stocks 14.38% 17.73% MSCI EAFE 7.67% 25.20% U.S. L- Government Bonds 8.70% 9.96% U.S. L- Corporate Bonds 8.78% 9.93% Domestic High-Yield Corporate Bond 9.76% 13.07% U.S. 30-Day reasury Bills 5.05% 14.49% NAREI-Equity 12.61% 17.73% NCREIF Property 9.15% 14.68% U.S. Inflation 3.74% 8.09% Average Coefficient * * excluding inflation and itself. Source: Morningstar and author's calculations. Correlation Coefficients Second lowest (after - Bills) U.S. Inflation

46 Finally: Infinite Holding-Period Efficient Frontier 45 16% Efficient Frontier and Selected Asset Classes for the hirty-six-year Period Using Estimated Infinite-Horizon Return 14% U.S. Small Stocks Average Return 12% 10% 8% 6% 4% U.S. L- Government Bonds U.S. L- Corporate Bonds S&P 500 Domestic High-Yield Corporate Bonds NCREIF Property Private Real Estate NAREI-Equity U.S. 30-Day reasury Bills MSCI EAFE Public Real Estate 2% 0% 0% 5% 10% 15% 20% 25% 30% Volatility

47 Finally: Components of the Efficient Frontier 46 As the efficient frontier becomes highrisk/highreturn, the allocation to small stocks becomes untenable (we will revisit). 6.8% he average Real Estate Allocation (no REIs)

48 Finally: Placing Allocation Constraints 47 What if the excessive allocations to small stocks (and/or any other asset class) are constrained? Changing Allocations to Real Estate as Maximum Weights Change for the Period Maximum Weight to Average (Public & Private) Real Estate Allocations Differences based on Using Using One-Year Returns Using Infinite-Horizon Returns One- v. -Year Returns: Any Asset Class Public Private otal Public Private otal Public Private otal 100% 8.1% 16.1% 23.5% 0.0% 6.8% 6.8% -8.1% -9.3% -16.7% 90% 8.1% 16.9% 24.5% 0.7% 7.0% 7.7% -7.4% -9.9% -16.8% 80% 8.5% 17.2% 25.6% 2.8% 7.2% 10.1% -5.6% -9.9% -15.6% 70% 9.9% 17.9% 27.8% 6.3% 7.4% 13.6% -3.6% -10.6% -14.2% 60% 12.5% 18.7% 31.2% 10.5% 7.4% 17.9% -2.0% -11.2% -13.2% 50% 15.6% 19.0% 34.6% 15.6% 7.4% 23.1% 0.1% -11.6% -11.5% Average 10.4% 17.6% 27.9% 6.0% 7.2% 13.2% -4.4% -10.4% -14.7% Standard Deviation 3.0% 1.1% 4.3% 6.1% 0.3% 6.3% 3.2% 0.9% 2.1% Coefficient of Variation 28.9% 6.4% 15.3% 102.1% 3.6% 48.0% -71.5% -8.5% -14.2% Both on a relative and absolute basis, the allocations to public real estate are more volatile than to private real estate. hough not directly observed here, the increasing allocations to public real estate reflect the tightening constraints on allocations to small-cap stocks While this figure is less than half that indicated when using oneyear returns, the allocation is still higher than the average (U.S.) pension plan s allocation to private real estate.