Constant Relative Risk Aversion - chunhualiao/public-docs GitHub Wiki

power utility function

Excellent question. You're now diving into the core economic theory that underpins Vanguard's entire approach to long-term, multi-asset investing.

The Constant Relative Risk Aversion (CRRA) utility function is a foundational concept from economics that Vanguard uses in its Vanguard Life-Cycle Investing Model (VLCM) to determine the optimal asset allocation, or "glide path," for its Target-Date Funds.

Let's break down what this means.

1. What is "Utility"?

In economics, "utility" is a way to represent a person's satisfaction or happiness. A utility function assigns a numerical value to different levels of wealth. The key idea is that everyone wants to maximize their utility.

A fundamental assumption is diminishing marginal utility of wealth. This means that an extra dollar is far more valuable to you when you have $1,000 than when you have $1,000,000. The first dollar provides a huge utility boost, while the millionth-and-first dollar barely moves the needle.

2. What is "Risk Aversion"?

Risk aversion describes how an investor feels about taking risks. A risk-averse investor, when faced with two options with the same average expected return, will always choose the one with less risk. The diminishing marginal utility of wealth directly leads to risk aversion. Because you feel the pain of losing a dollar more than the pleasure of gaining a dollar, you are naturally hesitant to gamble.

3. What is "Constant Relative Risk Aversion (CRRA)"?

This is the most crucial part of the term. It describes a specific, and very intuitive, type of risk aversion.

Relative Risk Aversion: This means your willingness to take risks is related to your proportion of wealth.
Constant: This means that as your wealth grows, your proportional risk-taking preference stays the same.

In simple terms, a CRRA investor who is comfortable investing 60% of their $100,000 portfolio in stocks would also be comfortable investing 60% of a $1,000,000 portfolio in stocks. Their desired asset allocation, as a percentage, doesn't change just because their wealth has increased.

This is contrasted with Constant Absolute Risk Aversion (CARA), where an investor would keep the same dollar amount in risky assets. The CRRA assumption is considered much more realistic for modeling investor behavior.

The Mathematical Formula for CRRA Utility

The most common mathematical representation of a CRRA utility function is the power utility function:

$$U(W) = \frac{W^{1-\gamma}}{1-\gamma}$$

Where:

$U(W)$ is the utility derived from a certain level of wealth, $W$.
$W$ is the investor's total wealth.
$\gamma$ (gamma) is the coefficient of relative risk aversion. This is the single most important parameter in the function.

The coefficient $\gamma$ determines an investor's level of risk aversion:

If $\gamma = 0$, the investor is risk-neutral.
If $\gamma > 0$, the investor is risk-averse.
A higher $\gamma$ means higher risk aversion. An investor with $\gamma=5$ is much more conservative than an investor with $\gamma=2$.

Vanguard, in its models, assumes a hypothetical average investor with a specific, proprietary $\gamma$ value that it believes best represents its target client base.

How Vanguard Uses CRRA for Target-Date Funds: The "Human Capital" Connection

This is where the theory meets practice. If an investor's relative risk tolerance is constant (CRRA), why does Vanguard aggressively change the asset allocation in its Target-Date Funds?

The answer lies in how Vanguard defines "total wealth." For the VLCM, an investor's total wealth is not just their financial portfolio; it is:

Total Wealth = Financial Capital (your portfolio) + Human Capital

Human Capital is the present value of all your future expected earnings. For a young person, their human capital is their biggest asset—it is stable and bond-like, as it represents a steady stream of future income.
As you get older, you "spend" your human capital and convert it into financial capital (savings and investments).

The CRRA assumption applies to this Total Wealth. To keep the risk level of the total wealth portfolio relatively constant, the allocation of the financial portfolio must change dramatically over time.

A Concrete Example of the Decision Process

Let's imagine two investors, both with the same CRRA utility function (the same $\gamma$).

Investor 1: A 25-Year-Old

Financial Capital: $25,000
Human Capital: $1,500,000 (huge, and behaves like a bond)
Total Wealth: $1,525,000
Decision: The investor's total wealth is overwhelmingly bond-like due to their massive human capital. To achieve their desired overall risk level (as defined by their CRRA preference), the VLCM will recommend that their financial capital be invested almost entirely in stocks (e.g., a 90% stock allocation). This aggressive stock position in the small financial portfolio is balanced by the stable, bond-like nature of their human capital.

Investor 2: A 65-Year-Old

Financial Capital: $1,500,000
Human Capital: $0 (they are at retirement)
Total Wealth: $1,500,000
Decision: This investor's total wealth is now composed entirely of their financial portfolio. To achieve the same desired overall risk level as their younger self (as per their constant relative risk aversion), the allocation of this portfolio must become much more conservative. The VLCM will recommend an allocation closer to 50% stocks and 50% bonds.

This is precisely the logic of the glide path. It is not that the investor's risk tolerance changes. Rather, the composition of their total wealth changes, and the financial portfolio must adjust to maintain a constant relative risk profile for the investor's total economic condition.

Here’s a step-by-step, intuition-first look at what “expected CRRA utility of terminal wealth for each simulated path” really means and why people bother to compute it.

1. Why utility instead of just dollars?

Raw wealth (or return) treats every extra dollar the same.
Most humans don’t. The first $100 000 feels more valuable than the next $100 000, and a $100 000 loss feels worse than the same-size gain feels good.
Utility functions translate “dollar world” into “happiness / satisfaction world,” capturing this diminishing-marginal-value and loss-aversion behavior.

2. What CRRA utility looks like

The Constant Relative Risk Aversion (CRRA) family is popular because it:

has a single parameter, $\gamma$ (“gamma”), that dials risk aversion:
- $\gamma = 1$ → $U(W)=\ln W$ (log utility)
- $\gamma = 2$ → square-root-shaped curve
- Higher $\gamma$ → steeper dislike of volatility
is scale-invariant—doubling all wealth outcomes doesn’t change the investor’s choices, which fits many real-world decisions.

Mathematically

$$ U(W)= \begin{cases} \dfrac{W^{1-\gamma}}{1-\gamma}, & \gamma \neq 1 \[6pt] \ln W, & \gamma = 1 \end{cases} $$

3. Where “simulated paths” come in

When you back-test or Monte-Carlo-simulate an investment strategy, you generate many possible future histories (paths). Each path ends in a different terminal wealth $W_i$.

Path (i)	terminal wealth $W_i$
1	$90 000
2	$110 000
…	…
N	$160 000

4. Step-by-step calculation

Compute utility for each path $U_i = U(W_i)$
Average them (a simple, equally weighted mean if paths are simulated with equal probability)

$$ \text{Expected utility} = \frac{1}{N}\sum_{i=1}^{N} U_i $$

That single number summarizes, in utility space, both the average outcome and the dispersion of outcomes your strategy produces.
(Optional but common) Translate back into dollars—certainty equivalent Solve for the sure amount $CE$ that gives the same utility:

$U(CE) = \text{Expected utility}$

If $CE$ is much lower than the average terminal wealth, the strategy is “too risky” for someone with risk-aversion $\gamma$.

Concrete mini-example

Assume three equiprobable paths:

Path	$W_i$	Utility when $\gamma = 2$
A	$80 000	$U = \tfrac{80{,}000^{,-1}}{-1}=1/80{,}000 = 0.0000125$
B	$100 000	$U = 1/100{,}000 = 0.00001$
C	$140 000	$U = 1/140{,}000 = 0.00000714$

Expected utility $= (0.0000125+0.00001+0.00000714)/3 ≈ 0.00000988$

Certainty equivalent when $\gamma = 2$: $$CE = 1 / (0.00000988) ≈ $101 200$$

Even though the average wealth of the three paths is $106 667, a risk-averse investor with $\gamma = 2$ would be indifferent between facing that risky distribution and simply receiving a sure $101 200.

5. What it tells you in practice

Compare strategies: Higher expected utility (for the same $\gamma$) → better for that investor.
Tune risk exposure: Vary allocation until expected utility stops rising, rather than maximising raw return.
Stress-test risk aversion: Check results for several $\gamma$ values (e.g., 1, 3, 5) to see how sensitive your conclusions are.

Key takeaways

Expected CRRA utility is just “take every simulated outcome, score it with a risk-adjusted happiness ruler, then average.”
It rewards higher average wealth and penalises dispersion in one coherent metric.
Converting the result back to a certainty equivalent makes the abstract utility number feel like dollars again.

(All formulas assume positive wealth; if paths can hit zero or negative values, you’d clip those or use another utility form to keep the math well-defined.)

Below is the “plumbing diagram” of where the CRRA utility function sits inside the Vanguard Life-Cycle Investing Model (VLCM) and what it is doing at each step.

1 VLCM in one sentence

VLCM searches across thousands of candidate glide paths, savings-/spending rules, and retirement ages and keeps the one that maximises the investor’s expected lifetime utility, subject to the capital-market forecasts generated by the Vanguard Capital Markets Model (VCMM). (institutional.vanguard.com)

The utility yard-stick at the heart of that optimisation is the CRRA function.

2 How the CRRA utility is wired in

Stage	What VLCM does	Where CRRA appears
A. Simulate economic futures	10 000 VCMM paths deliver annual real returns for each asset class.	—
B. Build personal cash-flow ledger	For each path and each future year, VLCM maps: pay-packet (human capital), contributions, portfolio returns, Social-Security & DB income, chosen spending rule.	—
*C. Convert each year’s consumption* and any end-of-life bequest into utility**	Uses the CRRA formula

$$ U(x)= \begin{cases} \dfrac{x^{1-\gamma}}{1-\gamma}, & \gamma\neq1\[4pt] \ln x, & \gamma=1 \end{cases} $$

to score that year’s real dollars. | CRRA encodes risk aversion (γ) and diminishing marginal value. | | D. Weight by survival & time-preference | Each year’s utility is multiplied by the probability you’re alive and by a discount factor β. | Utility is now “consumption-per-year” not just terminal wealth (institutional.vanguard.com) | | E. Average across Monte-Carlo paths | Add up steps C-D across years and across the 10 000 scenarios → expected lifetime utility of that candidate glide path. | Still CRRA-based but now an expectation. | | F. Apply behavioural tweaks | VLCM can tilt those utilities if income falls below a floor (income-shortfall aversion) or to overweight bequests, by re-weighting the low-tail or final-wealth utilities. The base CRRA formula is unchanged—the weights are. (institutional.vanguard.com) | | G. Pick the winner & translate to “certainty-fee equivalent” (CFE) | Candidate with the highest expected utility wins; the utility gap versus a benchmark glide path is restated as the maximum fee (in bp/yr) an investor with that γ would pay to switch. |

3 Why CRRA is the right tool here

Single dial for risk aversion (γ). Higher γ ⇒ steeper dislike of volatility ⇒ VLCM returns a more bond-heavy glide path. That matches the sensitivity analysis in Appendix 4 of the paper. (institutional.vanguard.com)
Scale-invariance. Doubling every cash-flow simply multiplies utility by a constant, so results aren’t distorted by the units of measurement.
Analytic convenience. CRRA keeps the optimisation convex enough for search algorithms to converge, even with Monte-Carlo noise.
Decomposable across time. Because VLCM cares about lifetime welfare, it needs a utility that can be summed year-by-year after discounting—exactly what CRRA allows.

4 What changes in a didactic re-implementation?

Vanguard production model	Teaching/DIY version
Full ledger of annual consumption + bequest utilities	Often simplified to utility of terminal wealth only (still CRRA) to keep run-times short.
Survival-probability weighting	Can be dropped or approximated with a fixed retirement horizon.
Behavioural add-ons (income-shortfall, myopic loss aversion, bequest tilt)	Optional extensions once the core CRRA engine is working.
Search over thousands of glide paths with constraints	Start with a grid search or gradient-free optimiser over equity share by age.
“Certainty-fee equivalent” translation	Nice teaching device: solve $U(CE)=\text{Expected utility}$ to express welfare gains in dollars or bps.

Key takeaway

In VLCM every design choice—how fast the glide path de-risks, how much equity to hold at age 75, whether to favour a bequest—rises or falls on a single scalar score: the CRRA-based expected lifetime utility. Master that utility pipeline and the rest of the model’s behaviour becomes transparent.