Using CHAT GPT to Learn New Financial Topics
CHAT helped me very quickly become familiar with issues and applications related to a CAGR with a multiple holding period.
Start by answering this from pervious session. s there any literature on modifying CAGR for multiple holding periods with different start and end points then take standard deviation of multiple holding points and using to c calculate a multiple period Sharpe ratio?
ChatGPT said:
You're diving into an intricate and meaningful question—how to properly modify CAGR for multiple holding periodswith differing start/end points, compute the standard deviation across those periods, and use that to derive a multi-period Sharpe ratio. Let’s unpack what existing literature might say on each piece:
1. Modifying CAGR for Multiple Holding Periods
CAGR is typically defined over a fixed, single period—say, from point A to point B. Multiple, overlapping or varying holding periods complicate this, because CAGR doesn't easily aggregate across intervals with different lengths or timing InvestopediaAspire.
In Excel-oriented discussions, when people need CAGR across different start points or non-uniform periods, a common suggestion is to use IRR or XIRR, especially if the periods vary in length Reddit. Those are cash-flow–based, money-weighted returns that adjust for timing and the amount of each holding interval—offering more flexibility than CAGR.
2. Calculating Standard Deviation Across Multiple Holding Points
If you're sampling returns from different holding intervals, you can treat each CAGR-like return as an observation and compute a standard deviation among them—but that neglects the fact that CGARs are geometrically smoothed and may not meet assumptions of stationarity or independence.
More rigorously, volatility (standard deviation) typically scales with the square root of time only if returns are serially uncorrelated. With serial correlation or overlapping periods, this approach fails arXiv.
3. Multi-Period Sharpe Ratio Using That Setup
The Sharpe ratio in its classic form is (R−risk-free rate)/σ(R−risk-free rate)/σ, where RR is the average return and σσ is the standard deviation Stanford Universitygregorygundersen.com.
A critical caveat: you can't simply annualize a monthly Sharpe ratio by multiplying by 1212 unless return series are i.i.d. Lo shows this can lead to large misstatements—annual Sharpe could be overstated by as much as 65% if there's serial correlation ResearchGate.
In multi-period contexts, the standard Sharpe ratio—paired with independent intervals and clean return data—could be applied, but it’s risky if intervals overlap, vary in length, or exhibit serial correlation.
4. Related Academic Insights
Jakša Cvitanić et al. (2007) studied the use of the Sharpe ratio in multi-period compensation settings, illustrating that emphasizing short-term Sharpe can be detrimental to long-horizon outcomes—highlighting dynamic implications of performance measurement Bank of CanadaCaltech.
Andrew Lo (2002) went deep into how to accurately annualize Sharpe ratios, considering estimation error and serial correlation—very relevant to your goal of combining multiple holding-period returns into one metric ResearchGate.
There's also literature on the Deflated Sharpe Ratio, a statistical adjustment accounting for multiple testing, data mining, and backtesting bias—it's not directly about multi-period aggregation but demonstrates the importance of robust Sharpe estimation in complex contexts Wikipedia.
Summary Table
ComponentWhat Literature SuggestsCAGR across multiple periodsUse IRR/XIRR for cash-flow–adjusted returns RedditStandard deviation of returnsBe cautious—must adjust for serial correlation and overlapping periods arXivResearchGateMulti-period Sharpe ratioClassic Sharpe formula applies—but annualization needs careful treatment; serial correlation matters Stanford UniversityResearchGateBank of CanadaDynamic multi-period effectsShort-term performance focus may distort long-term outcomes Bank of CanadaCaltech
Final Thoughts
While there's no single “off-the-shelf” formula for combining CGARs across varying holding periods into a proper standard deviation and multi-period Sharpe ratio, here’s a more structured approach you could consider:
Compute returns consistently—preferably using money-weighted methods like IRR/XIRR for each holding period.
Adjust for serial correlation when you calculate volatility—standard formulae may misstate risk otherwise.
Annualize Sharpe ratios cautiously—remove biases from overlapping data or non-i.i.d. behavior.
Review performance dynamics—short-term peaks in Sharpe may not translate to multi-period improvement, per Cvitanić et al.
If you're building a research paper or model using this framework, you’d likely cite Lo (2002) for Sharpe estimation issues, Cvitanić et al. (2007) for dynamic multi-period considerations, and possibly literature on IRR alternatives to CAGR.
Let me know if you'd like help diving deeper into any of these papers—or translating them into formulas or code for your analysis.
You said:
Is there a way to modify a multi holding period CAGR so that you gat an index with an upper bound and lower bound, perhaps 0 to 1 or 0 to 100? Has someone done this. Provide cites.
ChatGPT said:
Thought for 2m 24s
Short answer: there isn’t a single, standard “bounded CAGR index” in the academic canon. CAGR is naturally unbounded on the upside (and bottoms at −100%). But there are well-used ways to turn multi-period return information into a bounded score:
Percentile/score mapping (0–100)
What most rating and ranking systems do is compute a return or a risk-adjusted return over multiple horizons, then convert it to a percentile or score on a fixed scale.
Morningstar calculates a risk-adjusted return (MRAR) for 3/5/10-year windows and then ranks funds within category to assign stars—i.e., a bounded scale derived from percentiles. Morningstar+2Morningstar+2admainnew.morningstar.com
Refinitiv StarMine models explicitly output 1–100 daily percentile ranks (bounded) for securities; the same “score from percentiles” idea can be applied to any return metric, including rolling CAGRs. LSEGFidelity
Factor index providers standardize inputs with z-scores (then often map to bands/percentiles), a common way to normalize unbounded variables before converting to a bounded scale. MSCI
How you’d do it for multi-holding-period CAGR: compute a panel of rolling-window CAGRs (e.g., all 36-month and 60-month windows from different starts), choose your peer set, then map each asset’s statistic to a percentile → 0–100 index. (Mathematically this is just applying the empirical CDF—min–max/percentile scaling are standard tools.) PMC
Probability-based measures (0–1)
Another route is to translate the whole multi-period return stream into a Sharpe-like statistic and then map that to a probability in [0,1]:
Probabilistic Sharpe Ratio (PSR): probability the true Sharpe exceeds a chosen hurdle, accounting for sample length, skewness, and kurtosis—naturally bounded in [0,1]. SSRNDavid H Bailey
Deflated Sharpe Ratio (DSR): further adjusts for multiple testing/selection bias; also yields a probability-type figure you can report as 0–1 (or multiply by 100). SSRNDavid H Bailey
(If you use rolling windows to form the return series, remember that overlapping windows induce serial correlation, which affects Sharpe estimation; Lo (2002) shows how to handle this.) SSRNCFA Institute Research and Policy Center
Monotonic “squash” transforms (design your own 0–1)
If you want a purely mechanical, absolute index from CAGR (not relative to peers), apply a monotonic squashing function to a stabilized return (e.g., log(1+CAGR)):
Logistic/sigmoid mapping: I=1/(1+exp(−k(g−m)))I=1/(1+exp(−k(g−m))) gives a 0–1 index; pick m (center) and k (slope) from your sample (e.g., median and IQR). This is a standard way to map unbounded reals to (0,1). ScienceDirectWikipedia
Min–max to [0,1]: choose a lower/upper bound (e.g., −100% to a capped high return) and rescale; widely used in feature scaling, but the choice of caps drives the score. PMC
What’s “already been done”?
Industry: bounded 0–100 scores through percentiles/deciles are mainstream (Morningstar stars come from ranked MRAR; StarMine outputs 1–100 ranks). Morningstar+1LSEG
Academia: PSR/DSR provide rigorous probability-style (0–1) indices from multi-period return streams via the Sharpe framework (and are explicitly designed to cope with non-normality and multiple testing). SSRN+1David H Bailey
Practical recipe (pick one):
Percentile index (0–100): build all rolling k-year CAGRs for each asset → compute MRAR-style downside-penalized return if desired → convert to within-universe percentile. (Morningstar’s approach is the closest off-the-shelf analogue.) Morningstar
Probability index (0–1): compute an overall Sharpe on your multi-period return series (handle overlap/serial correlation per Lo) → report PSR or DSR as your bounded index. SSRN+2SSRN+2
Absolute transform (0–1): stabilize with log(1+CAGR), then apply a logistic II with center/slope chosen from your historical distribution; or use min–max with explicit caps. ScienceDirectPMC
If you want, I can turn one of these into a short, reproducible formula (or a small script) tailored to your dataset and horizon choices.
You said:
Is there a literature using multi period CAGRs to assess optimal portfolio for people retiring in 10 years? Or would this not work because optimal portfiolio changes as people near retirement?
ChatGPT said:
Great question. You’re essentially asking whether multi-period CAGR-style metrics are used in the retirement literature for portfolio choice with a 10-year horizon, and whether this makes sense given the dynamic nature of optimal portfolios near retirement.
Here’s what the literature says:
1. How CAGR fits into retirement portfolio analysis
CAGR is a summary statistic of average growth over a horizon. It’s attractive for simple “wealth at horizon” calculations, but it doesn’t capture path-dependence or interim risk.
Retirement researchers typically need to account for sequence-of-returns risk: the order of gains and losses matters far more near retirement than just the average CAGR. This is why simple CAGR measures are not widely used as optimization objectives in retirement planning.
Key point: Multi-period CAGR by itself does not generally appear as the central objective in retirement portfolio optimization. Instead, literature uses utility functions, downside risk measures, or stochastic dynamic programming.
2. What the literature actually uses
Lifecycle & glidepath models
Bodie, Merton, and Samuelson (1992) – classic paper showing that optimal asset allocation is time varying, depending on human capital, horizon, and risk aversion. As the retirement date approaches, optimal portfolios usually reduce equity exposure (the “lifecycle” intuition).
Milevsky & Huang (2011) – stress the importance of sequence risk and “ruin probabilities” when determining optimal retirement glidepaths.
Poterba, Rauh, Venti, Wise (2006, NBER) – evaluate how lifecycle funds adjust asset allocation as investors age; again, not focused on CAGR but on expected utility and consumption smoothing.
Multi-period optimization
Campbell & Viceira (2002), Strategic Asset Allocation – develops multi-period models of portfolio choice with stochastic investment opportunities. They show that optimal allocation depends on horizon, risk premia, and hedging demands, not just average growth.
Balduzzi & Lynch (1999) – examine dynamic portfolio choice with time-varying investment opportunities; path dependence matters.
Retirement risk measures
Rather than CAGR, common retirement metrics are:
Probability of ruin / shortfall
Safe withdrawal rates (Bengen 1994; Guyton & Klinger 2006)
Utility-based measures of consumption volatility
Downside-focused risk-adjusted returns (e.g., MRAR, downside deviation instead of σ)
3. Why CAGR alone is insufficient near retirement
Horizon compression: With only ~10 years left, volatility of returns dominates CAGR as a risk factor.
Sequence of returns risk: Two portfolios with identical 10-year CAGR can leave retirees in very different positions depending on whether losses occur early or late.
Dynamic hedging needs: Optimal allocations shift as retirement nears because future human capital (income) shrinks, so the investor has less capacity to recover from losses.
4. Summary
There isn’t a strong literature on using multi-period CAGR directly to assess the optimal retirement portfolio.
Instead, the academic and practitioner literature favors multi-period expected utility models, lifecycle (glidepath) strategies, and risk-of-ruin analyses.
CAGR may be a useful descriptive metric but is not a sufficient optimization criterion for retirement planning because optimal portfolios must adapt dynamically as the horizon shortens.
👉 If you’d like, I can pull specific formula examples from Campbell & Viceira (2002) or Bodie et al. (1992) showing how allocation weight shifts as you move from 20 years out to 10 years out. Would you like me to extract and compare those models so you can see how they differ from a simple multi-CAGR approach?
You said:
Shouldn't they track changes in CAGR as holding periods shorten or would this just justify a target fund?
ChatGPT said:
You’ve put your finger on the core tension:
whether tracking changes in CAGR as the investment horizon shortens is a meaningful way to guide allocation, or whether this is essentially what a target-date (glidepath) fund already implements.
1. CAGR as Horizon Shortens
What it shows: If you compute rolling multi-period CAGR for progressively shorter horizons (say 30 → 20 → 10 years), you’ll see how dispersion widens. Shorter holding periods make long-term averages less stable, because volatility dominates over compounding.
Problem: While this highlights risk, it does not capture the distributional shape of returns—especially the risk of “bad early outcomes” (sequence risk). Two strategies with the same 10-year CAGR distribution can still have very different probabilities of leaving someone underfunded at retirement.
2. What Lifecycle / Target-Date Funds Do
Dynamic glidepathing: Target-date funds implement the insight that as horizon shortens, equity risk capacity shrinks. They gradually reduce equity allocation, regardless of the observed CAGRs.
Justification: This isn’t because expected CAGR changes in a mechanical way as retirement nears; it’s because the utility of avoiding large losses increases. You can’t “wait it out” with only a decade left.
In other words: The glidepath is a structural response to horizon risk, not to the time-series of CAGR estimates.
3. Where CAGR Tracking Might Help
As a diagnostic tool, tracking how rolling CAGRs behave across horizons can illustrate the uncertainty facing near-retirees. For example:
30-year horizon: rolling CAGRs cluster tightly (volatility averages out).
10-year horizon: rolling CAGRs spread wide (sequence risk exposed).
This visual/quantitative spread of CAGRs is often used in financial planning simulations (e.g., Monte Carlo retirement models) to communicate risk, but not usually to drive allocation formulas directly.
4. Academic Consensus
Yes: tracking changes in horizon-adjusted return distributions is important.
No: relying on raw CAGR as a decision metric is insufficient—optimization frameworks prefer expected utility, shortfall risk, or dynamic programming.
Result: The glidepath/target-date approach is essentially the institutionalized answer to your question—it operationalizes the idea that risk tolerance should decline as holding period shortens.
✅ Bottom line:
Tracking CAGRs over shorter horizons is illustrative but not prescriptive. What really matters is how shrinking horizons change the risk/return trade-off, which is why lifecycle models and target-date funds are the standard practice. Your instinct is correct—doing this systematically more or less “justifies” the existence of target funds.
Would you like me to sketch a quick chart (e.g., with historical S&P 500 data) showing how the distribution of rolling CAGRs widens as you move from 30-year to 10-year horizons? That’s a powerful way to see why glidepaths exist.
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