---
title: "KOSPI·KOSDAQ Efficient Frontier"
language: "en"
canonical: "https://kred.dev/en/stocks/frontier"
lookback: "1y"
window_start: "2025-09-08"
window_end: "2026-09-04"
eligible_issues: 892
---

# KOSPI·KOSDAQ Efficient Frontier

KOSPI and KOSDAQ apart, this page draws each issue's place, the ex-post frontier and the tradable headline index on windows from a year to the whole record.

## What does the efficient frontier show?

It is a map with volatility on the horizontal axis and mean return on the vertical, both measured from the chosen window's weekly returns, and one dot per issue. The curve is the highest mean return those issues, mixed together, could have reached at each level of volatility. Everything it draws has already happened, so it holds nothing about the returns to come.

One dot is one issue, and the colored line wrapping the upper left of the dots is the ex-post frontier.

## How are the crosshair and the range buttons read?

The crosshair runs through the headline index that can actually be traded and cuts the plane into four regions. Left of it are issues that swung less than that index, above it issues that earned more on average. Changing the range button moves the issues and the curve together, and because each window is a different sample of a different period, that movement carries both estimation error and genuine change in the market.

The same issue sitting upper-left of the crosshair and lower-right of it. Changing the window moves the position.

## How does it matter for financial markets?

It is used to look back at how well diversification worked over the window. The distance between the individual dots and the curve is the gain from mixing, and the minimum-variance combination, which uses no mean-return estimate, is the most stably estimated point on the curve. A curve drawn from a sample tends to sit above what was actually attainable, so the shape of the curve and its distance from the headline index are read rather than its height.

The gray dots are individual issues, the colored dot the minimum-variance combination of them. It sits to the left of every one of them, and that distance is the effect of diversification.

## Overview

A descriptive map of a past window places each listed KOSPI and KOSDAQ issue as a dot on the plane of volatility and mean return, and adds the ex-post efficient frontier, the boundary that mixing a market's issues could have reached. The two markets are estimated separately, so the screen shows one of them at a time. The values come from weekly returns over the chosen window and say nothing about returns to come. The same issue moves as the window changes from one year to the whole record, because each window covers a different stretch of market conditions.

This is not a tool for choosing issues. An issue on the curve or in the upper left is not thereby better than one elsewhere, and the crosshair is only a reference line through the headline index that can actually be traded.

## Definition

An efficient frontier is the locus of the portfolios that carry the least variance at each expected return, and what this screen draws is the after-the-fact version of that locus. Per-issue means and covariances are estimated on weekly simple returns over the selected window, and the minimum-variance boundary computed from those estimates, under the constraint that no weight falls below zero, is placed on the plane of annualized volatility and annualized mean return. KOSPI and KOSDAQ are estimated separately, each on its own partition of one issue set that passed the same eligibility screen, so the two curves are separate estimates drawn from different samples and the screen shows one market's plane at a time. The object is the efficient set that has been the starting point of asset-pricing theory since Markowitz (1952) organized portfolio choice around the two moments of mean and variance, except that it is computed from a past window's sample rather than from true parameters.

The scatter spreads the same set of issues over the window and the plane the curve is drawn on, one dot per issue. A dot's horizontal coordinate is the annualized sample standard deviation of the issue's weekly returns and its vertical coordinate the annualized sample mean, and the curve draws the upper-left boundary of the combinations that can be built from those issues on the shrunk covariance. Marked on the curve are the minimum-variance combination, which uses no mean-return estimate at all, and the volatility-matched combination, the point whose in-sample volatility is closest to equal weighting. The same plane also shows the equal-weight comparison that holds those eligible issues at equal weights, the market's composite index together with the headline index that tradable products track, and the short-term risk-free rate placed at zero volatility. The crosshair runs through the headline index whenever its source data cover the window.

Where the curve runs is where sample estimates sit, not where returns will land. The sharpest statement of that distinction is the three curves of Broadie (1993). The curve drawn from the true parameters, the curve drawn from estimates and the curve actually earned by combinations chosen on those estimates are three different objects, and the estimated curve sits systematically above the other two. The statistical fragility of sample frontiers was recorded early by Jobson and Korkie (1981), and the fact that estimation error is amplified rather than averaged away as it passes through an optimizer became widely known once Michaud (1989) described optimization as an error-maximizing device. The size of that sensitivity was measured by Best and Grauer (1991) for small changes in means and by Chopra and Ziemba (1993) across errors in means, variances and covariances, and the conclusion that the mean is the most fragile input weighs more heavily the shorter the sample, as Merton (1980) showed. The same fragility makes the minimum-variance combination the most stably defined point on the curve, and it is also the reason the covariance enters as the shrinkage estimator that Ledoit and Wolf (2003) brought to portfolio selection and Ledoit and Wolf (2004) gave its constant-correlation form, not as the raw sample covariance. Jagannathan and Ma (2003) show that a lower bound of zero on the weights is itself a regularization equivalent to shrinkage, and this screen accordingly constrains the weights to be nonnegative. The equal-weight comparison sits beside the curve, carrying onto this plane the comparison of DeMiguel, Garlappi, and Uppal (2009), in which optimized combinations tend to trail plain equal weighting out of sample.

## Methodology

The computation is repeated from scratch for each selected window, and no parameter is shared between windows.

**(a) Returns.** For issue $i$ in week $t$ the return $r_{i,t}$ is the weekly simple return obtained by compounding the holder returns of the trading days that fall in that week. Weeks are cut on the trading calendar as calendar weeks beginning on Monday, and the last trading day of the week is the week's date. Returns are price returns that leave out cash dividends, so issues with large dividends, and preferred shares above all, sit lower on the vertical axis than their realized holding performance. A week in which trading is halted and no new price prints carries a return of zero and the accumulated move is realized all at once in the week trading resumes, an accounting that holds the halted balance as zero-return cash. Days on which the reference price is reset are looked up in the capital-change ledger and treated differently according to whether a ratio was confirmed or the gap was judged a halt without a capital change, and a delisted issue is valued at its last observed close and then held as zero-return cash. Since eligibility requires listing throughout the window, however, no delisted issue ever reaches this plane. The treatment of the bias that arises when delisting returns are truncated follows Shumway (1997). Only simple returns make a portfolio return exactly the weighted sum of its parts, so log returns are not used, and a single issue therefore shows different figures here and on the per-issue detail screen, which measures annualized volatility from daily log returns. The weekly frequency is chosen to lessen, at the level of the covariance, the correlation bias that nonsynchronous trading loads onto daily returns.

**(b) Eligible set.** As of the window's last trading day, an issue is kept only if it was listed before the window began and was not delisted within it, if the share of the window's weeks in which it traded clears its floor, if the median traded value over the window's trading days clears its floor, and if it belongs to the KOSPI or the KOSDAQ market. Preferred shares are included. Index membership plays no part in the screen and no issue's eligibility depends on any other issue, so the market split leaves eligibility untouched. When a market's eligible count falls short of the minimum, that market's curve is not drawn for that window. The longer the window, the more the sample tilts toward survivors, since only issues listed for the whole span remain.

**(c) Moments.** The window's weekly sample mean $\bar{r}_i$ and sample standard deviation $s_i$ are annualized arithmetically by the number of weeks in a year $h$.

$$\hat{\mu}_i = h\,\bar{r}_i,\qquad \hat{\sigma}_i = \sqrt{h}\,s_i$$

Because the vertical axis is an arithmetic mean, the compounded cumulative return falls further below it as volatility rises, so a higher vertical position does not imply higher cumulative performance. The sampling error of a mean estimated from a short sample was formalized by Merton (1980), and it shrinks only with a longer window, not with more frequent observation.

**(d) Covariance.** The issue count rivals the number of observed weeks and on the shorter windows exceeds it, so the sample covariance $S$ is noisy, and on those windows singular, and it is therefore shrunk toward a constant-correlation target $F$ as a convex combination.

$$\hat{\Sigma} = \delta\,F + \bar{\delta}\,S$$

Here $\bar{\delta}$ is the remaining weight, one minus the shrinkage intensity $\delta$. The target $F$ is the matrix that sets every pair's correlation to the single average of the sample correlations while keeping the sample variances, and $\delta$ is obtained from the constant-correlation formula of Ledoit and Wolf (2004) and truncated to lie between zero and one. Shrinkage was brought to portfolio selection by Ledoit and Wolf (2003), and the large-dimensional version with an identity target was formalized by Ledoit and Wolf (2004, Journal of Multivariate Analysis), but an identity target pulls correlations toward zero and makes the diversification effect look larger than it is, which is why this screen uses the constant-correlation target. Under either target the estimator remains a simplified approximation of the correlation structure among issues.

**(e) Boundary.** The target mean $m$ is moved on an even grid from the mean of the minimum-variance combination to the largest sample mean, and a quadratic program is solved at each grid point.

$$\min_{w}\; w^{\top}\hat{\Sigma}\,w \quad \text{s.t.}\quad w^{\top}\hat{\mu} \ge m$$

The weight vector $w$ is constrained to have nonnegative components summing to one. The solution comes from an active-set quadratic program started from the previous grid point's solution, with sequential quadratic programming as the fallback when it does not converge. The minimum-variance combination solves the same problem without the target constraint, and the volatility-matched combination is the point on the curve whose in-sample volatility is closest to equal weighting. That the lower bound on the weights acts as a regularization equivalent to shrinkage was shown by Jagannathan and Ma (2003), and since this procedure solves the problem of Markowitz (1952) on sampled inputs it carries the amplification of estimation error that Michaud (1989) named. The sensitivity analyses of Best and Grauer (1991) and Chopra and Ziemba (1993) measured how large that amplification is.

**(f) Anchors and the comparison.** The market composite and the headline index take their returns from weekly close ratios of the index level on the same weekly grid, annualized the same way. An index has no reference-price resets, so its close ratio is the holder return as it stands. The headline index carries a point only when its source data cover the whole window, and when they do not, no point is made and the crosshair moves to the composite. The short-term risk-free rate is the latest observation on or before the window's last date, placed at zero volatility. The equal-weight comparison holds the same eligible set at equal weights and follows the comparison framework of DeMiguel, Garlappi, and Uppal (2009). The positions of the curve portfolios, the equal-weight comparison and the indices are all gross of transaction costs, taxes and market impact.

## Applications in Economics

The mean-variance plane is where economics first wrote the choice of accepting risk in pursuit of return as a single optimization problem. Before Markowitz (1952) diversification was the maxim about not putting all the eggs in one basket, and after it diversification became a geometry set by the covariance matrix. Capital-market equilibrium theory was built on that geometry, and the proposition that the market rewards only the risk that diversification cannot remove follows from the assumption that every investor faces the same efficient set. An ex-post frontier therefore does two jobs in economics, a ruler for the diversification benefit the market delivered over the window and at the same time a specimen of how little that ruler can be trusted.

As a ruler, the frontier measures the diversification effect realized over the window in the distance between the scatter and the curve. The farther the individual issues are strewn to the right of the curve, the lower the correlations among them and the larger the gain from mixing. In a span when one factor pulled the whole market along, those correlations rise and the distance narrows. KOSPI and KOSDAQ are estimated on separate planes, so the difference between the two markets is never folded into a single curve.

Merton (1980) showed that the precision of an expected-return estimate is governed by the length of the sample and is not improved by observing it more often, a result that leaves the vertical coordinates on a short window barely trustworthy. Jobson and Korkie (1981) recorded by simulation how far sample frontiers stray statistically from the true one, and Michaud (1989), by identifying the structure through which an optimizer picks out the inputs with the largest errors, explained why optimized portfolios were shunned in practice. Best and Grauer (1991) measured how much small changes in means upend the weights, Chopra and Ziemba (1993) quantified how many times larger the utility loss from errors in means is than the loss from errors in variances or covariances, and Broadie (1993) separated the true frontier, the estimated frontier and the frontier actually attained. Conclusions about asset allocation that rest on mean estimates are fragile for these reasons, above all when they are drawn from macro data, and the comparatively stable object is the minimum-variance combination, defined by the covariance structure alone.

Covariance estimation does not escape the limits of sample size either. In a panel whose issue count rivals its number of observations the sample covariance carries its most extreme correlations at face value, and the shrinkage estimators of Ledoit and Wolf (2003) and Ledoit and Wolf (2004) press those extremes toward a structured target and narrow the room an optimizer has to pick out error. Jagannathan and Ma (2003) showed that a constraint keeping weights from turning negative does the same work, which gives statistical meaning to the common practice of holding only long positions in the underlying shares. Even so, as DeMiguel, Garlappi, and Uppal (2009) confirmed across several data sets, plain equal weighting often beats elaborate optimization out of sample. That dropping delisted issues from the sample leaves only survivors and lifts the whole frontier belongs to the same family as the truncation bias shown by Shumway (1997), and that bias reappears on this screen, where longer windows hold fewer issues and tilt toward survivors.

## Applications in Financial Markets

In practice an ex-post frontier is a frame for diagnosing performance over a past window, not a tool for setting the allocation of the next one. A manager sets the position of the portfolio against the same window's curve, reads how far below it the portfolio sat at the same volatility, and splits that gap into the part that was missing diversification and the part that was issue selection gone wrong. The reference point on this screen is the headline index the crosshair runs through. The broad products that can actually be traded in the Korean market are the exchange-traded funds and futures that track KOSPI 200 and KOSDAQ 150 rather than the composites, so the four regions the crosshair cuts show whether each issue swung less than that index and whether it earned more on average. The composite is the second anchor, showing where the whole market sat once the small and mid caps that index leaves out are included.

Every attempt to carry the framework of Markowitz (1952) into practice ran into input estimation first. As Michaud (1989) pointed out, an optimizer piles weight onto the issues whose means happened to be estimated high and whose correlations happened to be estimated low, and the sensitivity measured by Best and Grauer (1991) and Chopra and Ziemba (1993) means that those weights turn over wholesale when the inputs move a little. Practical prescriptions therefore gathered into three strands, namely pressing the extremes of the correlation estimates with the covariance shrinkage of Ledoit and Wolf (2003) and Ledoit and Wolf (2004), keeping extreme weights out with the lower bound that Jagannathan and Ma (2003) justified, and removing the mean estimate from the inputs altogether through a minimum-variance strategy. That this screen marks only the minimum-variance and volatility-matched combinations on the curve and carries neither weights nor a tangency portfolio follows the same prescriptions. The upper right of the curve is driven by the issues whose means were estimated highest, as Michaud (1989) and Best and Grauer (1991) showed, so the uncertainty of a position grows in that direction.

Reading the height of the curve as attainable performance misuses the tool. Broadie (1993) showed that a curve drawn from a sample sits above the true curve, while the curve actually earned by combinations chosen in reliance on that sample falls below it, and Jobson and Korkie (1981) recorded by simulation how far a sample frontier strays from the true one. The equal-weight comparison of DeMiguel, Garlappi, and Uppal (2009) shows the form that gap takes in practice, where a combination that sat on the curve in sample commonly trails equal weighting in the next window. When the range button changes, the curve and every issue's position move together, and since the windows differ in sample and in span, that movement mixes estimation error and market change.

The accounting also limits how a position may be read. On a price-return basis issues with large cash dividends and preferred shares sit below their realized holding performance, and since the figures are gross of transaction costs, taxes and market impact, the gap to what can actually be earned is widest for the thinnest-traded issues. This screen's eligibility requires listing throughout the window, so the sample on a long window tilts toward survivors. The accounting that does not truncate delisting returns answers the bias Shumway (1997) documented, but no issue on this plane ever falls under that accounting.

## Statistical Tests

Because the curve is an after-the-fact description rather than an estimate that publishes a test statistic, testing is organized as gates that ask whether the computation was carried out to the pre-registered specification, not how it performs out of sample. The gate criteria were registered before the computation and scored regardless of direction.

The integrity gate checks exhaustively, over the baseline construction and every sensitivity cell, that the smallest eigenvalue of the shrunk covariance is positive, that the quadratic program converged, that the weights sum to one within tolerance, that no weight is negative and that the attained mean does not fall short of its target. The same predicate is applied without sampling to every window of the serving grid, and no window is written until it passes. Every window passed. The windows the page draws are serving-layer output rather than the evidence grid the battery scores, and they are written under that same precondition.

The accounting audit recomputes the out-of-sample path of the latest completed vintage with independent code inside the battery and checks that the weekly return errors lie within floating-point tolerance. It passed. The solver-equivalence check sets the quadratic-program solutions at the grid points against the solutions of an independent second optimizer and requires the objective gap and the feasibility violations to be at floating-point level. It passed.

The landing record writes, for every completed vintage, where the minimum-variance combination, the volatility-matched combination, the equal-weight comparison and the indices actually landed in the next window. It is a prior commitment to publish whether the curve wins or loses out of sample, and a record that rules on nothing.

The sensitivity cells change one factor at a time, lengthening and shortening the estimation window, lowering the liquidity floor, raising the observation requirement to full coverage and switching the shrinkage target to a scaled identity matrix. The same gates are applied to each of them. A deliberately wrong construction is added beside them, one that contaminates the universe with the issues surviving at the data cut-off, as a negative control for how far survivorship bias pushes the curve up. Every cell and artifact is bound to the hash of the pre-registered specification, and a change to the specification marks all of them stale.

Some things are not tested. How far the estimated curve lies from the true one is the question Jobson and Korkie (1981) and Broadie (1993) treated, and it is not measured here. Whether optimized combinations beat equal weighting out of sample is the question of DeMiguel, Garlappi, and Uppal (2009), on which the landing record leaves data and gives no answer. The sampling error of the mean estimates is set by the window length, as Merton (1980) showed. The gate outcomes and the per-cell figures are in the gate record.

## Key Figures

Values for the default window, the most recent year. The chart's range button does not move this table. Positions drawn from data between 2025-09-08 and 2026-09-04

### Key figures for KOSPI

| Portfolio or index | Annualized volatility | Annualized mean return |
|---|---|---|
| Eligible issues | 385 |  |
| Minimum variance | 8.3% | 2.5% |
| Volatility-matched | 28.1% | 137.2% |
| Equal weight | 29.9% | 37.2% |
| KOSPI Composite | 38.6% | 81.3% |
| KOSPI 200 Index | 42.6% | 98.3% |
| Risk-free short rate | 0.0% | 3.1% |

### Key figures for KOSDAQ

| Portfolio or index | Annualized volatility | Annualized mean return |
|---|---|---|
| Eligible issues | 507 |  |
| Minimum variance | 15.4% | -21.1% |
| Volatility-matched | 40.1% | 151.7% |
| Equal weight | 41.0% | 35.3% |
| KOSDAQ Composite | 36.2% | 6.6% |
| KOSDAQ 150 Index | 42.8% | 10.8% |
| Risk-free short rate | 0.0% | 3.1% |

## Frequently Asked Questions

### Does the efficient frontier say anything about future returns?

No. It sets out, after the fact, where the chosen window's data landed, and a curve drawn from a sample tends to lie above the curve that was actually attainable.

### What is the crosshair on the efficient frontier drawn through?

The position of the headline index that tradable products track. Over long windows that the source data do not cover, the composite index takes its place.

### Why does the efficient frontier volatility differ from the per-issue detail?

This plane annualizes the standard deviation of weekly simple returns while the per-issue detail screen measures from daily log returns. Neither value is wrong.

### Why do longer efficient frontier windows hold fewer issues?

Only issues listed throughout the window are eligible, so the sample tilts toward survivors.

### Which point on the efficient frontier is estimated most stably?

The minimum-variance combination. It is determined by the covariance alone, so the mean-return estimate, whose errors are largest in a short sample, never enters it as an input.

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