EV-LP Review And Update
Expected Value tells us whether an edge exists. Likely Profit tells us whether our bankroll can sustain the way we choose to bet it.
Sports bettors often ask the question: Is this bet profitable (+EV) on average? This is the right place to start, but it is not the right place to finish.
In three earlier posts, we developed our approach incrementally:
- Pairing Expected Value (EV) with Likely Profit (LP).
- Separating the Bookmaker (Additive) and Bettor (Multiplicative) Playbooks.
- Converting both EV and LP into practical probability thresholds.
This review follows that same progression, from finding value to protecting growth.
Starting Point: Expected Value and Likely Profit
Suppose a bettor risks a fraction \(F\) of their current bankroll at decimal odds \(O\), with an estimated win probability \(P\).
The bankroll has two possible one-bet multipliers:
Here, \(WB\) is the Win-Balance multiplier and \(LB\) is the Loss-Balance multiplier. Decimal odds include the returned stake, so the winning net profit is \(F(O-1)\), not \(FO\).
The original equations deliberately use the same outcomes and probability weights:
This parallel structure is central to the framework. EV takes the probability-weighted arithmetic mean of the two bankroll outcomes; LP takes their probability-weighted geometric mean. The inputs are symmetrical in form, while the different averaging operation is precisely what makes LP more sensitive to losses and stake size.
Expected Value: Is There An Edge?
The EV equation simplifies to:
Therefore, \(EV>0\) means that the bettor's estimated probability is high enough to make the offered odds profitable on average.
EV is the arithmetic view. It answers:
How much should this bet return on average?
Likely Profit: Does The Stake Imply Positive Geometric Growth?
The companion LP equation can equivalently be written as an expected log-growth rate:
Because \(LP=e^g-1\), \(LP\) and \(g\) always have the same sign. A positive LP indicates positive long-run geometric growth under repeated bets with the same probability, odds, and bankroll fraction.
LP is not the probability of finishing ahead over a fixed number of bets, and it does not describe the path, drawdowns, or risk of ruin. It is a transformed expected log-growth measure whose long-run interpretation assumes that the estimated probabilities remain valid and that repeated outcomes behave consistently with them.
LP is the compounding view. It answers:
Does this stake imply positive geometric growth under the model?
That distinction produced the original decision rule:
| EV | LP | Interpretation |
|---|---|---|
| Positive | Positive | The edge and stake size support bankroll growth |
| Positive | Zero or negative | The bet may have value, but the stake is too aggressive |
| Zero or negative | Any | No estimated edge at the offered odds |
Positive EV is necessary. Positive LP asks whether the bettor is using that edge to generate bankroll growth.
Two Playbooks
The second stage of the framework explained why EV can be enough for one side of the market but incomplete for the other.
Bookmaker's Additive World
At the level of a broad, well-capitalized, and actively risk-managed book, bookmakers seek to collect small pricing advantages across many bets. Diversification and volume can make aggregate results more predictable, although correlated liabilities, model error, and capital constraints still require risk management.
Their playbook is therefore additive:
Take a small edge repeatedly and let volume accumulate the profit.
For pricing and aggregate margin, EV is a central metric. Individual outcomes matter less than the average margin across the full book, but they do not cease to matter for liability management.
Bettor's Multiplicative World
A bettor faces a different constraint: finite capital. A loss reduces the base available for the next stake, while a win increases it. When stakes are fractions of the current bankroll, returns compound multiplicatively.
That makes sizing inseparable from selection. Two bets can offer the same +EV and still have very different consequences for long-run growth (-LP).
Consider a $1,000 bankroll with a $100 stake:
| Bet X: High Risk | Bet Y: Moderate Risk | |
|---|---|---|
| Decimal odds | 11.0 | 2.0 |
| Estimated win probability | 10% | 55% |
| Expected Value (EV) | +$10 | +$10 |
| Win-Balance multiplier | 1.9 | 1.1 |
| Loss-Balance multiplier | 0.9 | 0.9 |
| Likely Profit (LP) | -3.02% | +0.50% |
Correction to the earlier post: Recalculation gives \(LP_X=-3.0174\%\) and \(LP_Y=+0.5021\%\). These replace the previously published values of \(-1.32\%\) and \(+0.27\%\).
Both bets have identical positive EV. Bet X, however, combines a low hit rate with a 10% bankroll stake. Its negative LP says that repeatedly taking this risk at that size would shrink the typical compounded bankroll. Bet Y clears both tests.
This is the bettor's playbook:
Find positive EV, then size the bet so that expected bankroll growth remains positive.
LP does not replace EV, and it does not turn a poor probability estimate into a good one. It simply exposes the interaction between edge, odds, and stake size.
From Metrics to Probability Thresholds
The third stage made the process directly actionable. Instead of calculating EV and LP after choosing \(P\), we can solve for the minimum probability needed to clear each test.
EV Threshold
Setting \(EV=0\) gives:
The result is the market's implied probability. Ignoring commission and other costs, positive EV requires:
This threshold depends on the odds, but not on the stake fraction. Changing the stake changes the amount won or lost; it does not create an edge.
LP Threshold
Setting \(LP=0\), or equivalently \(g=0\), gives:
Unlike \(P_{EV}\), this threshold depends on both the odds and the stake fraction through \(WB\) and \(LB\). For any non-zero stake below the full bankroll, the LP threshold is stricter than the EV threshold. The larger the stake, the more certainty is required to overcome volatility drag.
Canonical Example
Take decimal odds of 1.9091 and a stake equal to 1% of bankroll:
The thresholds are:
An estimated probability of 55% clears both. An estimate between 52.38% and 52.62% identifies a positive-EV bet that is nevertheless oversized at a 1% bankroll stake.
This leads to a practical sequence:
- Estimate the outcome probability conservatively.
- Compare it with \(P_{EV}\) to test whether an edge exists.
- Compare it with \(P_{LP}\) to test whether the proposed stake supports geometric growth.
- Reduce the stake or pass when the LP test fails.
Closing Line Value
Closing Line Value (CLV) asks whether the price taken by the bettor was better than the market's closing price. It is usually interpreted as a process metric: consistently beating a liquid closing market suggests that the bettor is identifying information or mispricing before it is fully absorbed.
There is no single universal CLV equation. Let:
- \(O_B\) be the decimal odds when the bet was placed;
- \(O_C\) be the closing decimal odds; and
- \(q_B=1/O_B\) and \(q_C=1/O_C\) be their raw implied probabilities.
For a back bet, common definitions include:
| CLV definition | Equation | Interpretation |
|---|---|---|
| Odds difference | \(O_B-O_C\) | Simple price movement, but difficult to compare across odds ranges |
| Odds-ratio CLV | \(\frac{O_B}{O_C}-1\) | Percentage improvement in gross payout relative to the close |
| Probability CLV | \(q_C-q_B\) | Change in implied probability, measured in percentage points |
| Relative probability CLV | \(\frac{q_C}{q_B}-1\) | Relative repricing; algebraically equal to odds-ratio CLV |
| Log CLV | \(\ln(O_B/O_C)\) | A symmetric, additive measure that is convenient for aggregation |
Positive back-bet CLV means that the taken odds were higher than the closing odds: \(O_B>O_C\). For a lay bet, the direction reverses because laying at lower odds than the eventual close is favorable.
Closing-Price EV Proxy
The connection to DMDA is direct if we treat the de-vigged closing probability \(p_C\) as the market's best available estimate of the true probability. The expected return per unit staked at the original price is then:
If the closing market is fair, so that \(p_C=1/O_C\), this becomes:
Under those assumptions, odds-ratio CLV is not merely correlated with value: it is the bet's closing-price EV estimate per unit staked. In the bankroll-scaled DMDA notation:
This also links CLV to the first probability threshold:
Positive CLV therefore says that, under the closing-price proxy, the closing market supports the conclusion that the entry price cleared the EV threshold.
DMDA-Qualified CLV
Positive CLV does not automatically mean that the stake cleared the stricter LP threshold. DMDA can extend the usual CLV test by evaluating the original bet at the closing probability:
where:
Within this framework, we call the result DMDA-qualified CLV when the closing-price proxy supports both positive EV and positive LP. This is a framework-specific classification rather than a universal industry term. The closing-price proxy supports positive geometric growth only when:
This creates three useful post-bet classifications:
| Closing assessment | Interpretation |
|---|---|
| \(p_C\le P_{EV}\) | Negative or zero CLV; the close does not support the entry edge |
| \(P_{EV}<p_C\le P_{LP}\) | Positive CLV, but not enough to support the chosen stake |
| \(p_C>P_{LP}\) | DMDA-qualified CLV; the proxy supports both the edge and positive geometric growth at the chosen stake |
For the canonical bet at \(O_B=1.9091\) and \(F=1\%\), consider a closing price of \(O_C=1.905\):
| Measure | Result |
|---|---|
| EV probability threshold, \(P_{EV}\) | 52.380703% |
| LP probability threshold, \(P_{LP}\) | 52.618839% |
| Closing implied probability, \(p_C=1/O_C\) | 52.493438% |
| Odds-ratio CLV, \(O_B/O_C-1\) | +0.215223% |
| Closing-price EV per unit staked | +0.215223% |
| Closing-price LP, \(LP_C\) | -0.002395% |
The close moved the implied probability by approximately 0.112735 percentage points and produced positive odds CLV. Under the closing-price proxy, it therefore supports the entry price against the EV threshold. However, \(52.493438\%<52.618839\%\), producing \(LP_C=-0.002395\%\). The proxy therefore does not support positive geometric growth at the chosen 1% stake.
Converting the LP probability threshold back into closing odds gives:
At \(O_C=1.900460\), closing-price LP is approximately zero. Strictly positive LP requires the back selection to close at odds below this boundary, subject to de-vigging and commission adjustments.
Measurement Cautions
CLV is only as reliable as the closing benchmark:
- In multi-outcome bookmaker markets, use de-vigged closing probabilities rather than raw \(1/O_C\) values.
- On betting exchanges, the final liquid price or Betfair Starting Price may provide a cleaner benchmark, but thin markets and late shocks can still make it noisy.
- Commission should be included in DMDA calculations. For a simplified back-bet calculation with commission rate \(c\) on net winnings, use \(O_{\text{eff}}=1+(1-c)(O-1)\); actual exchange commission is settled on net market winnings.
- Compare like with like: same market, selection, rules, and settlement terms.
- One bet's CLV proves little. The signal becomes useful through repeated, preferably liquidity-weighted observations and calibration against outcomes.
CLV does not replace the bettor's own probability estimate. It supplies a second, later estimate from the market. That makes it a valuable audit trail for DMDA:
Pre-bet DMDA asks whether our probability clears the EV and LP thresholds. Post-bet CLV asks whether the closing-price proxy supports that assessment.
Full DMDA Framework
The three stages fit together as a single discipline:
First, identify positive expected value bets (+EV). EV compares the bettor's probability estimate with the price offered by the market.
Second, respect your bankroll constraints. A finite bankroll compounds, so variance and stake size cannot be ignored.
Third, demand enough growth. The EV threshold establishes the minimum for an edge; the higher LP threshold establishes the minimum for growth at the chosen stake.
In compact form:
For real betting-exchange decisions, these thresholds should be calculated after allowing for commission on net market winnings. Commission reduces the effective winning return, raises the break-even probability, and narrows small apparent edges.
No formula can rescue a badly calibrated probability estimate. DMDA is a decision framework, not a prediction engine. Its value is that it forces the bettor to state three things explicitly: the estimated probability, the offered price, and the fraction of bankroll at risk.
Conclusion
Expected Value remains the foundation of rational betting, but a bettor cannot live on arithmetic averages alone. Bankrolls grow geometrically, losses reduce future capacity, the sequence of wins and losses exposes us to volatility drag, and excessive sizing can turn a genuine edge into negative long-run growth.
The completed DMDA framework therefore asks two questions in order:
- Is the price favorable (+EV)? That is the EV test.
- Is the stake sustainable (+LP)? That is the LP test.
The objective is not merely to find bets that look profitable on paper. It is to manage risk and stake sensibly enough to give a real edge the opportunity to compound.
EV finds the opportunity. LP determines whether the bankroll can sustain it.
Note: Sanity Check of final draft using ChatGPT.
