風險評估模型蒙特卡洛模型

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A common problem when evaluating a portfolio manager is that the history of returns is often so short that estimates of risk and performance measures can be highly unreliable. A similar problem occurs when testing a new trading strategy. Even if you have a fairly long history for the strategy’s performance, often you only have observations over a single market cycle which can make it difficult to evaluate how your strategy would have held up in other markets. If you trade stocks you have probably heard the refrain: “I’ve never seen a bad back test”.

在評估投資組合經理時，一個常見的問題是收益的歷史通常很短，以至于風險和績效指標的估計可能非常不可靠。 測試新的交易策略時會發生類似的問題。 即使您對策略的執行歷史已有很長的歷史，但通常您只能在單個市場周期中觀察到，這可能使您難以評估您的策略在其他市場中的表現。 如果您交易股票，您可能會聽到這樣的話：“我從未見過糟糕的回測”。

One method to address this deficiency is through Factor Model Monte Carlo (FMMC). FMMC can be used to estimate a factor model based on a set of financial and economic factors that reliably explain the returns of the fund manager. We can then simulate returns to determine how the manager would have performed in a wide variety of market environments. The end result is a model that produces considerably better estimates for risk and performance than if we simply used the return series available to us.

解決此缺陷的一種方法是通過因素模型蒙特卡洛(FMMC) 。 FMMC可用于基于一組可靠地解釋基金經理收益的財務和經濟因素來估計因素模型。 然后，我們可以模擬收益，以確定經理在各種市場環境中的表現。 最終結果是，與僅使用可用的收益序列相比，該模型可以更好地估算風險和績效。

任務和設置 (The Task and Set Up)

For this case study, we will be analyzing the returns for the new hedge fund Aric’s Hedge Fund; hereafter known as AHF. The hedge fund case is particularly interesting because hedge funds can use leverage, invest in any asset class, go long or short, and use many different instruments. Hedge funds are often very secretive about their strategy and holdings. Thus, having a reliable risk model to explain the source of their returns is essential.

在本案例研究中，我們將分析新對沖基金Aric's Hedge Fund的回報； 以下簡稱AHF 。 對沖基金的案例特別有趣，因為對沖基金可以利用杠桿，投資于任何資產類別，做多或做空以及使用許多不同的工具。 對沖基金通常對其策略和持股非常保密。 因此，擁有可靠的風險模型來解釋其收益的來源至關重要。

Keep in mind that Aric’s Hedge Fund is not a real hedge fund (I’m Aric, I don’t have a hedge fund), but this is a real series of returns. I obtained the returns for a hedge fund in operation that we invest in where I work so the results of this study are applicable to a real-world scenario.

請記住，Aric的對沖基金不是真正的對沖基金(我是Aric，我沒有對沖基金)， 但這是一系列真實的回報。 我獲得了對沖基金的回報，該對沖基金投資于我工作的地方，因此本研究的結果適用于現實情況。

We have data for Aric’s Hedge Fund from January 2010 to March 2020. For the purpose of this post and evaluating the accuracy of our model we will pretend as though AHF is pretty new to the scene and that we only have data from January 2017 through March 2020. To overcome the data deficiency, we will build a factor model on the basis of this “observed data” and then utilize the entire data series to evaluate the accuracy of our simulation for assessing the risk and performance statistics.

我們擁有2010年1月至2020年3月Aric對沖基金的數據。出于這篇文章的目的，并評估我們模型的準確性，我們將假裝 AHF尚不成熟，并且我們僅提供2017年1月至3月的數據。 2020年。為克服數據不足，我們將在此“觀察數據”的基礎上建立一個因子模型，然后利用整個數據系列來評估模擬的準確性，以評估風險和績效統計數據。

The below graph shows the cumulative return of AHF since January 2010. The data to the right of the red line represents the “observed period”.

下圖顯示了自2010年1月以來AHF的累計收益。紅線右側的數據代表“觀察期”。

We will be conducting the analysis in R using the extensive library of packages available therein including: PerformanceAnalytics and quantmod. Aside from the hedge fund returns series, all of the factor data can be obtained freely from Yahoo! Finance, the Federal Reserve Bank of St. Louis FRED Database, and the Credit Suisse Hedge Fund Indices; you have to sign up for Credit Suisse to access the indices, but still…free.

我們將使用其中提供的廣泛的軟件包庫在R中進行分析，包括： PerformanceAnalytics和quantmod 。 除了對沖基金收益系列以外，所有因子數據都可以從Yahoo!免費獲得。 金融 ，圣路易斯聯邦儲備銀行FRED數據庫和瑞士信貸對沖基金指數 ； 您必須注冊瑞士信貸才能訪問該指數，但仍然…免費。

模型估計 (Model Estimation)

A common technique in empirical finance is to explain changes in asset prices based on a set of common risk factors. The simplest and most well-known factor model is the Capital Asset Pricing Model (CAPM) of William Sharpe. The CAPM is specified as follows:

經驗金融中的一種常見技術是基于一組常見風險因素來解釋資產價格的變化。 最簡單和最著名的因素模型是William Sharpe的資本資產定價模型(CAPM)。 CAPM指定如下：

Where:

哪里：

• ri = Return of asset ‘i’
  ri =資產“ i”的收益
• m = Return of market index ‘m’
  m =市場指數“ m”的回報
• ∝= Excess return
  ∝ =超額收益
• ? = Exposure to the Market Risk factor
  ?=暴露于市場風險因素
• ?i = Idiosyncratic error term
  = i =特異誤差項

Market risk, or “systematic” risk, serves as a summary measure for all of the risks to which financial assets are exposed. This may include recessions, inflation, changes in interest rates, political turmoil, natural disasters, etc. Market risk is usually proxied by the returns on a large index like the S&P 500 and cannot be reduced through diversification.

市場風險或“系統性”風險，是金融資產??所面臨的所有風險的匯總度量。 這可能包括衰退，通貨膨脹，利率變化，政治動蕩，自然災害等。市場風險通常由標普500指數等大型指數的回報所替代，并且無法通過多元化降低。

? (i.e. Beta) represents an asset’s exposure to market risk. A Beta = 1 would imply that the asset is as risky as the market, Beta >1 would imply more risk than the market, while a Beta < 1 would imply less risk.

?(即Beta)代表資產的市場風險。 Beta = 1表示資產的風險與市場一樣，Beta> 1的風險比市場高，而Beta <1的風險比市場低。

? is idiosyncratic risk and represents the portion of the return that cannot be explained by the Market Risk factor.

?是特質風險，代表不能由市場風險因素解釋的收益部分。

We will extend the CAPM to include additional risk factors which the literature have shown to be important for explaining asset returns. Aric’s Hedge Fund runs a complicated strategy using many different asset classes and instruments so it’s certainly plausible that it would be exposed to a broader set of risks beyond the traditional market index. The general form of our factor model is as follows:

我們將把CAPM擴展到包括其他風險因素，這些因素已被文獻證明對于解釋資產收益很重要。 Aric的對沖基金使用許多不同的資產類別和工具來執行一項復雜的策略，因此，它可能會面臨比傳統市場指數更廣泛的風險。 我們的因子模型的一般形式如下：

All the above says is that returns (r) are explained by a set of risk factors j=1…k where r j is the return for factor ‘j’ and ? j is the exposure. ? is the idiosyncratic error. Thus, if we can estimate the ? j, then we can leverage the long history of factor returns (r j) calculate conditional returns for AHF. Finally, if we can reasonably estimate the distribution of ? then we can build randomness into AHF’s return series. This enables us to fully capture the variety of returns that we could observe.

上面所說的全部是，回報(r)由一組風險因子j = 1…k解釋，其中rj是因子“ j”的回報，而?j是風險敞口。 ?是特質錯誤。 因此，如果我們可以估計?j，那么我們就可以利用長期的要素收益率(rj)計算AHF的條件收益率。 最后，如果我們可以合理估計distribution的分布，則可以將隨機性構建到AHF的收益序列中。 這使我們能夠充分捕捉我們可以觀察到的各種回報。

The FMMC method will take place in three parts:

FMMC方法將分為三個部分：

• Part A: Data Acquisition, Clean Up and Processing
  A部分：數據采集，清理和處理
• Part B: Model Estimation
  B部分：模型估算
• Part C: Monte Carlo Simulation
  C部分：蒙特卡洛模擬

A部分：數據采集，清理和處理 (Part A: Data Acquisition, Clean Up and Processing)

For the factor model I will be using a set of financial and economic variables aimed at measuring different sources of risk and return. Again, all the data used in this study are freely available from Yahoo! Finance, the FRED Database, and Credit Suisse.

對于因子模型，我將使用一組金融和經濟變量，旨在衡量風險和回報的不同來源。 同樣，本研究中使用的所有數據均可從Yahoo!免費獲得。 金融，FRED數據庫和瑞士信貸。

We’ll begin with the FRED data. Next to each variable I have placed the unique identifier that you can query from the database.

我們將從FRED數據開始。 在每個變量旁邊，我放置了可以從數據庫查詢的唯一標識符。

FRED Variables:

FRED變量：

• 5-Year Inflation Expectations, 5-Years Forward. (T5YIFRM)
  未來5年通脹預期，未來5年。 (T5YIFRM)
• Term Spread: 10-Year minus 3-month Treasury Yield Spread. (T10Y3M)
  期限利差：10年期減去3個月的美國國債收益率利差。 (T10Y3M)
• Credit spread premium: Moody’s Baa corp bond yield minus 10-year Treasury yield. (BAA10Y)
  信用利差溢價：穆迪的Baa公司債券收益率減去10年期美國國債收益率。 (BAA10Y)
• 3-month T-bill rate. (DGS3MO)
  3個月期國庫券利率。 (DGS3MO)
• TED Spread: 3-Month LIBOR Minus 3-Month Treasury Yield. (TEDRATE)
  TED利差：3個月倫敦銀行同業拆借利率減去3個月國庫券收益率。 (TEDRATE)
• International bond yield: 10-Year government bond yields for Euro Area. (IRLTLT01EZM156N)
  國際債券收益率：歐元區10年期政府債券收益率。 (IRLTLT01EZM156N)
• Corporate Bond Total Return Index: ICE BofAML Corp bond master total return index; in levels. (BAMLCC0A0CMTRIV)
  公司債券總收益指數：ICE BofAML Corp債券主總收益指數； 在水平上。 (BAMLCC0A0CMTRIV)
• CBOE Volatility Index (i.e. the VIX). (VIXCLS)
  CBOE波動率指數(即VIX)。 (VIXCLS)
• CBOE Volatility Index of US 10-Year Treasuries (i.e. the Treasury VIX). (VXTYN)
  美國10年期美國國債(即美國國債VIX)的CBOE波動率指數。 (VXTYN)

The FRED API leaves something to be desired and does not allow you to pull data in a consistent way. The returns of AHF are monthly so our model will need to be estimated using monthly data. However, FRED retrieves data at the highest available frequency so daily data always comes in as daily. Furthermore, the data is retrieved from the beginning of the series, so you end up getting a lot of NAs. As such, we will need to do a little clean up before we proceed.

FRED API有一些不足之處，并且不允許您以一致的方式提取數據。 AHF的回報是每月的，因此我們的模型需要使用每月的數據進行估算。 但是，FRED會以最高可用頻率檢索數據，因此每日數據始終以每日形式出現。 此外，數據是從本系列的開始檢索的，因此您最終會獲得很多NA。 因此，在繼續之前，我們需要進行一些清理。

The following segments of R code show loading the identifiers into variables and separate queries to FRED for the daily and monthly data. The daily data is cleaned and converted to a monthly frequency. I’ve tried to comment the code as much as possible so you can see what’s happening.

R代碼的以下各節顯示將標識符加載到變量中，并分別向FRED查詢每日和每月數據。 每日數據將被清理并轉換為每月一次。 我嘗試了盡可能多地注釋代碼，以便您可以看到發生了什么。

The FRED data is good to go. The other set of variables that we will need are financial market indices. Growth, Value and Size indices feature prominently in asset pricing models such as the Fama-French 3-Factor Model and I take the same approach here. Returns from financial indices are obtained from the venerable Yahoo! Finance.

FRED數據很好。 我們將需要的另一組變量是金融市場指數。 增長，價值和規模指數在諸如Fama-French 3-Factor Model之類的資產定價模型中具有突出的地位，在此我采用相同的方法。 財務指數的收益來自古老的Yahoo! 金融。

Yahoo! Finance Variables:

雅虎！ 財務變量：

• Value: Russell 1000 Value Index (^RLV)
  值：羅素1000價值指數(^ RLV)
• Growth: Russell 1000 Growth Index (^RLG)
  成長：羅素1000成長指數(^ RLG)
• Size: Russell 2000 Index (^RUT)
  大小：羅素2000指數(^ RUT)
• Market: S&P 500 Index (^GSPC)
  市場：標普500指數(^ GSPC)
• International: MSCI EAFE (EFA)
  國際：MSCI EAFE(EFA)
• Bonds: Barclays Aggregate Bond Index (AGG)
  債券：巴克萊綜合債券指數(AGG)

Lastly, we’ll load in the hedge fund specific indices courtesy of Credit Suisse (CS). Obtaining the index requires a few extra steps as the data needs to manually downloaded to Excel from the Credit Suisse website and then loaded into R. Each index corresponds to a specific hedge fund strategy.

最后，我們將根據瑞士信貸(CS)的數據加載對沖基金的特定指數。 獲取索引需要一些額外的步驟，因為需要將數據從瑞士信貸網站手動下載到Excel，然后加載到R中。每個索引對應于特定的對沖基金策略。

Credit Suisse Variables:

瑞士信貸變量：

• Convertible Bond Arbitrage Index (CV_ARB)
  可轉換債券套利指數(CV_ARB)
• Emerging and Frontier Markets Index (EM_MRKT)
  新興和前沿市場指數(EM_MRKT)
• Equity Market Neutral Index (EQ_Neutral)
  股市中性指數(EQ_Neutral)
• Event Driven Index (EVT_DRV)
  事件驅動索引(EVT_DRV)
• Distressed Opportunities Index (DISTRESS)
  苦惱機會指數(DISTRESS)
• Multi-Strategy Event Driven Index (MS_EVT)
  多策略事件驅動索引(MS_EVT)
• Event Driven Risk Arbitrage Index (RISK_ARB)
  事件驅動風險套利指數(RISK_ARB)
• Fixed Income Arbitrage Index (FI_ARB)
  固定收益套利指數(FI_ARB)
• Global Macro Index (GL_MACRO)
  全球宏觀指數(GL_MACRO)
• Equity Long-Short Index (EQ_LS)
  股票多空指數(EQ_LS)
• Managed Futures Index (MNGD_FT)
  管理期貨指數(MNGD_FT)
• Multi-Strategy Hedge Fund Index (MS_HF)
  多策略對沖基金指數(MS_HF)

B部分：模型估算 (Part B: Model Estimation)

Recall that for the purpose of this case study we are “pretending” as though we only have data for AHF from January 2017 through March 2020 (i.e. the sample period). In reality we have data going back to January 2010. We will use the data in the sample period to calibrate the factor model and then compare the results from the simulation to the long-run risk and performance over the full period of January 2010 to March 2020.

回想一下，就本案例研究而言，我們“假裝”為好像我們只有從2017年1月到2020年3月(即采樣期)的AHF數據。 實際上，我們擁有的數據可以追溯到2010年1月。我們將使用采樣期間的數據來校準因子模型，然后將模擬結果與2010年1月至3月整個期間的長期風險和績效進行比較2020年。

Model estimation has 2-steps:

模型估算分為兩步：

Estimate a Factor Model: Using the common “short” history of asset and factor returns, compute a factor model with intercept, factor betas j=1…k, and residuals

估計因子模型：使用常見的資產和因子收益的“短”歷史記錄，計算具有截距，因子beta j = 1…k和殘差的因子模型

Estimate Error Density: Use the residuals from the factor model to fit a suitable density function from which we can draw.

估計誤差密度：使用因子模型中的殘差來擬合一個合適的密度函數，我們可以從中得出該密度函數。

I have proposed 27 risk factors to explain the returns of AHF, but I don’t know ahead of time which form the best prediction. It could be that some factors are irrelevant and reduce the explanatory power of the model. In order to select an optimal model, I use an Adjusted-R2 based best-subset selection algorithm available through the package. leaps performs an exhaustive, regression-based search across the proposed variables and selects the model with the highest Adjusted-R2. The algorithm proposes the following 14-factor model with Adjusted-R2 of .9918:

我已經提出了27個風險因素來解釋AHF的回報，但是我不知道是什么構成最佳預測。 可能某些因素無關緊要，從而降低了模型的解釋力。 為了選擇最佳模型，我使用了可通過軟件包使用的基于Adjusted-R2的最佳子集選擇算法。 jumps對建議的變量進行詳盡的，基于回歸的搜索，并選擇具有最高Adjusted-R2的模型。 該算法提出以下14因子模型，其中Adj??usted-R2為.9918：

• Russell 1000 Value (RLV)
  羅素1000值(RLV)
• Russell 2000 Index (RUT)
  羅素2000指數(RUT)
• S&P 500 (GSPC)
  標普500(GSPC)
• MSCI EAFE (EFA)
  MSCI EAFE(EFA)
• Barclays Aggregate Bond Index (AGG)
  巴克萊綜合債券指數(AGG)
• Corporate Bond Total Return Index (Corp.TR)
  公司債券總回報指數(Corp.TR)
• VIX
  VIX
• Treasury VIX (T.VIX)
  財政部國庫券(T.VIX)
• Convertible Bond Arbitrage Index (CV_ARB)
  可轉換債券套利指數(CV_ARB)
• Equity Market Neutral Index (EQ_Neutral)
  股市中性指數(EQ_Neutral)
• Multi-Strategy Event Driven Index (MS_EVT)
  多策略事件驅動索引(MS_EVT)
• Equity Long-Short Index (EQ_LS)
  股票多空指數(EQ_LS)
• Managed Futures Index (MNGD_FT)
  管理期貨指數(MNGD_FT)
• Multi-Strategy Hedge Fund Index (MS_HF)
  多策略對沖基金指數(MS_HF)

Now that we have selected our variables, we can estimate the calibrated factor model and see how it does.

既然我們已經選擇了變量，我們就可以估計校準因子模型并查看其效果。

Based on the results of the regression, we observe that AHF is significantly exposed to traditional sources of risk. Specifically, AHF appears to trade equity and debt and may employ derivatives to either hedge or speculate.

基于回歸結果，我們觀察到AHF明顯暴露于傳統風險源。 具體而言，AHF似乎在買賣股票和債務，并可能使用衍生工具對沖或進行投機。

Positive exposure to both the S&P 500 (GSPC) and MSCI (EFA) indices suggests that AHF trades global equity and has a long bias. The positive value for AGG further suggests they trade fixed income but may have a slight preference for Treasuries based on the negative coefficient for the corporate bond total return index (corp. tr). The generally significant results for the various hedge fund strategies suggests that AHF employs a complex trading strategy and may use derivatives such as futures (highly significant value for the CS Managed Futures Index, MGND_FT). Futures may be used to either hedge positions or target access to a specific market.

標普500(GSPC)和MSCI(EFA)指數的正面敞口表明AHF交易全球股票并且具有長期偏見。 AGG的正值進一步表明它們交易固定收益，但基于公司債券總回報指數(corp。tr)的負系數，可能會稍微偏愛國債。 各種對沖基金策略的總體意義重大，這表明AHF采用了復雜的交易策略，并可能使用諸如期貨之類的衍生產品(CS管理期貨指數MGND_FT的顯著價值)。 期貨可用于對沖頭寸或目標進入特定市場。

The plot of AHF’s realized returns v. the fitted values from the model demonstrates a high degree of fit and explanatory power.

AHF的已實現回報與該模型的擬合值的關系圖顯示出高度的擬合度和解釋力。

C部分：模擬 (Part C: Simulation)

Parametric and non-parametric Monte Carlo methods are both widely applied in empirical finance, but either presents challenges for estimation.

參數和非參數蒙特卡羅方法都廣泛應用于經驗金融中，但都給估計帶來挑戰。

Parametric estimation of factor densities requires fitting a large multivariate, fat-tailed probability distribution; which in our specific case would contain 14 variables. Correlations can be notoriously unstable and inaccurate estimation of the variance-covariance matrix would bias the distribution from which we will draw the factor returns. This problem may be overcome by employing copula methods, but this adds to the complexity of the model. On balance, we would prefer to avoid parametric estimation if possible.

因子密度的參數估計需要擬合較大的多元胖尾概率分布； 在我們的特定情況下，它將包含14個變量。 相關性可能是非常不穩定的，方差-協方差矩陣的不正確估計會偏向于分布我們將得出因子收益的分布。 通過使用copula方法可以解決此問題，但這增加了模型的復雜性。 總而言之，如果可能的話，我們寧愿避免參數估計。

A potential alternative is non-parametric estimation. To conduct a non-parametric simulation, we could bootstrap the observed, discrete empirical distribution that assigns a probability of 1/T to each of the observed factor returns for t=1…T. This would serve as a proxy to the true density of factor returns and allow us to bypass the messy process of estimating the correlations. However, bootstrap resampling can result in the duplication of some values and the omission of others and while this may be appropriate for inference, it does not provide an obvious advantage in our application.

潛在的替代方法是非參數估計。 為了進行非參數模擬，我們可以引導觀察到的離散經驗分布，該分布為t = 1…T的每個觀察到的因子收益分配1 / T的概率。 這可以代替要素收益率的真實密度，并允許我們繞過估計相關性的麻煩過程。 但是，自舉重采樣可能導致某些值重復，而導致其他值遺漏，雖然這可能適合推斷，但在我們的應用程序中并未提供明顯的優勢。

A more efficient method is simply to take the relatively long history of factor returns as given and add each of the residuals. Simply put, we have 123 months of factor returns (January 2010-March 2020) and 39 residuals (based on the results of the calibration portfolio which spans January 2017-March 2020). If we add each of the 39 residuals to the 123 factor returns, we can produce 123×39 scenarios for the return of AHF (4797 observations in total). This large sample should be capable of providing us with good insight into the tails of AHF’s return distribution and has the advantage of utilizing all of the observed data.

一種更有效的方法就是簡單地考慮給定的相對較長的要素收益歷史并添加每個殘差。 簡而言之，我們有123個月的因子回報(2010年1月至2020年3月)和39個殘差(基于跨越2017年1月至2020年3月的校準產品組合的結果)。 如果將39個殘差中的每個殘差加到123個因子回報中，就可以為123個AHF回報生成123×39個場景(總共4797個觀測值)。 這個大樣本應該能夠為我們提供對AHF收益分布的尾巴的良好洞察，??并具有利用所有觀測數據的優勢。

The simulation proceeds as follows:

仿真過程如下：

• Use the calibrated model and factor returns to form predictions for the returns of AHF for January 2010 through March 2020.
  使用校正后的模型和因子收益形成對2010年1月至2020年3月AHF收益的預測。
• For each of the “i”, i = 1…123, estimated returns, add each of the “j”, j=1…39, residuals.
  對于每個“ i”，i = 1…123，估計收益，將每個“ j”，j = 1…39，殘差相加。

績效分析 (Performance Analysis)

Recall that when we introduced this exercise we pretended as though we only had the performance history of AHF from January 2017 through March 2020. Such a short history of performance alone provides only limited insight into the risk/return features of a fund manager over a relatively narrow window of market conditions. To address this shortcoming and provide a more accurate picture of performance we have proposed using Factor Model Monte Carlo (FMMC). The factor model was calibrated using the short, common history of factor and fund returns. The Monte Carlo experiment used factor returns over a longer horizon (January 2010 through March 2020) and the realized factor model residuals to construct 4797 simulated returns for AHF.

回想一下，當我們引入此練習時，我們假裝我們只有從2017年1月到2020年3月才有AHF的業績歷史。僅憑如此短暫的業績歷史，就只能相對有限地了解基金經理的風險/收益特征。市場條件窗口狹窄。 為了解決此缺點并提供更準確的性能描述，我們建議使用因素模型蒙特卡洛(FMMC)。 因子模型是使用較短的常見因子和基金收益歷史進行校準的。 蒙特卡洛實驗使用更長范圍內(2010年1月至2020年3月)的要素收益率和已實現的要素模型殘差來構造AHF的4797個模擬收益率。

To evaluate the performance of our model we will focus on the results for the mean annual return and volatility as well as the venerable Sharpe and Sortino Ratios. Let’s see how we did.

為了評估模型的性能，我們將重點關注平均年收益率和波動率以及可追溯的夏普和索蒂諾比率的結果。 讓我們看看我們是如何做到的。

1. Average Return

1.平均回報

The table below depicts the mean (i.e. average) annual return for the factor model Monte Carlo (FMMC), full history of AHF (January 2010-March 2020) and the truncated/”observed” history (January 2017-March 2020):

下表描述了因子模型蒙特卡洛(FMMC)，AHF的完整歷史記錄(2010年1月至2020年3月)和截斷/“觀察到的”歷史記錄(2017年1月至2020年3月)的平均(即平均)年收益：

Immediately we can see the improvement that the FMMC model has over the Truncated period. The FMMC model is able to fully capture the return dynamic of AHF while the Truncated return substantially underestimates full history mean.

馬上我們可以看到FMMC模型在截斷期間的改進。 FMMC模型能夠完全捕獲AHF的返回動態，而截斷的返回則大大低估了整個歷史均值。

2. Volatility

2.波動性

Accurate estimation of the mean alone cannot support the claim that our model is robust. Of equal importance is the volatility. The below table shows the annualized volatility (i.e. standard deviation) for each of the periods under consideration:

僅憑均值的準確估計就不能支持我們的模型穩健的說法。 波動同樣重要。 下表顯示了所考慮的每個時期的年度波動率(即標準差)：

Both the FMMC and Truncated estimates slightly undershoot the realized volatility of AHF over the full period. However, both estimated are very close.

FMMC和“截斷”估計都略微低于AHF在整個時期內已實現的波動性。 但是，兩者估計都非常接近。

3. Sharpe Ratio

3.夏普比率

With the mean and volatility estimates in hand, we can now calculate the Sharpe Ratio. The Sharpe Ratio is calculated as follows:

有了平均值和波動率估算值，我們現在可以計算夏普比率。 夏普比率計算如下：

For most of the test period (January 2010-March 2020) the risk-free rate as proxied by the yield on the 3-month T-Bill was very close to 0%. For simplicity we will adopt 0% as the risk-free rate for our calculations. The below table shows the results:

在大多數測試期間(2010年1月至2020年3月)，由3個月國庫券收益率所代表的無風險利率非常接近0％。 為簡單起見，我們將采用0％作為無風險利率進行計算。 下表顯示了結果：

The FMMC estimate shows dramatic improvement over the Truncated for estimating the Sharpe Ratio of AHF. This is not necessarily surprising as above we showed the mean return for the Truncated period to be poor while the estimate for the FMMC was quite close. Naturally this will feed into the results for Sharpe, but, again, the results show the utility of the FMMC approach.

FMMC估計值比截斷值(用于估計AHF的Sharpe比率)顯著提高。 這不一定是令人驚訝的，因為上面我們顯示了截斷期的平均收益很低，而FMMC的估算卻非常接近。 自然，這將被納入Sharpe的結果中，但是結果再次顯示了FMMC方法的實用性。

4. Sortino Ratio

4. Sortino比率

Finally, we turn to the Sortino Ratio. Sortino is similar to Sharpe, but instead of total volatility, it is focused on what is termed “downside volatility”; or the standard deviation of returns below a stated threshold. Typically, the threshold is set to 0%; the idea being that volatile, positive returns are not considered “bad” because you are still making money, but volatile negative returns suggest an outsized chance of large losses. A higher ratio is considered better. The Sortino Ratio is calculated as follows:

最后，我們轉到Sortino比率。 Sortino與Sharpe類似，但不是總波動率，而是著眼于所謂的“下行波動率”。 或低于規定門檻的收益標準偏差。 通常，閾值設置為0％； 之所以這樣的想法是，因為您仍在賺錢，所以波動的正收益不被認為是“壞”的，但是波動的負收益表明出現巨額虧損的可能性很大。 比率越高越好。 Sortino比率計算如下：

The table depicts the results:

下表描述了結果：

The FMMC estimate is very close to the full period and accurately expresses the volatility of the downside returns. We see marked improvement over the Truncated estimate which is lower due to a combination of a lower average return and more downside volatility.

FMMC的估計非常接近整個周期，并準確表示了下行收益的波動性。 我們看到，由于平均收益較低和下行波動較大，兩者的總和比截斷后的估計要低得多。

結論性意見 (Concluding Comments)

Manager evaluation is one of the oldest and most common problems in investment finance. When the track record of the manager is short it can be difficult to assess the efficacy of the strategy which has ramifications for both fund managers and fund allocators.

經理評估是投資金融中最古老，最常見的問題之一。 如果經理的業績記錄很短，則可能難以評估該策略的有效性，這對基金經理和基金分配者都有影響。

In this post, we introduced Factor Model Monte Carlo (FMMC) as a possible solution to the short history problem and used the real world example of Aric’s Hedge Fund (AHF) to demonstrate its efficacy. By using a factor model and the common, short history of fund and factor returns, we estimated the exposure of AHF to different sources of economic and market risk. We were then able to simulate the returns of the AHF to construct a longer history of returns with the goal of gaining improved insight into the fund’s long term performance.

在本文中，我們介紹了因素模型蒙特卡洛(FMMC)作為短期歷史問題的可能解決方案，并使用了Aric對沖基金(AHF)的真實示例來證明其有效性。 通過使用因子模型以及常見的短期資金和因子收益歷史，我們估計了AHF暴露于不同的經濟和市場風險來源的風險。 然后，我們能夠模擬AHF的收益，以構建更長的收益歷史，目的是獲得對基金長期業績的更深入了解。

The results from the FMMC method showed dramatic improvement over using the short history of returns in isolation. Using the full history of returns for AHF as a comparison, we see that the FMMC method accurately models the return, volatility, Sharpe and Sortino Ratios of the fund. By comparison, the truncated history of returns severely underestimated the performance of AHF which would have the consequence of misleading investors.

FMMC方法的結果表明，與單獨使用較短的收益歷史相比，有了顯著的改進。 使用AHF的全部收益歷史作為比較，我們可以看到FMMC方法可以準確地模擬基金的收益，波動率，夏普和Sortino比率。 相比之下，截短的收益歷史嚴重低估了AHF的業績，這可能會誤導投資者。

Factor Model Monte Carlo has proven to be an effective technique for modeling risk and return for complex strategies and serves as a powerful addition to the fund analyst’s tool kit.

因子模型蒙特卡洛(Monte Carlo)已被證明是對復雜策略的風險和回報建模的有效技術，并且是基金分析師工具包的有力補充。

Until next time, thanks for reading!

直到下一次，感謝您的閱讀！

Aric Lux.

阿里克斯·勒克斯(Aric Lux)。

Originally published at http://lightfinance.blog on August 28, 2020.

最初于 2020年8月28日 發布在 http://lightfinance.blog 上。

翻譯自: https://towardsdatascience.com/better-portfolio-performance-with-factor-model-monte-carlo-in-r-3910d0a6ceb6

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