Optimization By Simulated Annealing


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My first paper, published in GARP\'s flagship magazine.

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Optimization By Simulated Annealing

  1. 1. G L O B A L A S S O C I AT I O N O F R I S K P R O F E S S I O N A L S P O R T F O L I O O P T I M I Z AT I O N Optimization by Simulated Annealing If you are searching for portfolios that offer optimal combinations of risk and return, simulated annealing (SA) can prove a useful tool. Vallabh Muralikrishnan defines SA, explains the challenges posed by portfolio optimization and provides step-by-step instructions for implementing an SA algorithm. he introduction of the Markowitz Efficient tify mathematically the portfolio (i.e., the combination of T Frontier more than 50 years ago popularized the use of mathematical optimization techniques in finance. The central idea of portfolio optimiza- tion is that if the risk and return of available financial instruments could be quantified accu- rately, then portfolios that minimize risk for target levels of return could be identified. Thus, portfolio optimization comprises two distinct problems: the first problem is to decide on measures of risk and return and to quantify the available instruments) that has optimal risk-return charac- teristics. One challenge in addressing the second problem is that the number of potential portfolios grows exponentially with the number of available instruments. For example, a portfolio manager with only 200 instruments can construct 2200 potential portfolios. Enumerating and considering each potential portfolio is impractical because of computing time constraints. Also, the complexity of the risk-return risk-return relationship of a portfolio; the second is to iden- function could exclude the possibility of effectively using GLOBAL ASSOCIATION OF RISK PROFESSIONALS 45
  2. 2. G L O B A L A S S O C I AT I O N O F R I S K P R O F E S S I O N A L SP O R T F O L I O O P T I M I Z AT I O Nthe gradient search method (the most popular non-linear ing to a worse portfolio is controlled by the “cooling sched-optimization technique). ule,” and as the cooling schedule reaches completion, the The gradient search method tends to converge on local SA algorithm evolves into a simple greedy algorithm. Theoptimums and requires matrix inversions that can be time following example illustrates how the SA algorithm can beconsuming or impossible if the objective function is not deployed.smooth or differentiable. Therefore, search algorithms canbe used to scan the vast space of potential portfolios effi- A Simple Illustrationciently for optimal combinations of instruments. First, we must set up the problem. For the purpose of illus- Simulated annealing is one such algorithm. This article tration, we will consider the problem of selecting an opti-illustrates how simulated annealing (SA) can be used to mal overlay portfolio of credit default swaps (CDS) toidentify portfolios with optimal combinations of risk and hedge and improve returns on a bank’s loan portfolio whilereturn. managing the overlay portfolio’s mark-to-market (MTM) volatility. Based on fundamental analysis, internal con- straints and personal judgment, we will assume that the portfolio manager has developed a list of 200 potential CDS transactions from which to construct a portfolio (“the “When applied to portfolio con- CDS universe”). Presumably, he has restricted this universe struction, the SA algorithm begins to CDS positions that he would feel comfortable transact- ing based on his view of idiosyncratic risks. One measure with a randomly selected portfolio. of return to be used in this context is economic value added (EVA). The portfolio manager will only want to take CDS It then iteratively jumps to other positions with positive EVA. portfolios in the neighborhood of After screening for idiosyncratic risks, the portfolio man- ager may also focus on minimizing the expected MTMits current position in search of bet- volatility of the overlay portfolio. The MTM volatility of the CDS portfolio is directly related to the volatility of the ter risk-return metrics.’ credit spreads associated with each position in the portfo- lio. Therefore a crude measure of this risk is the historical standard deviation of credit spreads weighted by portfolio notional (STD), as calculated below. The portfolio manag- SA is an optimization heuristic inspired by the cooling of er will want to minimize the expected volatility of the over-solids. As a substance cools, its molecules arrange them- lay portfolio.selves into a low energy state (as measured by an “energy The risk-return relationship of potential portfolios canfunction”). The energy function could have several local now be measured by the following ratio: EVA/STD. This isminima. The cooling process is akin to minimizing the the objective function to be maximized. Clearly, EVA mayenergy function. It is an empirical fact that the rate of cool- not be a desired measure of return, and the historical stan-ing determines the final structure of the solid, and the solid dard deviation of credit spreads may not fully measure theformed by rapid cooling is usually not as stable as the solid expected MTM volatility of a CDS portfolio.formed by a slower cooling schedule. This is because rapid Nevertheless, these simple measures will be used to illus-cooling forces the molecules into the first locally minimal trate the simulated annealing technique. The choice ofenergy state that they happen to chance upon. A slower appropriate risk and return measures will depend on thecooling schedule increases the probability that the mole- goals and context of portfolio optimization, which is notcules converge on the globally minimal energy state. the subject of this article. When applied to portfolio construction, the SA algo- The EVA for a particular CDS can be calculated usingrithm begins with a randomly selected portfolio. It then the following formula:iteratively jumps to other portfolios in the neighborhood ofits current position in search of better risk-return metrics.However, in order to avoid converging to a local minimum,which is the bane of greedy algorithms, the SA heuristic In the above formula, the “Expected Revenue” is sim-allows a non-zero probability of moving to portfolios with ply the annual spread multiplied by the notional. Theworse risk-return metrics. This prevents the heuristic from “Capital” term can refer to either the regulatory or eco-getting stuck at a local minimum. The probability of jump- nomic capital to be held by the institution for a particular46 GLOBAL ASSOCIATION OF RISK PROFESSIONALS J U N E / J U LY 0 8 I S S U E 4 2
  3. 3. G L O B A L A S S O C I AT I O N O F R I S K P R O F E S S I O N A L S P O R T F O L I O O P T I M I Z AT I O N transaction. If $10 million notional in protection is sold (long position) for 50 basis points (bps), then the annual expected revenue is simply $50,000. If protection is bought (short position) on the same terms, the annual expected revenue (i.e., cost) is $50,000. Of course, long positions incur a positive capital charge, while short posi- The above random portfolio consists of all swaps with tions incur a negative capital charge (i.e., capital release). an indicator value of 1. The EVA/MTM value associated The portfolio EVA is simply the sum the EVA of compo- with this portfolio can be easily calculated using the nent swaps. appropriate formulas. On iteration i, a uniform random The historical standard deviation (in dollars) of a portfo- number between 0 and 1 is generated and assigned to lio of CDS can be calculated using the following formula: each swap in the universe. That number is then compared with 0.5*schedule(i). If the random number assigned to a particular swap is less than 0.5*schedule(i), then the indi- cator for that deal is switched from its current state. For In the above formula, “w” is a vector containing the each iteration, this function initially generates vastly dif- notional amounts of CDS in the portfolio, and “™” is the ferent portfolios. However, the variation in portfolio covariance matrix of historical spreads for the swaps. STD composition between iterations decreases as the heuristic is in dollar terms. winds down. Finally, in each step, an accep- Implementation Steps tance probability is calculated to Now that the problem is set up, we can implement the SA determine whether to keep the cur- algorithm. The first step is to define the following three rent configuration of swaps (i.e., functions: the annealing schedule, the neighbor generator the current portfolio) or to switch and acceptance probability. These functions must be tested to the configuration proposed by and refined empirically for each problem. the neighbor generator. This proba- The annealing schedule is used to control the path the bility depends on the value of the heuristic takes as it searches for the global optimum. Its objective function corresponding to value depends on how far along the SA algorithm has pro- the current and proposed configu- gressed. Initially, it allows the heuristic to search the space Vallabh rations, as well as on schedule(i). of portfolios randomly. As the routine progresses, however, Muralikrishnan In our implementation, we set the annealing schedule forces the heuristic to move toward the acceptance probability to 1 if the maximum. For our illustration here, we use a simple the proposed configuration yields a higher objective. linear annealing schedule represented by the following Otherwise, we set the acceptance probability to a random function: number between 0 and 1 multiplied by schedule(i). Clearly, the heuristic will always choose the proposed configuration if it yields a better objective. However, there is initially a non-zero probability of choosing a proposed configuration, In the above formula, “N” is the total number of itera- even if it yields a worse objective. Still, since schedule(i) tions over which we wish to run the SA heuristic, and “i” goes to 0 as the heuristic winds down, the probability of represents the current iteration. Clearly, the annealing accepting a configuration with a worse objective also schedule has the property that it produces values close to 1 decreases to zero. initially and then converges to 0 as the heuristic runs its It is also important to note that in the implementation course. The choice of annealing schedule is subjective and we’ve outlined, the algorithm initially makes bigger must be chosen based on the design of neighbor generator moves (as evident from the neighbor function) and has a and acceptance probability functions. greater probability of selecting worse outcomes (as evi- The neighbor generator is used in each step to generate dent from the acceptance function). sample portfolios to consider on the next iteration. Each The initial large moves are not necessary, but they do transaction in the swap universe is assigned an indicator allow for the algorithm to experiment with vastly differ- variable (1 or 0) that indicates whether or not the swap is ent combinations from the initial starting point. included in a portfolio. This can be accomplished by using However, the probability of selecting worse outcomes a random number generator to assign a vector of indicators decreases over time. to potential transactions: Given the three functions we’ve discussed (the annealing GLOBAL ASSOCIATION OF RISK PROFESSIONALS 47
  4. 4. G L O B A L A S S O C I AT I O N O F R I S K P R O F E S S I O N A L S P O R T F O L I O O P T I M I Z AT I O Nschedule, the neighbor generator and the acceptance proba-bility), it is clear that the SA algorithm can be implementedusing the following steps:1. Decide on the number of iterations over which to run the algorithm. This is subjective and can be set based on empirical tests.2. Select a random starting portfolio and set it as the “best portfolio.”3. Calculate the objective function value for this port- folio and store it. case. This conclusion can be easily duplicated. However, even more interesting is the variability in the results of the4. Generate a neighbor portfolio and calculate the simulated annealing algorithm. To illustrate the variability objective function value for this new portfolio. of results, the two algorithms were run 10 times, and ele- mentary statistics were computed:5. Based on the acceptance probability, accept or reject the new portfolio. If accepted, set the new portfolio as the “best portfolio” and update and store the corresponding objective function value.6. Repeat steps 4 and 5 until the maximum number of iterations is reached. For a given number of iterations, there seems to be sig-Simulated Annealing vs. Greedy Algorithm nificant variability in the final objective value found by theThe benefits of using SA over a greedy search algorithm implementation of the SA algorithm.can be demonstrated empirically. Unlike SA, a greedy algo-rithm will only select portfolios that have higher values of Further Study: The Challenge of Convergencethe objective function. We can modify the SA algorithm SA is a search algorithm and therefore is subject to theoutlined earlier in this article to create a greedy algorithm same criticism as all search algorithms. In our experi-and then compare the results of optimization by the two ment, we used SA to search a space of 2200 (over 1060)algorithms. A greedy algorithm can be created by simply potential portfolios, with a maximum of 50,000 itera-setting the acceptance probability to zero, if the objective tions. Due to the vast space of potential portfolios, con-of the proposed portfolio is less than the objective of the vergence is understandably difficult. Several options are“current” portfolio on all iterations. available to address the problem of convergence; none, To test the effectiveness of the two search algorithms, however, is perfect.we start with identical swap universes and identical start- One option is to increase the number of iterations, buting points. This ensures that the value of the objective this will be restricted by available computing time. Anotherfunction will be the same for identical portfolios regardless option is to start from several different starting points andof which algorithm is used. It also enables fair compar- compare the results. A third option is to use SA to identifyisons because both algorithms start from the same point. one solution and then use that solution as the starting pointThe following table summarizes the results from the afore- for a gradient search.mentioned experiment. Much experimentation is required to customize the SA The result of the above experiment clearly shows that the algorithm for a particular problem. Nevertheless, SASA search algorithm beats the greedy algorithm in every remains a useful tool in the search of optimal portfolios. ■FOOTNOTES:1.“Portfolio Selection,” Journal of Finance 7, no. 1 (March 1952): 77-91.✎ VALLABH MURALIKRISHNAN is an associate in the asset portfolio management group at BMO Capital Markets. He has a masters in mathematical finance from the University of Toronto. He can be reached at vallabh.muralikrishnan@bmo.com48 GLOBAL ASSOCIATION OF RISK PROFESSIONALS J U N E / J U LY 0 8 I S S U E 4 2