Higher moment Portfolio models. Skewness preference. So far in the portfolio optimization course our focus has been on single dimension analytics. With both risk and performance we have only looked at one metric at a time. While our Solver models have worked with multiple constraints,

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Bank consolidation and M&A drivers It is that time of the year again within the banking industry in the Middle East. The move for bank consolidation and the need to roll out and brush the dust off our bank M&A models.  Like every other industry

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Bank consolidation and M&A drivers It is that time of the year again within the banking industry in the Middle East. The move for bank consolidation and the need to roll out and brush the dust off our bank M&A models.  Like every other industry

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Unexpected Loss, Expected Loss & Economic Capital. A follow up post on our Economic Capital series where we looked at an alternate approach for calculating Economic Capital using accounting data rather than the BIS guidelines. Within that discussion it was felt that we needed a

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Calculating Economic Capital – Using Leverage ratio So far we have presented two methods for estimating Economic Capital. The first uses the worst case change in Shareholders equity, the second the volatility of the same changes. The challenge with method one and two is that

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Seven new free risk and portfolio management case studies – Value at Risk, ALM, Greeks & Monte Carlo Simulation tweaks

We have been busy over the last three months.

As part of the effort around the re-launch of FinanceTrainingCourse.com website on Amazon Web Services and the be a Finance Rock Star campaign we put together seven new case studies for our risk, investment and treasury management customers.

The seven risk, treasury and investment management case studies cover Value at Risk (VaR), Asset Liability Management (ALM), Jet Fuel Hedging, Barclays Libor Scandal, Fixed Income Portfolio Optimization, Option Price Sensitivity – Greeks & a advance Monte Carlo Simulation tweak for increasing the speed of convergence without increasing the computational overhead.

The Value at Risk (VaR), Asset Liability Management (ALM) and Fixed Income Portoflio Optimization case studies do a step by step walk through of building the model using MS Excel. We haven’t held anything back and all you need is your laptop, a cup of coffee and a little bit of time to rebuild the same model on your laptop.

The Greeks (Delta and Gamma) dissection uses graphical tools to define behavior of the two Greeks across Spot, Time, Volatility, Strike and Interest Rates. Jet Fuel Hedging, Barclays LIBOR and Monte Carlo Variance Reduction tweaks are contextual case studies without any excel model building effort or documentation. Greeks, Fuel Hedging and Monte Carlo, if you are not careful will give you a run for your money.

The Barclays LIBOR piece is the easiest read of them all. Just background and insights, no models, equations or twisted concepts.

Understanding Value at Risk (VaR) – A detailed step by step study guide

Understanding Asset Liability Management (ALM) – Simple balance sheet, simple examples, core ALM concepts

Jet Aviation Fuel Hedging Case Study

The Sales & Trading Interview Guide to the Barclays Libor scandal

Fixed Income Investment Portfolio Management & Optimization – Using Excel Solver to rebalance and fine tune your portfolios

Options and Derivative pricing – Understanding Greeks -A graphical dissection of Delta & Gamma

Monte Carlo Simulation – Antithetic Sampling and Variance Reduction Techniques

The case studies were put together as part of the Derivative pricing and Risk Management course series Jawwad teaches at the SP Jain campus in Dubai and Singapore and were tested by EMBA and GMBA students in class. The more advance concepts and topics were developed as part of the treasury and corporate treasury solutions sales workshop we have been running for our treasury and banking customers in the Middle East.

If you like what you see here or would like us to build a detailed step by step case study on a topic that you feel has a large audience, please drop us a note. We will add it to our task list.

Enjoy.

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Fixed Income Investment Portfolio Management using duration, convexity and Excel solver

It doesn’t matter if you manage a pension fund, a life insurance trust fund or the proprietary book of an investment bank, at some point in time you hit your allocation and risk limits and need to rebalance your portfolio.

In most instances your limits and target accounts focus on interest rate sensitivity, volatility, Yield to risk ratios, liquidity and concentration limits. Your objective is to create the most efficient fixed income investment portfolio that balances an optimal mix of the above constraints against yield to maturity. The time tested, risk versus reward tweak.

In our new risk training workshop for fixed income portfolios case study we will build a simple model using Excel solver that shows how to handle the fixed income portfolio optimization problem. The model can be easily extended to handle larger portfolios and additional constraints around liquidity, factor sensitivity, volume concentration, value at risk and volatility.

For the purpose of this case study we will assume that we are advising a large pension fund who is re-evaluating fixed income portfolio allocation due to its new investment policy. The assets under management at the fund are US$500 million. We want to recommend:

  • Portfolio allocation that minimizes duration
  • Portfolio allocation that maximizes convexity

The liabilities are also equal to $500 million with a weighted average maturity of 20 years. Modified duration or interest rate sensitivity of liabilities was last measured in the monthly risk report at 9%.

Fixed Income Portfolio Management: Introducing Duration and Convexity

Duration is a measure of how prices of interest sensitive securities change as the underlying rate of interest changes. For example, if duration of a security works out to 2 this means roughly that for a 1% increase in interest rates price of the instrument will decrease by 2%. Similarly, if interest rates were to decrease by 1% the price of the security would rise by 2%.

Here is the numerical approximation for modified duration.

Figure 1 Fixed Income Portfolio management. Numerical approximation for duration

Convexity: The Duration approximation of change in price due to changes in the yield works only for small changes. For larger changes there will be a significant error term between the actual price change and that estimated change using duration.

Convexity improves on this approximation by taking into account the curvature of the price/ yield relationship as well as the direction of the change in yield. By doing so it explains the change in price that is not explained by Duration.

A positive convexity measure indicates a greater price increase when interest rates fall by a given percentage relative to the price decline if interest rates were to rise by that same percentage. A negative convexity measure indicates that the price decline will be greater than the price gain for the same percentage change in yield.

Duration and Convexity together are used to immunize a portfolio of assets and liability against interest rate shock.

Figure 2 Fixed Income Portfolio Management. Numerical approximation for Convexity

Fixed Income Portfolio Management: Introducing the Optimization model

Our first scenario assumes a rising interest rate outlook. Ignoring liabilities and maturity mismatch for now, our fund manager would like to rebalance the portfolio to minimize duration so that the value of assets do not fall significantly due to changes in interest rates. We assume:


Figure 3 – Fixed Income Investment Portfolio – Date, Rate shift, size.

Fixed Income Investment Portfolio Management: Breaking down the optimization model


Figure 2 – Fixed Income Investment Portfolio Management: The securities analytics model

There are four parts to this model:

  • Part 1- The securities universe specification: This is the pink-shaded area and defines the complete investment universe. You can only allocate a security if it is described in universe. Assets are classified in buckets of 20, 15, 10, 5 and 3 year maturities. We have assumed that current date (the valuation date) is the same as date of purchase (the settlement or value date) for all assets in all buckets.
  • Part 2 – The securities pricing model: This calculates the price and yield and is shaded brown. Current price is calculated using the Excel price function as illustrated below:


    Figure 3 – Price calculation

     

    The excel price (bond pricing) function is based on the data inputs of settlement date, date of maturity, coupon rate, yield to maturity, frequency and basis. Frequency here is 2 which mean that coupons are paid semi-annually. Cell $D$4 is the current date used in the input parameters in Figure 1.

     

    Price changes just add or subtract the specified interest rate shocks and recalculate new prices for use in duration and convexity calculations. The rate shocks are 1 basis points (1/10,000).

     

  • Part 3 – Portfolio Duration Calculation: this is shaded blue and shows duration calculations. Duration is calculated using the duration approximation formula introduced above:

    Figure 4 Fixed Income Investment Portfolio: Duration approximation

    In the context of the Analytics Model, this is calculated as follows:

     

     

    Figure 4 – Duration calculation

     

    In calculation of Duration-down, Cell G44 is replaced by G45 and F44 is replaced by F45. Note that the general form of the formula is applied but instead of just calculating duration in one line, duration up and down are calculated respectively and the average of both is taken.
    This average of the two durations will be used in our model.

     

  • Part 4 – Portfolio Convexity Calculation
  • The final part of the model calculates convexity and is highlighted in purple. The applicable convexity formula is:

    Figure 5 Fixed Income Portfolio Investment – Convexity calculation

    The calculation is as under:

    Figure 5 – Convexity calculation

    The convexity adjustment is calculated using the formula:

     

    Fixed Income Investment Portfolio Management: Summarized Portfolio Analytics

     

    We now need a summarized portfolio analytics table that can be used in our optimization process. The results derived by combining the actual portfolio allocation and the portfolio analytics generated above would appear as shown below:

     


    Figure 6 – Fixed Income Investment portfolio management. Portfolio analytics results

    How are these results calculated? The answer is through the Analytics Model and the allocation of assets followed currently for each bucket. The allocation table is shown below:

     

 

Figure 7 – Portfolio allocation

Notice that the total bond portfolio allocation is 97% not 100%. 3% of the allocation is held in cash and/or non-interest sensitive securities.

Portfolio Duration is calculated by using the Excel sum-product function.

Sum-product is simply the combination of two operation that involves multiplying the individual cells in two vectors (Portfolio Allocation, Security Duration) and then summing the resulting product across all cells.

For instance (10%*duration average for 15 year bond) + (10%*duration average for 10 year bond)….. And so on.

Portfolio Convexity is calculated in the same manner by using the Excel sum-product function. (10%*convexity for 15 year bond) + (10%*convexity for 10 year bond)….. And so on.

And ditto for portfolio yield calculations. (10%*portfolio yield for 15 year bond) + (10%*portfolio yield for 10 year bond)….. And so on.

Figure 8 – Fixed Income Investment Portfolio Management: Calculating portfolio yield, portfolio duration and portfolio convexity

Portfolio sensitivity of -0.028600% is calculated in the following way:

Figure 9 – Fixed Income Investments Portfolio Management. Calculating portfolio sensitivity

IR shift is the interest rate shift. It is measured in bps (basis point shift).

Fixed Income Investments Portfolio Management: Portfolio Optimization using solver

If we had a single linear equation representing a single constraint and a single position, the Excel Goal seek function would be sufficient. However a multi position fixed income investment portfolio has many constraints and many positions. In addition because you are dealing with bonds, the underlying model is no longer linear. You need a non-linear tweak to make it work.

The Excel solver function helps us optimize our portfolio allocation model with a few tweaks. We demonstrate the simplest of scenario in this write up but they can very easily be extended. As is the case with all optimization models, the trick is in designing the constraints. While there can be only one objective function (minimize or maximize a specific portfolio metric), with the right constraint design you could get close to a near optimal solution reasonably quickly. While the current model focuses only on fixed income investment portfolio, the design of the model can very easily be extended to multi-class portfolios. In addition new target accounts and risk constraints can be added just as easily.

Fixed Income Investments Portfolio Optimization. Optimizing the base case – Minimizing duration

The trustees of our pension fund have given a target to the investment fund manager to earn at least 3%. Bond proportion should be 99% of the fund, with the remaining for cash. Risk management and diversification targets specify that no greater than 13% of the total fund be allocated to any given asset bucket.

Given these objectives, how should the investment manager set out to minimize duration?

The targets are effectively constraints. Once we have defined them correctly, the solver function takes these constraints into account, evaluates the target optimization cell (minimize duration), and searches for an optimal solution. Since the layout of the spreadsheet has been described above, all we know need to do is to define the solver model and click solve.

Figure 10 – Fixed Income Investments Portfolio Management. Using Excel Solver for minimizing duration for a fixed income portfolio

Pick ‘Min’ as your objective and then click ‘Solve’. Solver will work through the model till it reaches the optimal solution. The revised fixed income portfolio allocation is as follows:

Figure 11 – Portfolio allocation

Note that none of the asset bucket has higher than 13% proportion of assets. Also 99% is invested in bonds, rest in cash. The revised portfolio analytics summarizing our target account is also shown below:

Figure 12 – Fixed income investments portfolio management – Revised portfolio analytics

Fixed Income Investments Portfolio Optimization – Maximizing Convexity

Positive convexity is generally a desirable attribute in a portfolio. In addition to minimizing duration, an alternate case could be made for maximizing convexity. If you expect rates to decline, a more convex fixed rate asset would rise by more compared to a less convex asset.

All it will take is set the Target Cell at portfolio convexity instead of duration. Note that in solver we click on ‘max’ instead of ‘min’ this time. The revised allocation is as follows:

Figure 13 – Fixed Income Investments Portfolio Management – Revised optimal portfolio allocation for maximizing convexity

And the revised portfolio analytics results for both the maximized convexity and minimized duration scenarios are presented below:

Figure 14 – Fixed Income Investment Portfolio Management – Optimized portfolio analytics results

Figure 15 – Fixed Income Investments Portfolio Management – Consolidated results

Fixed Income Investments Portfolio Optimization. Next steps

You can easily extend the model to include constraints for value at risk, volatility, interest rate mismatch, gap management, concentration, portfolio liquidity, daily, monthly and weekly turnover, credit ratings and grades. A sample sheet showcasing some of these variations will be available for sale early next week at our store.

If you need more help beyond the sample portfolio, we also help customer build customized portfolio builds and solver models.

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Sales and Trading Interview Guide: Understanding Greeks

Trading requires a combination of intuition, discipline and process. Of the three intuition is the most difficult to teach, since discipline and process is an incentives and control game. While individual intuition can be built over years of experience there are rules that make it easier to pick up that intuition faster.

Institutional intuition gets passed on between generation of traders through shadowing, standards, processes and controls. This passage of rites becomes easier if you have a knack for the subject, if you already know some of the rules or if you are familiar with the trading language.

The sales and trading language has many dimensions dealing with execution, trading strategy, customer behavior and product variations. This book only focuses on one very limited aspect of that language – the aspect dealing with risk management, hedging and Greeks.

The challenge with this part of the language lies partly with the terminology (a range of Greek symbols), partly with the presentation (partial differential equation), with calculations (a combination of Greek symbols and partial differential equations) and with interpretation (can you please say that again in a language that we can all understand).

Most business school derivative courses run out of time before the product universe has been covered, let alone spend time on teaching how to read, predict or forecast the behavior of exotic Greek symbols. .

Advance derivative courses cover pricing and if we are lucky spend limited time on sensitivities and Greeks because of conflicts with other topics in the outline. Sometime as business school students all we get are case notes and text references that are long on definitions and calculations but short on guidance and practical applications.

Which is unfortunate because the option price sensitivity topic is difficult to grasp for most audiences given its very non-linear nature. It takes time to think comfortably in the non-linear world. We understand simple straight forward, single dimensional relationships very well. When you ask us to envision a new dimension or even worse collate reactions from multiple dimensions into a single trading decision, our mental frameworks breakdown.

To develop an appreciation for this topic you need at least a few days of hands on or modeling experience followed by active application of the same concepts. The reason why you have purchased this book is because you don’t have a few days. You possibly have a few hours or a night before that interview or presentation is due.

So we have tried to compress primary lessons into short bite sized pieces. There are some equations but we don’t spend time on them or their derivations. We do spend time on ground rules, behavior and intuition. As a trader I am more likely to ask you about how Gamma is going to behave under a given scenario and how that is different from Vega’s reaction.

Our assumption is that you have some familiarity with Options, Black Scholes and the derivative pricing world. If this is not the case you need more help which is available on our partially free site at FinanceTrainingCourse.com

This book is based on a four part MBA course on derivative pricing and risk management that I have taught in Dubai and Singapore and the risk and treasury management practice I have run since 2003. The material is based on training tools we developed to teach advance treasury concepts to our students using our signature hands on, equations off mode.

And now let’s go waltz with some Greek symbols.

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Preparing for the quantitative portion of a sales and trading interview for a main street bank is a nightmare. Specially if the bank is an active derivative trader and wants it intake class of interns and full time analysts and associates to hit the Sales and Trading desk running.

While basic option concepts generally get covered quite well in the MBA curriculum, when it comes to option price sensitivities and Greeks, our understanding remains rudimentary and superficial. One reason is the focus on formulas and calculation rather than intuition and understanding. Most courses have run out of time when it comes to delta, gamma, vega, theta and rho and stop after a basic rudimentary coverage of the material.

There is a lot of good material available on basic quantitative and numerical techniques tested in a Sales and Trading interview. But when it comes to option price sensitivities or Greeks, available material generally looks like this.

As part of the work we do with customers and students, our Apple iPad iBooks team is working on two very interesting and exciting titles.

The first is the Sales and Trading Interview Guide – Understanding Greeks for Dummies. Using the interactive iBook template we will help you master your Greeks to such a level where future mention of delta, gamma, vega, theta and rho would no longer break you out in cold sweat and palpitations. The iBook will cover Greeks behavior across time, volatility, spot and strike prices using easy to understand language, graphs and self assessment quizes.

But it’s the second iPad iBook title that we are really excited about. Sales and Trading Interview Guide – Delta Hedging and other higher dimensions, will help you build your own delta hedging sheet in excel using Monte Carlo Simulation. Both iPads books will have options for purchasing supporting excel spread sheets that extend the concepts covered in the iBook.

Planned for release in mid September, the two books will increase our inventory of interactive iBook for iPad titles to 5. Please feel free to drop me a note if you would like to learn more about the release dates and table of content for both titles.

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