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Portfolio Planning and Valuation of Equities – Financial Assets, Investments
written by : William F Bryant When most people think of a portfolio, their thoughts immediately jump to equities, stocks, mutual funds, ETFs and perhaps, for some, bonds. The fact is that any holding of assets cab be valued as a portfolio. Cash flow goes out for an investment that has risk and over time a commensurate cash flow, or series of cash flows, back in. A private equity fund might hold a portfolio of companies, an industry firm might hold vested interest in R&D projects, completed patents and new product lines or an individual might hold varying levels of real assets and financial assets. The point is that anything that has value with cash flows used to purchase and anticipated cash flows in the future from its sale could be considered a holding in a portfolio. Our financial system recognizes real assets and financial assets. This is an important distinction and one that I would say becomes even more important in times of economic uncertainty and rising inflation. Real assets are land, physical structures, machinery, inventories and even intellectual property, all things that make real products via real work and real commodities. Financial assets can be thought of as contracts that entitle the holder to a cash value or potential cash flow that is secondarily or further removed from the real assets. These things might be equities, bonds, insurance, pensions or certain derivatives. As you can see, a company or individual could place a value on each of these items and all of these items require certain outflows with commensurate inflows of cash at points in time. Quite a diversified portfolio of holdings could be put together with these items mentioned and what’s more is that each of these items can be further subdivided into even more categories of varying types. So how does one work out a diversification and what does it entail. Diversification is a concept of investment rooted in correlations of different asset classes and types. Correlation is, simply, how two assets’ valuations move, be it together as in they both appreciate, or separately, as in one depreciates while the other appreciates. Now, as you might believe, there are varying degrees to which asset values will move, while both may increase, one may increase more than the other and vice versa while decreasing or any other arrangement you can think of in the tracking of the assets value. There is a measure of the individual asset movement, increasing and decreasing, over time known as its volatility. The correlation coefficient is a measure of this movement using historic values of the assets and their corresponding volatilities. This value can be useful when planning investments when you want to be sure you don’t “have all your eggs in one basket” or are carrying all the same type of investments in one basket so they all depreciate at the same time. Correlation is the easily recognizable “face”, but there is a behind-the-scenes influencer or asset values and correlations, that of risk. Risk is any event that might negatively influence the expected values of assets. This is, in fact, what you are attempting to diversify against so it makes sense that it is the behind-the-scenes influencer. The lower the risk associated with an asset, the lower the volatility of the value and the lower the correlation it will have with an asset that has higher risk and higher volatility in the movement of its value. As an example Treasury Bills, in our financial system, are considered risk-free or guaranteed repayment at the T-Bill rate. This is an important piece of information since the risk-free rate lays the foundation to measure all other riskier investments and their corresponding rates of interest (repayment amounts and timing). Since these investments are risk-free it would make sense that there is no volatility in their value. Unfortunately, this isn’t the case. While you are guaranteed to receive your anticipated return, interest rates do not remain static and the interest rate of your bond may be better or worse that the changing interest rate on the market. Interest rates themselves also have volatility to add some complexity. A short bit about interest rates and the relation to risk. Interest rates, obviously, are compensation for investing in an asset. The compensation covers several types of risk and is the reason that interest rates seem to vary over assets and securities. Risks might include default risk (also known as credit risk) inflation risk, liquidity risk (how easily a commitment to an investment can be exchanged back into cash), unsecured vs secured risk, debt ranking in the levels of invested capital (subordinated) and so on. The riskier the investment, or the less likely you are to get your anticipated cash flows, the higher the interest rate. Ideally, the workings of the market will value investments of similar risk at similar interest rates, however, if there is a high demand for investment in a particular asset or security the interest rate will actually reduce. This event can be seen in the U.S. Treasury Bill auctions and it should be fairly intuitive. If someone does not want to invest in some investment, the interest rate will rise to entice you to invest. The opposite is also true. If there is demand competing for an investment, the interest rate can be lowered until the demanders willing to get the lowest interest are served. As you can see, interest rates may have some volatility and due to this, if the interest rate on an investment, you have committed to, is lower than an investment of similar risk the value of your investment is worth less on the market. It is for this reason that time of commitment of the investment also becomes important. With this information briefly covered, a look at portfolio planning and diversification is better understood. We know that diversification and risk are linked. We know investments vary based upon risk and that risk is compensated by interest rates. Now we will look at how these interest rates correspond to financial asset classes. Financial assets are broadly divided into fixed-income, equities and derivatives. Each of these can be further divided into securities by type, fixed-income (debt securities) can be federal, municipal or corporate bonds of different types, equities can be preferred or common stock of varying types and derivatives can be broken down into futures, options, tranches of SPVs, swaps and so on. Fixed-income and certain types of tranched SPV investments are set payments to the investor at a predetermined interest rate. These are straight forward to value with the exception of corporate debt that may include warrants or convertibles. This type of asset class is also rated for credit risk and often backed by the assets of the company, bearing in mind there are no senior secured loans or other senior claims on company assets. It is for these reasons that fixed income is the lowest risk of the asset classes, just above the risk-free Treasury Bonds. Equities are the next level up. As you may know, equities are shares in the net income of a firm and often traded on market exchanges. Many people restrict their diversification to holding equities in various industries. As you may already see, this could be considered a naïve interpretation of diversification given the numerous securities in each of the asset classes, or even in equities themselves since equities can be subdivided into preferred and common stocks. The uniqueness of this asset class is that the risk and commensurate interest rate on your investment is not known, only estimated based on historic values. Your risk is based upon a company’s anticipated ability to meet its obligations (liabilities) and your gained interest is in terms of increasing value of the company which is most often heavily weighted to increasing net income. In terms of financial ratios this can be quickly/loosely derived from the ROE, return on equity, although the actual cost of equity can be regressed from the movement of the stock price. The riskiest level is, generally, that of derivatives. Derivatives, in my opinion, are best used as hedging devices that provide insurance on asset values, however derivatives are also used in a speculative manner because of the benefit of greater returns through the use of leverage. But, as you should know by now, greater potential returns also imply a greater risk of not receiving anticipated cash flows. Derivatives are most commonly, options or futures, but are contracts that attempt to lock in future values of an asset or cash flows in the present time and thus eliminate the dynamics of future cash flows associated with that of equities, while at the same time, hoping to make a profit from the locking of the value. But, if you recall with the example of the T-Bill, just because you lock in a future value of an asset, it does not mean that the value will be worth the paper the contract is printed on in the future. As a simple example, if you lock in a call option (right to buy) on a stock at a price of $20 in the future because you feel the stock is undervalued, but upon the expiration of the contract the stock is only priced at $19, the call option is worth anything to you. I understand that this is a very simplified analogy to a contract, but option pricing and theory and finance’s “law of one price” need significant explanation that I will not go into for the topic of portfolio diversification. What should be gleaned thus far is the relationship of risk to interest rates and cash flows, different types of assets and each having different levels of risk, and finally that diversification exists in how these different assets and risk levels move with or against each other (correlation), in terms of value, over time. Now, if you recall, the volatility aspect for each of the assets, there is a statistical component that incorporates the volatility of the individual assets and the correlation of the said assets. This component is called the covariance and can be used to calculate the volatility of the entire portfolio as we will see later. For now, I think conceptual understanding of the basis for the organization of a portfolio is more important because financial investment advisors will as questions to determine risk tolerance and attempt to approximate returns from the investment. If you understand why this is happening and the relationship of risk to returns then perhaps you will be better equipped to make informed decisions. At this point it should make sense that different combinations of the aforementioned asset classes will result in different levels of risk and different levels of returns. This relationship of risk to return is known as the Sharpe ratio and essentially measures expected return per unit of risk so that there is a method of comparison amongst different portfolios. But how does one go about selecting a relationship that is perceived to be most beneficial to an investor’s preference while incorporating risk and return? A concept from economics called utility is used. A Utility Function attempts to create a utility metric that uses the expected return on an assembled portfolio of investments and scales it down by the investor’s risk assessment (a measure of risk aversion) and the unbiased estimation of the variance of the portfolio’s returns to find the ideal portfolio. Of special note are two considerations, the first is that this method and equation is specific to returns that are normally distributed and the second is that the measure of risk aversion is an attempt to quantify a reactional behavior to risk. The concept of risk itself if difficult enough for most people to understand and so often portfolios have gone to rating the risk and return by how many years the investment must be maintained and how much of a gain is desired upon retirement of the investment. Those people familiar with IRAs will recognize this concept from the selection of available portfolios for your company. 
The ‘A’ is the risk metric. As you can see from the equation, the greater the A, all else equal, the more the expected return is penalized, lowering the utility. It can then be surmised that higher A values indicate less tolerance for risk. This value will change different investors’ perception of the same portfolio given different risk tolerance.
We have looked at some of the basic components, broadly, of investments, but I am sure you want to get into allocation of capital. It was hinted at earlier, but there is an allocation to the risk-free asset and an allocation to any other assets known as the risky portfolio. To be sure, there really is no risk-free asset and no risk-free return, to say nothing of real returns derived from the nominal returns, but most financial calculations make use of the going rate on Treasury bills as the risk-free rate and the risk-free asset. Something that adds complexity to this is both liquidity risk and, in our current economic environment, inflation risk. These risks will both have an affect on what I would term as ‘real value’. ‘Real value’ is dependent upon time period and its corresponding economic influences on the purchasing power of the total value of a portfolio and how easily that purchasing power can be transformed into other desired assets. I make mention of this because almost no financial assets make an adjustment for inflation and the inflation that we are going to be stepping into is going to be substantial, 1970s substantial. Given this event, the only assets that will have real value are those with productive capacity aka the real assets. This is something to keep in mind when assessing a portfolio of investments. In order to get in capital allocation, a fair amount of math is needed. Math is an amazing toolbox, useful in many situations since the key concepts and their methodologies are transferable. In the case of portfolios and investments this includes statistics, algebra and calculus. Usually, math examples being with the simplest scenarios. In the case of portfolios this means that a basic portfolio would consist of one risky asset and one risk-free asset and the allocation of capital would be relative to the expected returns of these assets in order to determine expected returns for the entire portfolio. This formula should look familiar if you have ever used the Capital Asset Pricing Model (CAPM). 
Note that b% would be the percent allocated to the risky asset portfolio and it is also a slope akin to the Beta in the CAPM and, appropriately enough, the plotted line of this equation is called the Capital Allocation Line (CAL).
Remember rise over run? b% , the slope from the equation, is then also equal to the Expected return of the risky portfolio minus the risk-free rate, divided by the standard deviation of the risky portfolio, the Sharpe ratio. Then, using the variance property of Var[bX] such that ‘b’ is a constant and X is the random variable, that equals , the standard deviation of the complete portfolio including the risk-free portion is
We now see a visual relationship of an allocation to risk and return and this understanding can be used to compute any feasible risk to return relationship for a desired allocation.
Once we understand the capital allocation line, the issue then becomes which portfolio, with a given risk to return ratio, do we choose. Well, we haven’t yet implemented the risk aversion metric from the utility function. Both functions together: 
We should see that we can use the CAL in the Utility function, in addition to the relationship of the standard deviation of the complete portfolio as proportional to the standard deviation of the risky portfolio. Since our measure to determine which portfolio we would choose will be determined by the utility, because it is the only function that incorporates, portfolio risk, portfolio return and investor risk tolerance, we need to maximize the utility.
Recall from calculus that maximizing a function implies derivation of the function, in this case with respect to b (our allocation percentage to the risky asset). This then allows a solution to an optimal allocation percentage given a set of portfolio risk, portfolio expected returns and an investor risk tolerance metric.
Interestingly, this theorizes that an investor is indifferent to portfolio risk and return pairings that have the same utility and same risk aversion metric, A. If these are plotted, you would see what is called the indifference curve. The comparison of the slopes of these curves (rates of change) at varying pairings of Utility and risk aversion indicate additional information about risk aversion, however the usefulness of these curves exists more so when the CAL is plotted with these curves, revealing which utility to aversion rating meets the desired allocation.
Live Run Portfolio: Choosing 3 Equities for a Risky Portfolio (no prior research or evaluation, simply chosen) Find the individual Expected Returns through Portfolio expected returns and standard deviation. 1) Grab the closing stock prices, determine time period: daily, weekly 2) Transform to normal data by using returns: LN[price at period(t)]/LN[price at period(t-1)] 3) Find the Beta. option 1 (excel) use the slope command of the returns against a “market portfolio” like S&P option 2 regress the market returns on each individual stock returns 4) CAPM model to determine expected return on equity Formula adjustments to annual returns for standardized financial comparison Determine risk-free rate and the risk premium 5) Calculate the Covariances, Correlations amongst the individual equities. 6) Calculate the optimal weights to minimize the standard deviation of the portfolio. Stock C, Stock D, Stock E Assuming a risk-free of 2% and a risk premium of 8.5% 
We can use Excel to optimize our weights in each stock in order to minimize our portfolio standard deviation. Just note the covariance matrix is still set in weekly and an adjustment must be made to the sigma to scale to an annual standard deviation. Of course we could have also used the solver to choose weights that would have maximized the return and then calculated the standard deviation of that portfolio, but since I tend to be risk averse myself I look to minimize risk.
But we could take this one step further and simply make use of the Sharpe ratio mentioned in the beginning and attempt to maximize the return per unit risk of the portfolio by optimizing the stock weights. The previous arrangement has a Sharpe ratio of 0.375, and we can see the optimized weights for a maximum Sharpe ratio is 0.43.
You can see how simply varying the weights of the assets themselves can change the expected return and standard deviation of a portfolio of only 3 equities. Now imagine mixing in a researched set of equities, fixed income instruments and risk-free treasuries. But lets use this maximized Sharpe ratio as a risky portfolio to demonstrate the CAL and the indifference curve. If we simply add a portion of risk-free treasuries to our risky portfolio and recall that the capital allocation of the risky portfolio is the slope of the capital allocation line, which is also the Sharpe ratio we can plot the CAL. This will give a 43% allocation to the risky portfolio and a 57% allocation to the risk-free asset and an expected return on the complete portfolio of 5.17% and standard deviation of 7.47%, right on the CAL.
Remember that the risk aversion metric needs to be incorporated to account for an investors risk tolerance and the utility function can map a set of portfolios for a given risk metric, A, if you were to plot the combinations of risk and returns that met the required A at the level of specified utility. Let’s use a risk aversion of A = 2 and find the utility (U) for this complete portfolio. Calculation reminder: whenever using A in an equation, always just use the percentages as decimals. We can calculate the utility for the slope we derived from the CAL, 0.43. The Utility at this allocation is 0.046 But because the CAL did not incorporate yet incorporate the risk metric for an investor, the utility is not maximized at this allocation for this portfolio. If we plot different allocations to the risky portfolio we can see that this is the case. 
We see the that greatest utility is found at a 60% allocation (remember the derivation to maximize earlier) for a risk metric A = 2, a U= 0.043 and the given risky portfolio of E[r] = 9.39%, Sigma = 17.39% and Sharpe = 0.43. If you calculate this out to check, it maximizes the return to the complete portfolio giving a complete E[r] = 5.23%, Sigma = 10.43%, Sharpe = 0.31. Laying the CAL on the indifference curse suggests this is optimal given the information that we laid out.
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