10 Financial Formulas I Wish I Had Known When I Started Investing

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Enhancing your analytical toolset beyond the basics

When one is just starting to deal with markets and portfolios, all the information out there could be overwhelming and confusing. Especially without prior knowledge of key concepts or experience in using them. Online research usually produces unsatisfactory results, as most widely available posts offer mediocre learning resources at best. You will almost always see authors advising you to adhere to such concepts as compound returns, standard deviations and Sharpe ratios, etc. And while mastering those may be fundamental, useful knowledge and results often come from other, more practice-oriented and real-data-tested tools. In this story, I will focus on 10 financial formulas/models that are not so well known to most novice investors but I still find quite useful for constructing and managing one’s portfolio.

The content of this post does not constitute financial or investment advice. It is intended for information and education purposes only. The author is not affiliated with any institution and does not earn income from any company or financial product that might be mentioned. References to names of institutions are not recommendations.

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1. Long-term expected stock returns

One of the most important metrics involved in asset allocation decisions and portfolio optimization is the expected return of an asset. It is notoriously difficult to forecast, especially over very long time horizons. And given that in many markets around the world (especially in the US), stock repurchases are a key part of a cash distribution policy, such estimates need to factor their impact more explicitly. This is what the Grinold-Kroner model does by disaggregating the expected return into 3 major components: the income return (dividend yield corrected for the expected change in outstanding shares), the nominal earnings growth (inflation and real earnings growth), and the repricing return.

The model is also useful for estimating the equity risk premium for the stock market. A rise in share repurchases will lead to a decrease in the expected equity market premium. An important note is that this model is based on a simplified view of expected returns and as such relies on assumptions that are relevant to the Gordon Growth model used for equity valuation, namely stability of the business model and constant growth and leverage. Therefore, this model also inherits some important limitations. Nevertheless, it can be utilized for analysis, simulation and stress testing of expected returns, and provides a useful understandable link between the equity market and the macro economy.

2. Factor-based variances and covariances

Another key input for a portfolio construction problem is the variances and covariances of the assets, available in the investment opportunity set. If we only had to estimate a few of those, that would not be a big issue, but in reality, the set of available investments can be rather large. This could require enormous computational effort and, if the number of assets exceeds the number of available observations, this can lead to a situation where some portfolios seem erroneously riskless. These issues can be addressed by reducing the dimensionality of this problem: that is, using the dependence of asset returns on common factors to impose a more concise structure on the variance-covariance matrix.

For this to work, the common factors K need to be identified, the regressions of asset returns need to be estimated, and the resulting sensitivities are to be used to calibrate a variance-covariance matrix, using the covariances between the much smaller number of factors themselves. As a result, instead of estimating N(N — 1)/2 elements of the matrix, where N is the number of assets, the factor-based matrix only requires N x K factor sensitivities plus K(K + 1)/2 elements of the smaller matrix plus N estimates of unique component variances v² (K < N). With a proper selection of the K factors, the number of estimated elements and the estimation errors would be significantly lower. This method becomes very useful in handling a significant number of assets and improving the cross-sectional consistency of the variance-covariance estimates.

Note, that the use of the factor-based model means that the variance-covariance matrix will have a bias, as such a model can never be completely specified and therefore there will be a difference between the estimates and the true variances and covariances (which are unobservable). But it allows for a relatively precise estimate of a biased set of variables, rather than the very noisy and imprecise estimate of the unbiased variables.

3. Required trade capital

When executing trades, beginner traders could often doubt whether the amount of capital they are committing to an individual investment position at a given time is too high or too low. This uncertainty likely goes away with some experience but nevertheless, it is usually useful to consider some basic principles. After all, if a trade turns out to be undercapitalized, there is a high risk it might fail when volatility spikes, which may cause quite significant losses.

One approach is to think of trade capitalization as an attempt to cover both the cost of opening the position (and maintaining it) and the risk of loss. This does not mean that the full amount of capital is actually invested when the trade is executed, but rather that you need to have that capital, in addition to the amount committed to the trade, available in your account so that you can absorb losses. This concept is important for margin trades as they involve leverage — i.e., the invested amount is higher than the amount of equity provided by the investor and the difference comes as a loan from the broker. In such trades, a relatively small unfavorable move against the investor can generate such a loss that entirely wipes out the equity in the position.

Sophisticated traders, professionals and financial institutions would normally use more complex models to estimate potential future losses and ensure they hold capital to cover them. In the absence of such methods, the individual investor can use the so-called Maximum Drawdown (MDD) amount as a proxy for the maximum loss of a trade. Hence, a trading account must be able to sustain a maximum (or even greater) drawdown and still be in a position to continue trading. The total required capital is therefore equal to the sum of the MDD (currency amount) and the initial margin (currency amount). Prudently, there should be a buffer over the MDD to ensure that trading can continue even in the event that the position loses as much as the MDD. The buffer should ideally reflect the market views of the investor and their personal risk preference and tolerance.

4. Equity duration

In the world of fixed-income securities, such as bonds, the concept of duration is well-defined and clear: it measures the first-order sensitivity of the bond price with respect to parallel changes in the yield curve. And it can be thought of as the average time until an investment pays back cash flows. But unlike bonds, common stocks do not have fixed cash flows and maturity. Equities represent a potentially infinite stream of risky, unknown cash flows. There have been multiple propositions as to how equity duration could be measured based on these characteristics. The approach suggested here was developed by D. Schröder and F. Esterer.

If equity duration is defined in a way that resembles that for bonds, then it is also the cash-flow weighted average time at which shareholders receive the cash flows from their investment in a company’s share. Then the formula above in continuous time takes into account the fact that there is no maturity, that the cash flows are uncertain (hence, the expectation sign), and that a company-specific cost of capital k needs to be used to discount them (the implied cost of capital is used, i.e., the internal rate of return equating the current price to the sum of the discounted future cash flows, similar to a bond’s yield to maturity).

This measure provides information about the maturity of a stock:
– newer companies (especially growth companies) generally do not pay out dividends at present, hence their cash flows are (far) into the future and they are long-duration stocks;
– more mature companies usually pay higher dividends at present and are therefore short-duration stocks.

Using the Gordon Growth model and the representation of duration as a measure of the sensitivity of the stock price to changes in the implied cost of capital, a simplified formula can be derived. It shows that shares with a long duration are more sensitive to changes in the discount rate k than shares with a short duration, indicating that equity duration is also a measure of a firm’s discount rate risk. As this formula links duration to the assumed constant growth rate (of dividends), it follows that:
– companies with a high dividend growth rate g have a large part of their cash flows in the far future, and hence they exhibit a long equity duration: their share price is very sensitive to changes in the discount rate k;
– companies with a low cost of capital also exhibit a long duration: a change in the cost of capital k has a higher relative impact compared to companies with a high cost of capital.

The second formula can also be expressed in terms of the payout ratio, the return on equity, and the price-to-book ratio of the stock. In such a formulation, the equity duration is quite useful in deciphering the relation to fundamentals and return components as well. It is also consistent with the previous interpretations: stocks with a low P/B ratio (i.e., value stocks) are short-duration stocks while growth stocks exhibit a long duration.

In discrete time, the authors of the above-linked paper use an approximation derived from the first derivative of the present value formula for stocks with respect to k. This measures the equity duration as the slope of a share’s pricing formula with respect to the implied cost of capital, standardized by the factor −(1 + k)/P₀. Different valuation models can be utilized to estimate the implied cost of capital (the authors suggest the Residual Income model).

Their empirical analysis, using the duration measures just described, finds that short-duration stocks have, on average, both higher expected and realized returns than long-duration stocks. This cannot be explained by the market factor exposure and instead, the authors show that a firm’s average cash flow maturity is a priced risk factor. Since short-duration stocks have less exposure to discount rate risk, this additional return must be compensation for their higher exposure to cash-flow risk.

5. Standard of living risk

This metric is not related to managing one’s individual portfolio but rather to assessing whether one’s assets will be sufficient to meet one’s financial goals. In a typical portfolio optimization task, the investor’s specific goals do not play a direct role. But the truth is that portfolios are in fact constructed to meet those goals. The possibility that this might not happen is called standard of living risk (SLR). When it is high, the risk that the goals will not be met is also high. Why is this useful information? Because it gives an indication of whether an investor’s risk aversion needs to be adjusted — which is a key input for an optimal portfolio. The higher the SLR, the higher the risk aversion — as the investor will have a lower capacity to take on risks to achieve his/her goals.

To measure SLR, the investor needs a component of their personal financial plan — namely, the personal balance sheet. In addition to current assets and personal liabilities, they will need to factor in and discount the expected future assets and future payment obligations, including those in retirement (which is usually the most important long-term goal). The sums of the current items and present values of the expected future items are respectively the implied assets and implied liabilities. The difference between both implied totals is the so-called discretionary wealth, which measures how much is left of the assets net of the liabilities. That is, the amount of assets not needed to cover liabilities. If discretionary wealth is high (SLR is closer to 0%), then the assets are sufficient and the investor can afford to take on risk in their portfolio. Conversely, if it is low (SLR is closer to 100%), there are either insufficient or no buffers to deploy in risk assets. When it is negative, action must be taken as the investor is essentially insolvent — this could involve additional ways to earn income, later retirement, spending cuts, debt reduction, etc.

6. Diversification ratio

For a given portfolio to benefit from diversification, the asset allocation must take into account the assets’ covariances. This is what conventional portfolio theory dictates and while this approach may have its drawbacks and limitations, it does provide some useful insights about the portfolio. Using a ratio introduced by Choueifaty and Coignard, the investor can determine the extent to which their portfolio is indeed diversified or not.

The diversification ratio divides the weighted asset volatilities by the weighted covariances and as such it will be:
– higher, when either the concentration in individual assets (their weights) decreases, or the average correlation between assets decreases;
– lower, when either the concentration in individual assets rises, or the average correlation between assets rises.

Hence, a higher ratio would indicate lower concentration and higher diversification benefits, and vice versa. Its usefulness as a guide to whether the portfolio needs to be modified is obvious, but it can also be used in simulations and stress tests, as well as optimization programs.

7. Macroeconomic estimate of the Equity Risk Premium (ERP)

There are many models for the forward-looking estimation of the equity risk premium (ERP) — the incremental return investors require for holding equities rather than the risk-free asset(s). One of them that I have found particularly useful is the Ibbotson-Chen macroeconomic model as it allows linking the premium to expected macro variables, which are normally subject to extensive analysis and forecasting.

The model expresses the ERP as a function of expected inflation, expected real earnings-per-share growth, expected P/E growth, and the expected income return. Usage of this model requires an estimation of each input parameter for the respective country and is generally only recommended to be used for countries where the equity market is a relatively large share of the economy and hence the relationship between macro variables and the equity market variables is confirmed (e.g., mostly developed economies).

The estimate of inflation expectations is usually derived from the bond market (i.e., the compound difference between yields on government bonds, one inflation-protected and one not). Real EPS growth can be set equal to the expected real GDP growth rate, adjusted up/down for a representational scope gap. The P/E growth component is set depending on whether the investor perceives that the market is under- or overvalued (if it is neither, then this component is zero). The income component can be set equal to the expected dividend yield adjusted for expected reinvestment income.

This model is useful as it can be easily stress tested and the ERP can be evaluated under various scenarios, including market crashes or recessions. The resulting ERP can be used to estimate required rates of returns on stocks and other assets (e.g., via CAPM or other pricing models) or weighted-average cost of capital.

8. International Fisher effect

This is one of the simplest equilibrium formulas but many inexperienced investors are not aware of it, even though they may have exposures in multiple currencies. There is some degree of disagreement about what exactly is referred to as the ‘international Fisher effect’ as some economists use that term to explain another relationship: if the real interest rate parity holds (i.e., real rates are equal for the domestic and foreign economy), then the nominal interest rate differential is equal to the expected inflation rate differential between the economies. However, in the definition used here, we refer to the relationship between the change in the exchange rate and the nominal interest rate differential between two countries.

This relationship tells us that the country with the higher interest rate or money market yield will see its currency depreciate so that this change in the exchange rate will offset the initial higher yield and the expected all-in return on the two investment choices (i.e., foreign vs. domestic investment) is the same. Academic research has shown that over the short and medium term, the rate of depreciation of the high-yield currency is generally lower than what this formula would imply, and in many cases, high-yield currencies have strengthened, not weakened. This effect does, however, hold better for long-term horizons and hence it is relevant for long-term investors. And if the equilibrium predicted by this relationship failed, then investors would be able to take advantage of FX carry trade strategies, meaning: generating returns by going overweight high-yield currencies at the expense of (or borrowing in) low-yield currencies.

9. Portfolio with assets in multiple currencies

This topic is also relevant for investors whose portfolios hold assets in several currencies. The returns of such portfolios can be decomposed to separate their drivers. While domestic assets do not incorporate an exchange rate effect in their returns, foreign assets do. The above formula shows how to express a foreign asset’s domestic return as a product of the foreign-currency return and the exchange rate changes. Respectively, this can be done for a multi-asset portfolio, each of which can be in its own currency.

Note that for this to work correctly, the change in the FX rate must be expressed as the percentage change in the foreign currency against the domestic currency — hence, the investor must consider the convention used to quote the FX rate. These formulas are useful not just for assessing and disaggregating historical performance, but also for modeling and forecasting. Expected future returns can be estimated using the expectation for the price movement in each foreign asset (in the foreign currency) and the expectation for exchange rate moves. One important variable that needs to be taken into account in developing such projections, however, is the correlations between any of these variables.

10. Congestion index

Finally, many beginners attempt to apply technical analysis and other trading tools when designing and managing their portfolios but struggle when it comes to ‘reading’ the market phase: trending vs. range-bound. Simply put, in a trending phase, there is a clear upward or downward direction, while in a range-bound phase, there is no direction and the market cycles between two levels. Usually, trend-following indicators (such as moving averages or MACD) are used to analyze the moves in the former, while oscillators (such as the relative strength index) are used for the latter.

The question is, how to tell which phase we are in? Using each of the sets of indicators just mentioned cannot really help as they are either too early or not meaningful in advance. One indicator that is helpful, is the so-called congestion index (CI) which measures the relative change in price over a given period (x-days) relative to the extreme range (highest high vs. lowest low) in the same period. It is usually smoothed by a 3-day exponential moving average to minimize the impact of daily noise. The critical values are -20 and 20: crossing over 20 from below = start of a rising trend; crossing under -20 from above = start of a downturn; moves between -20 and 20 = congestion/oscillation. The CI is used by some as an overbought/oversold oscillator with critical values of 85 and -85, respectively, warning about an impending price reversal.