Return and Risk of a Portfolio

A portfolio is a collection of assets.

It is always better to spread one's investment over a group of assets in order to minimize risk. This is called diversification. How does the risk reduce by doing so? The risk reduces because the loss in one asset can be offset by the gain in another.

It is important to keep in mind that risk reduction will happen if the returns from the assets that comprise the portfolio are not positively correlated. For instance, it does not make sense to have a 2 - asset portfolio, where both assets are of the same category and nature. Then, rather than reducing the risk, we would be doubling it.

Also, the risk of a portfolio responds to the weights of individual assets. Therefore, the amount of money invested in each asset is important in reducing risk. Through portfolio optimization techniques (for example, Markowitz's mean  - variance model), one can decide exactly what proportion of the portfolio should be invested in each asset to arrive at minimum risk.

We should not get carried away by the idea of not keeping all our eggs in one basket. If we have too many baskets, it becomes progressively difficult to monitor each one of them. There is always an optimum number of assets that should form a portfolio. More than that will lead to superfluous diversification, meaning that the cost and effort to track the portfolio will be more than the benefit from it.

The following video demonstrates the computation of return and risk for a portfolio of assets:

Using Texas Instruments BA II Plus Financial Calculator

Well, I need not write much about how to use the Texas Instruments BAII Plus Financial calculator. I have put together a brief video, which further redirects you to the Texas Instruments BAII Plus atomic learning page and it is quite self explanatory. Have fun with your calculator and make your life easy!!!

Return and Risk of a Single Asset

The concept of Time Value of money says, and it is rational too, that a dollar (or whatever currency is relevant for you) received today is more valuable than receiving it tomorrow. We still find that a lot of people postpone current receipt in favor of a future one. We often commit our funds to one or the other investment. Why do we do that? Well, of course, we are expecting our money to grow with time and for that to happen, the sacrifice of current income that we make today, should provide us some reward in the future. We call this reward, return.

We all know that future is uncertain and though we would like to see everything proceeding as per plan, things do go awry often. The last couple of years are recent testimony to this. It means therefore, that whenever we invest our money in expectation of future returns, there is always a chance that those expectations can be belied. That is what is the risk involved with any investment, that is, the fact that our actual return can differ from our expected return is the risk. If the actual return turns out to be more than expected, we celebrate and if the reverse happens, we can be found moping.

Therefore, just as we can hope to earn a positive return on our investments, we must also be willing to take a chance that they might turn out to be negative.

We can refuse to take that chance and avoid investing. The good part of that is that our money remains safe but dead. Stagnant money is the bad part. If it does not grow, there is only one thing that can happen with it: it goes on losing value. If you have $100 today and you hide them under a mattress, you will find that a year later, their actual worth will be less than a $100.

It follows that our willingness to tolerate a certain degree of risk, translates into a possibility of earning a positive return and the more risk you can bear, the more return you expect to earn. Put differently, if you have to be persuaded to take on some risk, you are going to ask for a suitable return for it. Risk and return go hand in hand.

In this post, I am focusing on how to find out the return and risk if we invest in a single asset (which, of course is not as wise as investing in more than one asset). The videos below demonstrate this simple process. The first video deals with return and the second one deals with risk.

Return of a Single Asset





Risk of a Single Asset

Homemade Dividends

Dividends are the distributed part of a company's profit and are one of the major ways (apart from a share repurchase program) in which a company offers cash rewards to its shareholders. The alternative is to retain the earnings in a hope of faster growth or to save for the rainy day. The company can also choose to offer non - cash rewards to the shareholders through stock - dividends or stock - splits.

I am focusing only on cash dividends here and there are a number of school of thoughts that consider the dividend policy of a company as an (un)important factor affecting its value. It should be remembered from the outset that dividends are always important. The debate is about the dividend policy. What is dividend policy?

Dividend policy is the choice between paying the dividends now or later and the literature offers arguments both for the relevance (for example, the signaling theory, the Walter model, the traditional position advocated by Graham and Dodd) as well as irrelevance of dividend policy (for example, clientele effect, Miller and Modigilani argument). Apart from these clear - cut 'yay' and 'nay' positions, we have some arguments which seek a middle ground. For example, Myron Gordon's classification of firms as 'growth' firms, 'normal' firms and 'declining' firms and the (ir)relevance of dividend policy to each one of them. We also have J. F. Muth's Rational expectations hypothesis. It proposes that what is important is not what really happens but what was expected to happen. For instance, if you expected the company to announce a high dividend but it didn't do so, you may revise your assessment of the company downwards and vice - versa.

In this post, we talk about the concept of homemade dividend (Miller and Modigilani), which simply means creating a personal dividend policy and ensuring the desired cash inflow in a given period(s). For example, if your company is paying more dividend than you need (well, you may want a lesser dividend receipt due to tax implications), you can receive the dividend check and reinvest the money back in the company (by buying additional shares through a DRIPS program) or anywhere else (perhaps in a tax saving scheme). And if your company is paying you less dividend than you need, you can make up for the shortfall by selling some shares.

If you can achieve your desired cash flow position on your own, then it does not matter to you what kind of dividend policy the company is following and unless you are affected, you are indifferent between quitting or staying with the firm. So, you might as well stay [If you stay, you don't cause a ripple. If you leave (and like you, others do too), then there is a selling pressure on the company's stock, causing it to turn southward].

Of course, this is an over - simplified way of looking at things because it does not consider factors like transaction costs and differential taxation treatment of dividend and capital gain income. However, it is important in understanding just how much relevance dividend policy has on its own, without the distortions caused by market imperfections and for that purpose (which is quite important), the procedure is helpful. Why I say this is because once you establish the pure relationship between the dividend policy and the value of a firm, you can then begin to factor in the imperfections and see how this relationship changes as each imperfection creates its own effect. It may very well be that it is the imperfect market conditions that cause the dividend policy to be relevant and without them, any policy does not have enough teeth to make an impact.

One can get a little philosophical here and muse: In a perfect world, no policy is needed. Everything takes care of itself on its own. Wow!!!

In the following video, I demonstrate the concept of homemade dividend:

Homemade Leverage

In the world of finance, leverage means using either fixed cost assets or fixed cost funds. When fixed cost assets are used, the resulting leverage is called the operating leverage and when fixed cost funds are used, we have financial leverage.

We will refer to financial leverage in this post, which is created when a firm (or a person) uses borrowed funds (debt).

Homemade means something personal. Therefore, homemade leverage means personal leverage. Just like when companies borrow and create corporate leverage, individuals, when they borrow on their personal account, create homemade leverage.

Of course, a company can reduce its leverage by retiring some of its debt. Likewise, individuals can undo  the effect of corporate leverage (at least for themselves) by doing the exact opposite of borrowing, that is lending, which practically means investing in an interest bearing security.

The concept of homemade leverage emanates from the Miller and Modigilani propositions on capital structure, where they demonstrate that investors can substitute homemade leverage for corporate leverage, when they move from one firm to another to ensure that their return and risk exposure remains unchanged.

In an ideal world, investors can render the capital structure policy of a firm irrelevant through homemade leverage because they can create their own desired leverage status independent of what the company does.

However, we live in a world that is not perfect or ideal by any means. One of the major assumptions for homemade leverage to work is that individual investors can borrow and lend at the same rate of interest as corporations. We know that this is not possible. Also, the taxation rates applicable to corporate and personal incomes are different. With these imperfections entering in, the mechanism does not work as well as it is presented in theory.

Nevertheless, it should be kept in mind that the motivation for the homemade leverage argument is not to give us some unrealistic ideas but to suggest that the capital structure policy of a company on its own can not have any effect on its value. It is only when the market imperfections appear on the scene, the  capital structure policy of a firm starts making a difference to its value.

In order to impact the value, the corporate policy on how it finances it business must be something that can not be mimicked by individual investors for financing their personal investment in the company. If individual investors can undo the effect of corporate policy by their own actions, it loses relevance. As I said before, in real world, many imperfections enter the picture, preventing individual investors in imitating corporate leverage. If they are to do that, they can only do so at different borrowing and lending terms.

In this post, we are going to assume an ideal world, not because it exists, but because we wish to see if corporate leverage is potent enough on its own to affect a firm's value.  Or is it that the existence of an imperfect world makes the capital structure policy relevant. The underpinning is that if in an ideal world, investors can switch between companies and policies on their own without changing their risk - return profile, capital structure (the mix of debt and equity capital) would then be an irrelevant item on and of its own.

The following two videos explain how the mechanism of homemade leverage works in an ideal world. In the first video, we assume an investor who wants to move from a levered firm to an un - levered firm and in the second one, we look at the opposite case, where an investor wants to move from an un - levered to a levered firm without affecting his / her return, risk and proportional ownership in companies.

Moving from a Levered to an Un - levered firm:

Moving from an Un - Levered to a Levered firm:

Stock Valuation: The Variable Growth Case (Gordon Model)

Financial Instruments are valuable because we derive some benefit from them in the form of return. It goes to say that higher the expected benefit from an asset, higher should be its value. Shares of stock are no exception. We expect to earn a return on them in two ways: 1) Dividends and 2) Capital gains (difference between the buying and selling prices).

We can see capital gains also as a function of expected future dividends. We know that a capital gain arises if the selling price of a stock is more than its buying price. It should lead us to think in terms of what affects the price of a stock. Well, there can be a host of factors, one of which is the dividend that we expect to get on it. Intuitively, if we expect to earn a higher dividend, we believe that perhaps the company in which we have invested is a profitable one. How else would it be able to afford a higher dividend payment? This perception can have a favorable impact on the stock price of the company, thereby creating an opportunity for a capital gain.

In essence therefore, we can look at the value (price) of a stock as simply a function of its expected dividends. Higher expected dividends create an upward pressure on the price and vice - versa.

We also know that expected dividends are receivable on future dates. Therefore, the value (price) of a stock should be the discounted value of these dividends. What we have now is the understanding that if we find the present value of expected dividends of a company, we can have an estimate of the current stock price. It should be kept in mind that this argument is valid only for dividend paying companies and for this post, I am sticking to a dividend paying company.

Of course, the price estimate can vary from person to person because people have different estimates of expected dividends and the required return (discount rate). Therefore, what we get from a stock valuation model of this type is an estimate of the intrinsic value of a stock, which can differ from its market value. Depending on who had a better idea about future expectations and the required rate of return, he / she will be that much closer to the real intrinsic worth of the stock. Needless to say that if we provide inputs to a model that are not in touch with the pulse of the company and the market, we will find ourselves gaping at some strange results. The fault may not be that of the model but in the inputs provided to the model. It works the same way as a computer does, that is, on the GIGO (garbage - in - garbage - out) principle.

We also know that the world is a flux. Therefore, like everything else, dividends also keep on changing over time. For a given time period, they grow (or fall) at a certain rate and then this rate can change as we move into another time period. In short, the growth rate (either positive or negative) keeps varying and when that happens, we start talking in terms of a stock valuation model that can account for changing growth in dividends.

For the sake of simplicity, let us assume that the dividends on a certain stock grow at a 5% p.a. rate for the first two years from now. After the first two years, the growth rate will change to 7% p.a. for an indefinite period of time.

Let us say that the current year's dividend was $1 per share. It will mean that the dividend at the end of the next year will be 1 x 1.05 = $1.05 and the dividend at the end of the second year will be 1.05 x 1.05 = $1.1025. If we find out the present value of these two dividends at some discount rate k, we will get a part of our answer. Why a part? Because we still need to account for 7% growth in dividends from the third year onwards. Once we do that, we can have the final answer for what is the value of this stock.

In the following video, we take up some data and arrive at the value of a stock assuming varying growth rates in dividend:

Covered Interest Arbitrage

If you decide to invest some money in a foreign financial instrument, what do you need to do? First, you need to obtain some foreign currency and then invest it in the desired financial instrument. We call this as taking a long position.

What would you do when the foreign instrument matures? Obviously, you will convert the proceeds back to your own currency to see if you made a profit.

You will realize that this can involve exchange - rate risk because by the time the foreign instrument matures, the exchange rate might have moved against you. For example, if the foreign currency has depreciated in the mean time, you will get lesser units of the home currency when you convert your earnings from the financial instrument.

What can be done to hedge this risk?

The answer is simple. You go ahead and take a short position by entering into a matching forward contract.

For example, if you invested in a short - term financial instrument in the Euro - Zone at a time when the exchange rate was 1 Euro / $ and if you expect your foreign earnings to be 200 Euros (assuming your home currency is the US dollar), you will want to make sure that even if the euro depreciates against the dollar, you do not lose on that account. If the Euro depreciated to Euro 0.8 / $ by the time your investment matured, you will get only 200 x 0.8 = $160 upon conversion. To prevent that from happening, you can enter into a forward contract to sell Euros at a pre - specified exchange rate, say Euro 0.95 / $, thereby locking in the rate at which your Euros will be converted into dollars. No matter what happens to the exchange rate in the spot market now, you can be assured of receiving 200 x 0.95 = $190 at maturity.

This process is known as covered interest arbitrage. What acts as a cover? The forward rate of exchange pre - specified in the forward contract. If you did not use a forward contract, your position would remain uncovered and you will have to bear the exchange - rate risk.

It follows that when your arbitrage is covered, you are shielded from FOREX fluctuations and that means even if the exchange rate moves in your favor, you can not take advantage of that. When you remain uncovered, you assume the exchange - rate risk but the up side is that you remain available for any favorable exchange rate movement.

Higher risk can translate into a higher return. It is as simple as that.

The question arises as to why someone would want to invest in a foreign financial instrument after all. The answer is pretty simple. If you can fetch a better return abroad, you go there, provided that  the extra return is not eroded by currency conversion.

Is there something that can discourage you from doing so? Yes. You will not be interested if whatever extra you earn is lost in converting your money from one to the other currency. Or if you  can get a similar return on a domestic financial instrument, you will not take the trouble of going abroad with your money. When will this happen? When the Interest Rate Parity (IRP) condition holds. When it holds, then the difference in national interest rates for similar securities is equal (but opposite in sign) to the forward premium or discount on the foreign currency. So, whatever you gain through the interest rate differential is offset by the premium or discount on the foreign currency.

The following video demonstrates this process:

For a foreign exchange trader, no particular country is the home country. (S)he will invest in whichever currency offers a higher return on a covered basis.

For deciding which currency to start the arbitrage process from, we take the following steps:

1. Compute the annualized forward premium / discount on the foreign currency
2. Compute absolute annual interest rate differential between the countries concerned
3. If the interest rate differential is less than the forward premium / discount, borrow in the higher interest rate country and invest in the lower interest rate country and vice - versa.

The following video gives a brief demo of this process. In the last step (where arbitrage profit is computed), I have said $116.67 - 100, whereas it should be $116.67 - 110 [Slip of tongue :)]. Please be guided accordingly.

Yield to Maturity

Yield to Maturity (YTM) of a bond is simply its compounded annual rate of return that you would earn if you bought and held the bond till it matures. It is a function of the current bond price, face value, coupon rate and the time remaining till maturity.

We can also say that the YTM is the Internal Rate of Return (IRR) of a bond.

There are a couple of assumptions inherent in the YTM. First, we assume that the coupon and principal payments will be made regularly by the bond issuer and second, the amount received as coupon payments will be continually re - invested at a rate equal to the YTM.

When the YTM < coupon rate, we have a discount bond (a bond selling for less than its face value)

When the YTM > coupon rate, we have a premium bond (a bond selling for more than its face value)

When the YTM = coupon rate, the bonds sells at par (equal to its face value)

Therefore, there is an inverse relationship between YTM and the price of a bond.

The easiest way to calculate YTM is through a financial calculator or a spreadsheet because the manual computation requires going through a few hit and trial steps.

The short video below demonstrates the manual procedure of finding a bond's YTM:

Bond Valuation

We know that when a government or a company borrows money from the public, it does so by issuing bonds. Naturally, a bond is a piece of paper (legal of course), that signifies debt.

Bonds are usually long - term borrowings, say 10 - 15 years. Needless to say that during the term of a bond, the borrowing party will pay interest to the lending party. The rate of interest at which this interest is paid is known as the coupon rate of interest and it remains the same for the term of the bond (at least for a straight bond). The principal amount on which the interest is calculated is called the Face Value or the Maturity Value of the bond. The interest payments are made periodically, either annually or semi - annually and are called coupon payments.

The value (or price) of a bond responds to the market interest rate. If market interest rates rise, the value of a bond falls and vice - versa. This is pretty intuitive. Suppose, you have a bond that pays you a 7% p.a. interest rate. You are quite happy with it until one day you find that the going market interest rate on similar financial instruments is 8.5% p.a. You get grumpy and want to sell off your bond and re - invest your money at the higher interest rate. The only problem is that you may not find a ready buyer for a bond that pays 7% at a time when other instruments are offering 8.5%. Since you want to sell and the buyers are not that forthcoming, you may have to plead the buyers and maybe give them a bargain price for your bond. The price would therefore fall. The reverse is true for when the market interest rates are lower than the coupon rate on your bond. Then, you have something valuable because you have a financial instrument that gives you more interest income as compared to other instruments on the market. Should you need to sell such a bond, it can fetch you a lucrative price.

Simply put, the market interest rate can also be called  the Required rate of return.  Let us call it k. If the coupon rate is higher than k, the price of a bond rises and is above the Face value of the bond.(Premium Bond). If the coupon rate is less than k, the price of a bond falls and is below the Face value of the bond. (Discount Bond) And if by chance, the coupon rate is equal to k, the bond sells for its face value.

The premium or discount on a bond reduces with time, i.e.; as we move closer to the maturity date, the price of the bond gets closer and closer to its face value, such that on the maturity date, the price is equal to the face value.

Mathematically speaking, the value of a bond is the present value of its coupon payments and face value (maturity value). The video below demonstrates how to find a bond's value:

Triangular Arbitrage

Triangular arbitrage is a means of earning risk - free profit by converting an amount in the first currency to the second, then converting the amount in the second currency to the third and in the end, converting the amount in the third currency back to the starting currency.

Let us say we have $100 and we convert them to Euros at the going exchange rate, then convert  our Euros to, let us say Yens and then convert our Yens back to dollars. If we find that the ending amount of dollars is more than what we started from ($100), then we have made a profit which is risk - free and is called the triangular arbitrage profit.

Opportunities for triangular arbitrage arise when there is a temporary disequilibrium between exchange rates. This misalignment is pretty short - lived and usually it can not be spotted unless a trader is alert or, even better, has some automated process set on a computer, which recognizes an arbitrage opportunity and executes a sequence of trades instantly.

The video below demonstrates manually, what a computer is supposed to accomplish:

Currency Conversion

It does seem simple to convert an amount in one currency to another. The good news is that it is!!! And should you find it a little tricky, you can watch the little video demonstration below. Toggle for a full - screen view.

Cross Rate

Cross rate is the exchange rate between two currencies implied by the exchange rates of both currencies with a third currency. The computation of cross rate is demonstrated in the video below. You can toggle  the video for a full - screen view: