MONETARY VALUE OF TIME: UNDERSTANDING COMPOUNDING AND REAL CHANGES Compounding is the eighth wonder of the world. He, who understands it, earns it . . . he who doesn't . . . pays it. Compounding is the most powerful force in the universe. Compounding is the greatest mathematical discovery of all time. The above words, attributed to Einstein, allude to the mystical relationship between time and money. These statements tacitly suggest that a proper understanding of this relationship is not obvious and requires a process of discovery. The human mind finds it difficult to comprehend certain aspects of this association. Our judgment can be obscured by the limits of mental accounting. The problem is further aggravated by emotions that cloud our intellect. This article examines two particular aspects of time and money that can impede our understanding. First is the power of compounding that sets in the long run. Second is our bias for nominal as opposed to real changes. Each of these is dealt in turn in this article. MENTAL ACCOUNTING AND LINEAR CHANGE Every day people make decisions using mental accounting. From buying groceries to making payments at the petrol station, mental mathematics comes into use. Likewise, during investment decisions, people mostly employ mental accounting. For example, when the bank advertises an annual deposit rate of 10%, customers understand that they would receive Rupees (Rs) 10 on every Rs 100 invested. Similarly, the borrower understands that if the rate is 16% then on every Rs 100, it has to pay Rs 16. These are examples of simple mental accounting processes that most people can easily perform. Its effectiveness reduces with the increase in mathematical complexity. The example below illustrates this point. If asked to calculate the following steps mentally, most people would struggle to go beyond the third step. 5 X 5 = 25 5 X 5 X 5 = 125 5 X 5 X 5 X 5 = 625 5 X 5 X 5 x 5 X 5 = 3125 5 X 5 X 5 X 5 X 5 X 5 =? 5 X 5 X 5 X 5 X 5 X 5 X 5 X 5 = ? COMPOUNDING In financial decisions, as we increase the element of time, the thinking process can become complex. At one year increments, the mathematical process is linear but as the time period increases the problem becomes non-linear. This can be illustrated with the help of an example. Let's assume that a person invests Rs 100 for five years at the rate of 10%. At the end of year one, he makes a profit of Rs 10. If he decides to consume the additional Rs 10, then his investment at the start of the second year reduces to Rs 100. At the end of the second year, he again makes a profit of Rs 10. If he keeps on using the extra profit, by the end of the fifth year, he is left with Rs 100 only. During the five years, the total profit consumed amounts to Rs 50. This makes a total of Rs 150 (Rs 100 of principal and Rs 50 of additional profit). Now this is a simple linear problem because the investment is evaluated on a yearly basis, the excess amount is always consumed and only the principal is reinvested. The game changes, however, if the additional Rs 10 is not consumed but is Monetary Value of Time: Understanding Compounding and Real Changes 02-610-2015-2 - 2 - kept for reinvestment1 . At the end of the first year, he now has Rs 110. In the second year, he gets a return of Rs 11, Rs 10 on the principal and Rs 1 on the excess. At the end of the second year, his money amounts to Rs 121. If he keeps on repeating this, then by the end of the fifth year, he is left with Rs 161. Now the difference in linear and compounded sum in this particular case is only Rs 11 (161150 or 11 percent of the principal). If the time period increases, the difference in linear and compounded sum would become extremely significant. The diagram below compares the compounded and linear process for Rs 100 invested at 10%. Figure 1: Compounded vs. Linear Return The above diagram shows that Rs 100 invested for sixty years at 10% per annum yields Rs 30,448. The linear increase only amounts to Rs 980. The process of compounding depicted in Figure 1 teaches an important lesson that long term financial planning is required to benefit from compounding. The process of compounding is also well illustrated in mythical stories. According to a popular tale, the emperor of Persia, when first came across the game of chess, was extremely impressed and offered its inventor one wish. The inventor made a seemingly nave request. He asked the emperor for one grain of rice for the first square of the chess board and for each subsequent square wanted double the rice compared to the previous one. This means two grains on the second square, four on the third, eight on the fourth, and so on. The emperor was surprised as his mental calculation suggested that the inventor was not asking for much. He did not realise that compounding was at play, and for the sixty-fourth square, he had to give 18 million trillion grains of rice, an amount that would cover the entire surface of the earth2 . There is another interesting analogy that explains the compounding process. If one penny was invested on the birth of Jesus at a mere 4% interest per annum, it would buy one gold ball equal to the earths weight in the year 1750. In 1990 it would buy 8,190 such gold balls3 . Compounding and Consistency Consistent returns, as opposed to one off high returns followed by losses, is the other important aspect of compounding. Imagine a situation where a person invests Rs 100 and is a promised a 10% annual return for three years but suffers a loss of 10% in the fourth. Assume the cycle of three years gain and one year loss 1 = (1 + ) FV is the future value, PV is the present value, r is the rate of return and t is the time period. 2 See Appendix A 3 See Appendix A 0 5000 10000 15000 20000 25000 30000 35000 0 10 20 30 40 50 60 70 Compounded Return Linear Return years Value Monetary Value of Time: Understanding Compounding and Real Changes 02-610-2015-2 - 3 - repeat indefinitely. This new process along with the original one where the investor only makes gains of 10% is depicted in Figure 2 below. Figure 2: Compounding with Loss The diagram above shows that with the loss of 10% every fourth year, the investor's total amount is a meagre amount of Rs 1500 compared to Rs 30,448, which the investor receives if he makes a consistent gain of 10% every year. Financial Institutions and Compounding Financial Institutions also take advantage of their customer's inability to handle the mathematical complexity associated with compounding. For example, the insurance companies offer financial packages promising a future value significantly greater than the principal invested. These include financial plans related to retirement, children education, marriage, health care etc. On the face of it, many of these plansseem extremely lucrative. A closer scrutiny, however, suggests that the returns they offer are not very high. For example, some years ago a bank offered a deal that if someone invested Rs 0.1 Million for their child at the time of the birth, at the age of sixty the child gets Rs 6 Million. The sixty times increase seems exorbitant at first thought. It amounts to a compounded return of only 7% per annum. Similarly, an insurance company offered a marriage plan that included paying Rs 10,000 every month for twenty-five years. At the end the term, the company promised to pay Rs 10 million. On the face of it, this seems a reasonably attractive offer but what is missing here is that it constitutes a compounded return of only 8% a year, a rate that can be easily beaten by making personal investments4 . REAL AND NOMINAL CHANGES As argued above, a nominal increase that is linear in nature is easy to comprehend. If a person' salary has increased from Rs 100,000 to 110,000, the change is not difficult to understand and appreciate. The same goes for a decrease. If the same person after receiving his salary is pick pocketed, the measureable change should produce the associated emotion. Individuals find it difficult to make sense of changes that are real in nature. Take the example of the same person who is pick pocketed. There might be much hue and cry and even an attempt to retrieve the lost amount. If the value of his Rs 110,000 decreases because of 10% inflation, the anxiety and pain might not be as real. In fact most people view inflation as increase in prices, not as a 4 Here, since the payment is made every month for the given time period. Future value is given by the formula, = (1 + ) 1 where C is the periodic payment, r is the periodic rate of return, and n is the number of periods. Appendix B provides a detailed illustration of this example. 0 5000 10000 15000 20000 25000 30000 35000 0 20 40 60 80 Compounded Return Compounding with Loss Years Value Monetary Value of Time: Understanding Compounding and Real Changes 02-610-2015-2 - 4 - decrease in the real value of their wealth5 . Financial institutions take advantage of this phenomenon. Table 1 below shows the data for the deposit rates offered by private Pakistani banks and the inflation rate. Table 1: Real and Nominal Returns Year Deposit Rate Inflation Real Return 2004 1.35 7.44 -6.09 2005 1.74 9.06 -7.32 2006 3.63 7.92 -4.29 2007 6.50 7.60 -1.10 2008 7.31 20.29 -12.98 2009 11.05 13.65 -2.60 2010 9.05 13.88 -4.83 2011 9.09 11.92 -2.83 2012 8.63 9.69 -1.06 2013 7.08 9.20 -2.12 Source: World Development Indicators and State Bank of Pakistan The data suggests that Pakistani banks, in the last ten years, have offered deposit rates that are significantly less than inflation. The real rate of return, which is the difference between the nominal rate and inflation, is negative for the entire period. The data surprisingly is at odds with the commonly observed behaviour of investors. Bank deposits are the most popular investment choice in Pakistan. It is also considered the safest form of investment. It is common sight for retired individuals and widows to put their money in bank accounts and use the returns to fund their living expenditure with the sense of security that their principal is intact. This might very well be true in nominal terms but in real terms their original endowment decreases significantly. Figure 3 below shows how 10% inflation erodes the value of Rs 100 over time. Figure 3: Inflation and Money 5 The real rate of return is the inflation adjusted return. Real rate of return = Nominal rate of return - inflation 0 20 40 60 80 100 120 0 5 10 15 20 25 Inflation and Money Inflation and Money No. of years Value Monetary Value of Time: Understanding Compounding and Real Changes 02-610-2015-2 - 5 - Figure 3 suggests that at 10% inflation rate, the real value of Rs 100 would be half in six years. After twenty years, the value would be one-tenth of the original amount. CONCLUSION This article discusses some of the limits of the mental accounting process. Primarily, the focus is on the power of compounding and our inability to comprehend it mentally. The article also examines our bias for nominal as opposed to real changes. Given these human limitations, a thorough understanding of financial planning tools and models becomes significant for effective financial decision making.

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