LECTURE NOTES | TIME VALUE OF MONEY | GRC that you have a friend who needs P5 today and promises to pay you P10 next month. In your analysis, you would think: “If I use my P5 today, I’ll get three candies. But, if I let my friend borrow this P5 that I have, I’ll get P10 next month and by then I would get 4 candies kay because of inflation P5:2candies and since ni earn man ka og interest na P5, then I can buy 4. Or, if you would but 3 candies now, you get to spend around P7.50, mas mahal kaysa sa P5 last month but since you earned interest, you’re still P2.50 richer.) So you get to ask, in the perspective of your friend, why would he be willing to pay P10 next month in exchange of P5 today? There are a lot of factors. It could be that he has an emergency (see the danger of not setting aside emergency fund so as early as now, start saving na mo for rainy days) or he has a positive problem where he can invest the P5 and earn more. Example mag business siya which gives him a return of P20 next month. TIME VALUE OF MONEY AND INTEREST RATES In this topic, we look at the technical details of how interest rates affect the value of financial securities by reviewing time value of money concepts. For example, interest rates have a direct and immediate effect on the value of virtually all financial securities—that is, interest rates affect the price or value the seller of a security receives and the buyer of a security pays in organized financial markets. Time Value of Money Time value of money is the basic notion that a dollar received today is worth more than a dollar received at some future date. This is because a dollar received today can be invested and its value enhanced by an interest rate or return such that the investor receives more than a dollar in the future. Also with the concept of inflation, people normally choose to spend now rather than in the future. (Example, with the P5 that we have today, we can buy 3 candies. Let’s say that there is a hyperinflation and we project that our P5 next month can buy 2 candies only. Normally, we choose to buy today because our P5 is worth 3 candies rather than the value of 2 candies next month. Review concept of inflation please. If you do not buy the candies now, alkansi ka og 1 candy by next month diba. So this concept forces you to spend now rather than in the future because you’re 1 candy richer.) o But what if you do not really need the candies now? To compensate you for delaying consumption (i.e., saving), you are paid a rate of interest by those who wish to consume more today than their current resources permit (users of funds). (So since you do not really wish nor need to spend, you would rather invest your money at an interest rate more than enough to cover inflation para dili ka alkansi (refer to previous notes in nominal interest rate, inflation and real interest rate). In a layman’s perspective, let us say The time value of money concept specifically assumes that any interest or other return earned on a dollar invested over any given period of time (e.g., two, three, four, . . . years) is immediately reinvested—that is, the interest return is compounded. This is in contrast to the concept of simple interest, which assumes that interest returns earned are not reinvested over any given time period. o compound interest-interest earned on an investment is reinvested. o simple interest-interest earned on an investment is not reinvested. LECTURE NOTES | TIME VALUE OF MONEY | GRC Example 2–5 Calculation of Simple and Compounded Interest Returns CALCULATION OF SIMPLE INTEREST RETURN Suppose you have $1,000 to invest for a period of two years. Currently, default risk-free one-year securities (such as those issued by the U.S. Treasury) are paying a 12 percent interest rate per year, on the last day of each of the two years over your investment horizon. If you earn simple annual interest on this investment, or you do not reinvest the annual (12 percent) interest earned, the value of your investment at the end of the first year is: With simple interest, the $120 in interest earned in year 1 is not reinvested in year 2. Rather, you take the $120 in interest out of the investment account and hold it until the end of year 2. Only the original $1,000 investment is carried forward and earns interest in year 2. Thus, the value at the end of the two-year investment horizon is: CALCULATION OF COMPOUNDED INTEREST RETURN If, instead, the annual interest earned is reinvested immediately after it is received at 12 percent (i.e., interest is compounded), the value of the investment at the end of the first year is: Notice that after the first year of the two-year investment horizon, you have $1,120 whether the investment earns simple or compounded interest. With compounded interest, however, the $120 in interest earned in year 1 is reinvested in year 2. Thus, the whole $1,120 is carried forward and earns interest in year 2. In this case, the value of the investment at the end of the two-year investment horizon is: LECTURE NOTES | TIME VALUE OF MONEY | GRC The above illustrates the value of the investment over the two-year investment horizon using compounded interest. By compounding interest using time value of money principles, an investor increases his or her return compared to the simple interest return. In the example above using a two-year investment horizon, a 12 percent annual interest rate, and an initial investment of $1,000, the investment is worth $1,254.40 at the end of two years under compounded returns rather than $1,240 using simple interest to calculate returns. The time value of money concept can be used to convert cash flows earned over an investment horizon into a value at the end of the investment horizon. This is called the investment’s future value (FV) and is the same as that in the compounded return example above. LECTURE NOTES | TIME VALUE OF MONEY | GRC Conversion of present values into the future values Alternatively, the time value of money concept can be used to convert the value of future cash flows into their current or present values (PV) (i.e., future dollars converted into their equivalent present value or current dollars). Two forms of time value of money calculations are commonly used in finance for security valuation purposes: the value of a lump sum and the value of annuity payments. LECTURE NOTES | TIME VALUE OF MONEY | GRC LUMP SUM V. ANNUITY PAYMENT A lump sum payment is a single cash payment received at the beginning or end of some investment horizon (e.g., $100 at the beginning of the sixth year or $100 at the end of five years). here you need to be conscious that there is a big difference in your investment is it states beginning of the fifth year or end of five years Annuity payments are a series of equal cash flows received at fixed intervals over the entire investment horizon (e.g., $100 a year received each year for five years). In actual practice, “annuity” payments can be paid more frequently than once a year—so that the term annuity really means a constant payment received at equal intervals throughout an investment horizon (e.g., twice, three times, . . . a year). So in annuity, you can pay (invest) daily, monthly, quarterly, semi-annually and annually. Lump Sum Valuation Present Value of a Lump Sum. The present value function converts cash flows received over a future investment horizon into an equivalent (present) value as if they were received at the beginning of the current investment horizon. (It actually answers the question, “If I need $10,000 at the end of 6 periods, given this interest rate, how much should I invest today at year zero?”) This is done by discounting future cash flows back to the present using the current market interest rate. The present value of an investment is the intrinsic value or price of the investment. Present value ( PV ) of a lump sum received at the end of the investment horizon, or future value ( FV ): where PV =Present value of cash flows FVt = Future value of cash flows (lump sum) received in t periods r = Interest rate earned per period on an investment (equals the nominal annual interest rate, i, divided by the number of compounding periods per year (e.g., daily, weekly, monthly, quarterly, semiannually) t = Number of compounding periods in the investment horizon (equals the number of years in the investment horizon times the number of compounding periods per year) (Don’t be confused because other finance books use different letters or symbols to represent these variables but these are just the same.) LECTURE NOTES | TIME VALUE OF MONEY | GRC Calculation of Present Value of a Lump Sum You have been offered a security investment such as a bond that will pay you $10,000 at the end of six years in exchange for a fixed payment today. If the appropriate annual interest rate on the investment is 8 percent compounded annually, the present value of this investment is computed as follows: If the annual interest rate on the investment rises to 12 percent, the present value of this investment becomes: If the annual interest rate on the investment rises to 16 percent, the present value of this investment becomes: Finally, if the annual interest rate on the investment of 16 percent is compounded semiannually (that is, you will receive t=(6x2)=12 total interest payments, each calculated as r=(16 percent / 2) = 8 percent times the principal value in the investment, where r in this case is the semiannual interest payment) rather than annually, the present value of this investment becomes: Notice from the previous examples that the present values of the security investment decrease as interest rates increase. This is because as interest rates increase, fewer funds need to be invested at the beginning of an investment horizon to receive a stated amount at the end of the investment horizon. This inverse relationship between the value of a financial instrument—for example, a bond—and interest rates is one of the most fundamental relationships in finance and is evident in the swings that occur in financial asset prices whenever major changes in interest rates arise. Note also that as interest rates increase, the present values of the investment decrease at a decreasing rate. The fall in present value is greater when interest rates rise by 4 percent, from 8 percent to 12 percent, compared to when they rise from 12 percent to 16 percent— the inverse relationship between interest rates and the present value of security investments is neither linear nor proportional. Finally, from this example notice that the greater the number of compounding periods per year (i.e., semiannually versus annually), the smaller the present value of a future amount. LECTURE NOTES | TIME VALUE OF MONEY | GRC Future Value of a Lump Sum. The future value of a lump sum equation translates a cash flow received at the beginning of an investment period to a terminal (future) value at the end of an investment horizon (e.g., 5 years, 6 years, 10 years, etc.). Future value ( FV ) of a lump sum received at the beginning of the investment horizon: Calculation of Future Value of a Lump Sum You plan to invest $10,000 today in exchange for a fixed payment at the end of six years. If the appropriate annual interest rate on the investment is 8 percent compounded annually, the future value of this investment is computed as follows: Finally, if the annual interest rate on the investment of 16 percent is compounded semiannually rather than annually (i.e., r =16%/2= 8% and t= 6 x2= 12), the future value of this investment becomes: Notice that the future value of an investment increases as interest rates increase. As interest rates rose from 8 percent to 12 percent, the (future) value of the investment of $10,000 for six years rose by $3,869.49 (from $15,868.74 to $19,738.23). As rates rose from 12 percent to 16 percent, the (future) value of the investment rose $4,625.73 (from $19,738.23 to $24,363.96). Note also that as interest rates increase, future values increase at an increasing rate. This is because as interest rates increase, a stated amount of funds invested at the beginning of an investment horizon accumulates to a larger amount at the end of the investment horizon. This is due to the compounding of interest returns. By contrast, as stated earlier, as interest rates increase, the present value of an investment decreases at a decreasing rate. These nonlinear relationships are illustrated in Figure 2–15. Finally, notice that as the number of compounding periods per year increases, the future value of a present amount increases. If the annual interest rate on the investment rises to 12 percent, the future value of this investment becomes: If the annual interest rate on the investment rises to 16 percent, the future value of this investment becomes: Relation between Interest Rates and Present and Future Values LECTURE NOTES | TIME VALUE OF MONEY | GRC Annuity Valuation Present Value of an Annuity. Calculation of Present Value of an Annuity You have been offered a bond that will pay you $10,000 on the last day of every year for the next six years in exchange for a fixed payment today. If the appropriate annual interest rate on the investment is 8 percent, the present value of this investment is computed as follows: The present value of an annuity equation converts a finite series of constant (or equal) cash flows received on the last day of equal intervals throughout the investment horizon into an equivalent (present) value as if they were received at the beginning of the investment horizon. The time value of money equation used to calculate this value is illustrated in Figure 2–16 and is represented as follows: If the investment pays you $10,000 on the last day of every quarter for the next six years (i.e., r= 8%/4=2% and t= 6x4 = 24) the present value of the annuity becomes: Present value ( PV ) of an annuity stream ( PMT ) received in the future: If the annuity is paid on the first day of each quarter, an extra interest payment would be received for each $10,000 payment. Thus, the time value of money equation for the present value of an annuity becomes: which can be reduced to the simpler equation: The present value of this investment becomes: LECTURE NOTES | TIME VALUE OF MONEY | GRC Present Value of $10,000 Received on the Last Day of Each Quarter for Six Years (i.e., t = 6x 4 = 24 ) LECTURE NOTES | TIME VALUE OF MONEY | GRC Future Value of an Annuity. The future value of an annuity equation converts a series of equal cash flows received at equal intervals throughout the investment horizon into an equivalent future amount at the end of the investment horizon. Future value ( FV ) of an annuity payment stream received over an investment horizon: which can be reduced to the simpler equation: Calculation of the Future Value of an Annuity You plan to invest $10,000 on the last day of every year for the next six years. If the interest rate on the investment is 8 percent, the future value of your investment in six years is computed as follows: If the investment pays you $10,000 on the last day of every quarter for the next six years (i.e., r = 8%/4 = 2% and t = 6 x 4 = 24), the future value of the annuity becomes: LECTURE NOTES | TIME VALUE OF MONEY | GRC ANNUITY DUE If the annuity is paid on the first day of each quarter, an extra interest payment would be earned on each $10,000 investment. Thus, the time value of money equation for the future value of an annuity becomes: Thus, the future value of this investment becomes: Effective Annual Return The annual interest rate, i , used in the time value of money equations in the calculation examples above is the simple (nominal or 12-month) interest rate on the securities. However, if interest is paid or compounded more than once per year, the true annual rate earned or paid will differ from the simple annual rate. The effective or equivalent annual return ( EAR ) is the return earned or paid over a 12-month period taking any within-year compounding of interest into account. Specifically, the EAR can be written as follows: where c number of compounding periods per year. In the first calculation example above, the EAR on the 16 percent simple return compounded semiannually (i.e., r =16%/2 =8% and c= 2) is computed as: and in another calculation above, the EAR on the 8 percent simple return compounded quarterly (i.e., r= 8%/4= 2% and c=4) is computed as: Thus, for each dollar invested at the beginning of the year, at 16 percent (compounded semiannually) and 8 percent (compounded quarterly), respectively, you would have earned $0.1664 and $0.0824 at the end of the year. Accordingly, the EAR provides a more accurate measure of annual returns in time value of money calculations.
