Future Value of a Single Sum with Non-Annual Compounding Dr. Craig Ruff Department of Finance J. Mack Robinson College of Business Georgia State University © 2014 Craig Ruff

Future Value of a Single Sum: Non-Annual Compounding The issue of a non-annual compounding is very important to most of us, even if we are not working in the finance field. Home mortgages and car loans typically involve non-annual compounding. Yields on bank deposits are typically quoted with daily, monthly, and quarterly compounding. As a general rule, anytime cash flows are non-annual – like stock dividends, or bond coupons, or car payments – then you will have to deal with some aspect of non-annual compounding.

(2) Future Value of a Single Sum: Non-Annual Compounding Consider this simple example: Suppose you deposit $100 in a bank account that pays 10%, compounded semi-annually. How much will you have in the bank account at the end of two years? The key to understanding non-annual compounding is that 10%, compounded semi-annually, really means the bank is paying you interest at the rate of 5% every six months. In other words, your money is now growing at a rate of 5% every six months. To be clear…. If you deposit money in an account paying 10%, compounded annually, your money grows at a rate of 10% per year. If you deposit money in an account paying 10%, compounded semi-annually, your money grows at a rate of 5% every six months. And these are two different growth rates.

(2) Future Value of a Single Sum: Non-Annual Compounding . To see that these are two different growth rates, suppose Investor A deposits $100 in a bank account that pays 10%, compounded semi-annually, for one year and Investor B deposits $100 in a bank account that pays 10%, compounded annually. How much will each have in the bank account at the end of that one year?

(2) Future Value of a Single Sum: Non-Annual Compounding Investor A: . As noted, a rate of 10%, compounded semi-annually, really means the bank is paying you interest at the rate of 5% every six months. That is, your money is growing at a rate of 5% every six months. Thus, if Investor A deposits her money in this account, six months later she would have: And, if Investor A then keep the $105 in the account for the next six months, the bank would pay her 5% interest on this $105. Thus, at the end of the year, she would have:

(2) Future Value of a Single Sum: Non-Annual Compounding Investor B: . Investor B’s balance at the end of the year is simply:

(2) Future Value of a Single Sum: Non-Annual Compounding Investor A vs. Investor B . Investor A started the year with $100 and end the year with $110.25, that a 10.25% annual growth rate. Investor B started the year with $100 and end the year with $110.00, that a 10.00% annual growth rate. Point: 10%, compounded semi-annually, is not the same growth rate as 10%, compounded annually. Instead, 10%, compounded semi-annually, is really the same as 10.25%, compounded annually.

(2) Future Value of a Single Sum: Non-Annual Compounding . (2) Future Value of a Single Sum: Non-Annual Compounding We can easily develop the formula for the future value of a single sum with non-annual compounding following the same logic as developing the formula for the future value of a single sum with annual compounding. Recall that: and Remember that this ½ refers to N and represents ½ year. Likewise, this 1 refers to 1 year.

(2) Future Value of a Single Sum: Non-Annual Compounding . (2) Future Value of a Single Sum: Non-Annual Compounding We can recast the future value at one year as: The Old Switcheroo

(2) Future Value of a Single Sum: Non-Annual Compounding . (2) Future Value of a Single Sum: Non-Annual Compounding Another try… Time for the old switcheroo.

(2) Future Value of a Single Sum: Non-Annual Compounding . (2) Future Value of a Single Sum: Non-Annual Compounding This would keep going… at 1 ½ years, the account balance would be: The Old Switcheroo

(2) Future Value of a Single Sum: Non-Annual Compounding . (2) Future Value of a Single Sum: Non-Annual Compounding This would keep going… at 2 years, the account balance would be: The Old Switcheroo

(2) Future Value of a Single Sum: Non-Annual Compounding One can see the pattern. This leads to the general formula of… Future value of a Single Sum with Non-Annual Compounding   Where N = the # of years and m = # of compounding periods in a year. For instance, if semi-annual, then m =2; or, if quarterly, then m = 4, etc.)

(2) Future Value of a Single Sum: Non-Annual Compounding A Quick Comment: Sometimes, people will describe an account that pays you 6%, compounded semi-annually, as: “At ½ year, the bank “pays” you ½ year’s worth of interest and then the bank starts paying you interest on the interest.”   While I generally don’t use this description, I find this a useful way of thinking when we are dealing with, for instance, bonds with semi-annual coupon payments or car loans with monthly loan payments. Suppose you have two $1000 face-value bonds. Each has one year to maturity and an 8% coupon rate. One bond makes annual coupon payments and the other bond makes semi-annual coupon payments. Thus, the bonds’ remaining cash flows look like: Clearly, the second bond (the one with the semi-annual coupons) is offering a preferred set of cash flows to a bond owner, as the $4 coupon payment paid at mid-year can be reinvested for the second half of the year and earn the bond owner some extra interest.   Likewise, you can flip this logic around if you are borrowing money. As a borrower, I would rather make a single payment of $1080 at the end of a year than the $40, $1040 combination. With the $40, $1040, I would either have to fund that $40 for six months (if I don’t have the money now) or I would lose the six-months interest I could have otherwise earned on the $40 for the six months (if I do have the money now). N=0 N=1 $1080 8% coupon, paid annually: $1040 8% coupon, paid semi-annually: N=1/2 $40

Calculating the Future Value of a Single Sum with Non-Annual Compounding

Calculating the Future Value of a Single Sum with Non-Annual Compounding Working with your calculator on non-annual compounding is a bit trickier than working with annual compounding. The key point to keep in mind is that your calculator does not know years, months, days, etc. It only knows periods. So, you have to be careful what you tell your calculator.

Calculating the Future Value of a Single Sum with Non-Annual Compounding Calculating the future value of a single sum with non-annual compounding is similar to that of a future value of a single sum with annual compounding. But you need to keep a keen eye on the rate and the number of periods. Using the earlier example… Suppose you deposit $100 in a bank account that pays 10%, compounded semi-annually. How much will you have in the bank account at the end of two years? Whether you use the formula or the TVM buttons, the key thing is that this money is really being invested at 5% for four periods: 10%, compounded semi-annually, tells you it is really 5% every six months. Two years tells you that there are really four six-month periods. As a general rule, when using the TVM buttons, as soon as you see 10%, compounded semi-annually, you can hit 10 / 2 = and then hit I/Y. (Granted, on this one the math is easy… you can just hit 5 and then hit I/Y.) The next thing to determine is how many ‘periods’ are in two years. The length of the period is defined by the compounding frequency. Semi-annual tells you that the period is six months. Thus, there are four six-month periods in two years.

Calculating the Future Value of a Single Sum with Non-Annual Compounding Notice how this logic plays out with the formula from slide 43. Plugging in the numbers for the example of the FV of a deposit $100 in a bank account that pays 10%, compounded semi-annually…. Be sure to remember Aunt Sally and use parentheses as needed if you are using the formula on your calculator to find an answer. Notice that once we divide .1 by 2 and multiple 2 by 2, the formula is finding the FV using a 5% rate and four periods.

Calculating the Future Value of a Single Sum with Non-Annual Compounding Following the same logic, you would tell your calculator that this is really a FV using 5% and 4 periods. On the calculator, you would solve it as follows: PV -100 FV ???? 121.5506 IY 5 PMT N 4

Calculating the Future Value of a Single Sum with Non-Annual Compounding At the risk of massively belaboring the point…. PV -100 FV ???? 121.5506 IY 5 PMT N 4 Notice that the numbers going into the calculator are the same as the ones used in the formula above. And, again, keep in mind that the formula uses decimal numbers (.05) and the TVM buttons uses percentages (5).

Examples

Example: Future Value of a Single Sum: Non-Annual Compounding As an example…. Suppose you want to know the balance in your bank account of depositing $1000 in a bank account that pays 3%, compounded semi-annually, at the end of five years. Again, the key point is that your calculator does not know years, months, days, etc. It only knows periods. Since 3%, compounded semi-annually is really 1.5% every six months, then you need to tell your calculator that you are really depositing $1,000 at 1.5% for 10 periods. Again, your calculator does not know years, months, weeks, etc…. It only knows periods. So, you need to tell your calculator that this $1,000 is growing at a rate of 1.5% per period for 10 periods (there are 10 six-month periods in 5 years). Buttons Numbers to Enter   PV -1000 FV ???? 1,160.5408 I/Y 1.5 N 10 PMT When doing this type of problem, as soon as I see 3%, compounded semi-annually, I would hit 3 / 2 = IY. Next, I would determine the number of periods. The period length is defined by the compounding frequency…six months. And there are 10 six-month periods in five years. Another way to think about this is as: since the compounding period is twice a year, I want to use half the rate (3/2) for twice as long (2*2).

Example: Future Value of a Single Sum: Non-Annual Compounding As another example, suppose you deposit $1000 in a bank account paying 8%, compounded quarterly, for three years. What is the FV at t=3? Since 8%, compounded quarterly is really 2% every three months, then you need to tell your calculator that you are really depositing $1000 at 2% for 12 periods (three years = 12 quarters). Buttons Numbers to Enter   PV -1000 FV ???? 1268.241 I 2 N 12 PMT Again, your calculator does not know years, months, weeks, etc…. It only knows periods. So, you need to tell your calculator that this $1000 is growing at a rate of 2% per period for twelve periods (there are twelve quarters in three years). Or, since it is quarterly compounding, you could think in terms of ¼ the rate for 4 times as long.

Example: Future Value of a Single Sum: Non-Annual Compounding As another example…. Suppose you deposit $1000 in a bank account paying 8%, compounded weekly, for 7.3 years. Again, thinking in terms of periods, you are telling your calculator that the rate is 8%/52 = .15384615 and the number of periods is 7.3*52 = 379.6. Buttons Numbers to Enter   PV -1000 FV ???? 1792.392338 I/Y 8/52 =.15384615 N 52*7.3 = 379.6 PMT Again, as a rule, as soon as you see the 8%, compounded weekly, you know the IY will be 8/52. So, you hit 8 / 52 = IY. Also, since the rate is compounded weekly, you know that your period is weeks. So, the N will be the number of weeks that the money is growing. So, how many weeks are there in 7.3 years? There are 52 weeks in a year so there must be 7.3*52 = 379.6 weeks in 7.3 years. You would hit 52 * 7.3 = N. Or, again note that we are putting the money in at 1/52 the rate for 52 times as long.

Example: Future Value of a Single Sum: Non-Annual Compounding As a final example…. At the start of the year, you deposit $1,000 in an account paying 10%, compounded quarterly. What is your bank balance at the end of ¾ of a year (that is, three quarters later)?   Since the bank is paying 10%, compounded quarterly, you can think of it as having deposited your money at the rate of 2.5% per quarter for 3 quarters. So, at the end of the third quarter, you would have: Buttons Numbers to Enter   PV -1000 FV ???? 1076.890625 I 10/4 N 4*3/4 = 3 PMT What are you telling your calculator? “Calculator, the rate is really 2.5% per period and there are three periods.” Or, you can think in terms of ¼ the rate (10/4 = 2.5) for four times as long (4*3/4 = 3).

(2) Future Value of a Single Sum: Non-Annual Compounding - The Special Case of Continuous Compounding

(2) Future Value of a Single Sum: Non-Annual Compounding - The Special Case of Continuous Compounding Recall the basic non-annual compounding formula: Continuous compounding is just a special case of this basic formula. With continuous compounding, ‘m’ is very, very, very large; in fact, m is infinity. It is a hard concept to grasp. Imagine 10%, compounded at greater and greater frequencies: 10%, compounded annually (m=1); 10%, compounded monthly (m=12); 10%, compounded daily (m=365); 10%, compounded hourly (m=525,600), 10%, compounded secondly (m=31,536,000) and on and on and on. The length of the period keeps getting smaller and smaller and smaller and smaller until it is infinitely small. And, correspondingly, m keeps getting bigger and bigger and bigger and bigger until it is infinity. Continuous compounding is not typically used in day-to-day finance; however, it is used in a good bit in derivatives analysis (options and futures), as it has some very useful mathematical properties.

(2) Future Value of a Single Sum: Non-Annual Compounding - The Special Case of Continuous Compounding The basic formula for Future value of a Single Sum with Continuous Compounding is:   Where n = the # of years and e is approximately 2.71828 Example: What is the future value of $100 deposited in a bank account paying 6%, compounded continuously, for 5 years? 28

(2) Future Value of a Single Sum: Non-Annual Compounding - The Special Case of Continuous Compounding Going back to the same example….What is the future value of $100 deposited in a bank account paying 6%, compounded continuously, for 5 years? Suppose this were a multiple-choice question on a test; however, you have forgotten the continuous-compounding formula. Other than guessing, what can you do? One approach would be to use the non-annual compounding formula and a very large value for m. As an example, try m=10,000. You would have: Please note that your answer will not really be identical to that on the previous slide, which used the correct formula. Indeed, if you carry your answers out far enough, the two answers will, at some point, diverge: But you should at least get the multiple-choice question correct. Buttons Numbers to Enter   PV -100 FV ???? 134.98 I/Y 6/10,000=.0006 N 5*10,000 = 50,000 PMT Using e $134.985880757600 Using '10,000' $134.985759270703 If you get slightly different answers here, it might be because when calculating this, I took ‘e’ out to about twenty decimal places.