# The Present Value and Future Value of Money

< prev: Interest Rates How would you like to make more than 100% interest compounded annually, virtually guaranteed, and with very little risk? This is not a misprint, and it is not a lure to sell you something. I have nothing to sell. Read on. When you learn about the present value of a dollar and the future value of a dollar, you can see things that might not be so obvious at first.

Money makes money. And the money that money makes makes more money.

— Benjamin Franklin

Money has a time value because it can be invested to make more money. Thus, a dollar received in the future has lesser value than a dollar received today. Conversely, a dollar received today is more valuable than a dollar received in the future because it can be invested to make more money. Formulas for the present value and future value of money quantify this time value, so that different investments can be compared. If a saver deposits $100 in a savings account today, and it pays 5% interest, what will it be worth 5 years from now, or 10 years from now? If an investor buys stock for $25, then sells it 3 years later for $45, what was its rate of return? A business has money and many ways to spend or invest it. What is the best use of that money?

The present value and future value of money, and the related concepts of the present value and future value of an annuity, allow an individual or business to quantify and minimize its opportunity costs in the use of money. **Opportunity cost**, in terms of the use of money, is the benefit forfeited by using the money in a particular way. For instance, if I spend $100 instead of depositing it in a bank that pays 5% interest, I forego the interest that I would have earned in the savings account by spending it instead of saving it, and if I would have saved it, then I forfeit the benefit of what I purchased. Of course, it might be possible to buy some stock, instead, that may double or triple, showing that the opportunity cost was even greater than originally thought. However, the future value of a stock is unpredictable, and the true opportunity cost of anything is really not knowable. However, the opportunity cost can be compared among specific investments where the rate of return is dependent on an interest rate that is either known or can be reasonable estimated by using the formulas for the present value and future value of money. Or a reasonable interest rate can be assumed simply to compare different investments.

## The Future Value of a Dollar

The **future value** (**FV**) of a dollar is considered first because the formula is a little simpler.

*The future value of a dollar is simply what the dollar, or any amount of money, will be worth if it earns interest for a specific period of time.*

If $100 is deposited in a savings account that pays 5% interest annually, with interest paid at the end of the year, then after the 1^{st} year, $5 of interest will be added to the $100 of principal for a total of $105. In the 2^{nd} year, interest will be earned not only on the principal of $100, but also on the $5 of interest earned. Thus, at the end of the 2^{nd} year, there will be 5 more dollars of interest earned from the principal added to the account, plus 25¢ earned from the previous year's interest of $5. Thus, at the end of the 2nd year there will be $105 + $5 + $.25 = $110.25 total in the account. This is an example of **compounding interest**, interest that is paid on interest previously earned. This process can be continued for any number of years.

Expressing this as an equation, if **P** = principal and **r** = interest rate per year, then the amount of money in the account after the 1^{st} year can be expressed by the equation **P (1 + r) = P + r*P = $100 + .05 * 100 = $100 + $5 = $105**. To find the amount after the 2^{nd} year multiply **105** by the same factor—**(1 + r)**. This equation can be expressed in terms of the 1^{st} equation: **P (1 + r) (1 + r)**, which reduces to **P (1 + r) ^{2}**. This equation can be extended to

**P (1 + r)**, with the superscript

^{n}**n**equal to the number of years. Thus, the amount of money in the account after 3 years is

**P (1 + r)**. For this example,

^{3}**100 (1 + .05)**, rounded to 2 decimal places.

^{3}= 100 (1.05)^{3}= 100 * 1.157625 = $115.76FV | = | P(1+r)^{n} | FV = Future Value of a dollar P = Principal or Present Value r = interest Rate per year n = Number of years |

**Using a calculator to determine future value:**

If you have a calculator that has the exponential function—usually designated by the y^{x} key—then this equation is easy to solve. Add the interest rate in decimal form to **1**, then press y^{x}, then enter **3**, then press the = key. Take this product, the **interest factor**, and multiply it by the principal. So for our example, enter **1.05**, then press y^{x}, then enter **3**, then press = to arrive at the interest factor **1.157625**. Multiply this by **100** to get **$115.76**, the amount of money in the account after 3 years. Because exponentiation has priority over multiplication, you can also enter it this way: **100** X **1.05** y^{x} **3** = **$115.76**.

### Compounding Interest

In all formulas that compute either the present value or future value of money or annuities, there is an interest rate that is compounded at certain intervals of time. This interval of time is assumed to be 1 year, but, if it is less than 1 year, as it frequently is, then there are 2 adjustments that must be made to the formulas:

- The number of time periods must be changed to represent the number of times that interest is compounded.
*The number of years must be multiplied by the number of compounding periods within a year.* - The interest rate itself must be changed to reflect the interest rate per time period.
*The annual interest rate must be divided by the number of compoundings in a year.*

Note also that most of the solutions to these formulas are rounded.

#### Example 1 — Adjusting a Formula for Non-annual Compounding of Interest

If you put $100 in a savings account that pays 5% interest annually, but is compounded daily, how much will be in the account after 10 years?

**Solution:** This is finding the future value of a savings account, but since this account is compounded daily, the formula will have to be adjusted as follows:

FV = | P(1+ | r c | )^{n*c} | FV = Future Value of Savings Account P = Principal r = interest rate per year n = number of years c = number of compounding periods in a year |

Thus, we find the solution by plugging the values into the formula:

FV= | 100 * (1+ | .05 365 | )^{(10*365)} ≈ 164.8665 ≈ $164.87 |

Note that with compounding interest, doubling either the interest rate or the amount of time more than doubles the amount of interest earned. For instance, $100 earning 5% interest that is paid yearly would equal **$62.89** of earned interest after 10 years; after 20 years, earned interest would equal **$165.33**.

*Thus, the future value of a dollar is the value that it will have after a specific time earning a specific interest rate.*

## The Present Value of a Dollar

Suppose you buy a zero coupon bond that matures in 10 years, then pays $1,000. How much is that future payment of $1,000 worth today at a 5% interest rate? In other words, if the prevailing interest rate is 5%, how much should you pay for a zero coupon bond that is sold at a discount to its par value?

In determining the future value of money, we know how much money we are starting with, and we want to know how much it will be worth at some point in the future at a specific interest rate. When we know how much a future payment will be, then we want to determine what its value is today at a given interest rate.

The **present value** (**PV**) is the current value of a payment that will be received in the future. **Discounting** is the process of determining the present value of a payment from a known future payment, or future value. This is the reverse of determining the future value of a payment, because in this case, we already know the future value. It is found by dividing the future value by the same interest factor, **(1 + r) ^{n}**, used to determine future value. Since FV = PV × (1+r)

^{n}, then, dividing both sides by (1+r)

^{n}yields:

PV | = | FV (1+r) ^{n} | PV = Present Value FV = Future Value r = interest Rate per time period n = Number of time periods |

### Example 2 — Calculating the Worth of a Zero Coupon Bond

How much would a zero coupon bond sell today, that pays $1,000 in 10 years, assuming an interest rate of 5% that is compounded and paid annually?

**Solution:** The zero coupon bond pays $1,000 in 10 years, so that is its future value in 10 years. If the prevailing interest rate is 5%, then to find the present value of the zero:

PV = | 1,000 (1+.05) ^{10} | = $613.91 |

**Using a calculator to determine present value:**

Enter **1,000**, press the divide key, ÷ enter **1.05**, then press the exponential key, y^{x}, then enter **10**, then the = key. The calculator should do the exponentiation 1st, because exponentiation has priority over division, then the division to arrive at the correct answer of **$613.91**, rounded to 2 places.

Summary: **1,000** ÷ **1.05** y^{x} **10** = **$613.91**.

## PV and FV Using Continously Compounded Interest Rates

The formulas for present value and future value can be modified to calculate PV and FV for continuously compounded interest rates. We note that as n increases to infinity, the following reaches a finite limit:

As n → ∞, | ( | 1 + | 1 n | ) | n | → e | = | 2.718281828... | n = number of compounding periods |

Consequently:

As n → ∞, | ( | 1 + | r n | ) | ^{n} | → e^{r} | r = interest rate n = number of compounding periods |

Therefore, the following PV and FV formulas calculate the respective values using continuously compounded rates:

FV | = | PV | × | e^{rn} | FV = Future Value of a dollar PV = Principal or Present Value r = interest Rate per year n = Number of years |

PV | = | FV | × | e^{-rn} | = | FV e ^{rn} | PV = Present Value FV = Future Value r = interest Rate per time period n = Number of time periods |

### Example 3 — Calculating the FV for a Continuously Compounded Interest Rate

Sounds like you would make a fortune earning a continuously compounded rate, but not really. To show that there is little difference between a rate compounded daily and one that is compounded continuously, we calculate Example 1 using the continously compounded rate: $100 in a savings account earning 5% interest annually, but is compounded continuously, will yield the following FV after 10 years?

FV = | PV x e^{rn} = $100 × e^{(.05 × 10)} = $100 × e^{.5} ≈ $100 × 1.648721 ≈ $164.87 | FV = Future Value of Savings Account PV = Principal r = interest rate per year n = number of years c = number of compounding periods in a year |

If you use Microsoft Excel or the free OpenOffice Calc, the above is calculated thus: =100*EXP(0.5). Note that when rounded to the nearest penny, the continously compounded rate equals the daily compounded rate:

FV= | 100 × (1+ | .05 365 | )^{(10*365)} ≈ 164.8665 ≈ $164.87 |

## Calculating the Interest Rate of a Discounted Financial Instrument

To find the present value, we need to know the future value and the interest rate; to find the future value, we need to know the present value and the interest rate. But sometimes, both the present value and the future value are known, but not the interest rate. A good example of this problem is the zero coupon bond. A zero coupon bond pays no interest during its term, but is bought at a discount to its par value. Thus, in this case, the purchase price is known, which is its present value, and its future value is the par value of the bond, usually $1,000, which is paid when the bond matures. But what is the **equivalent interest rate**? As we will see below, even though a zero pays no interest, it still has an equivalent interest rate, which can be calculated and compared to other investments. (But, unfortunately, you still have to pay taxes on the interest every year, even though you don't actually receive it until the zero matures!) How would it compare to a savings account that pays 5% interest compounded annually, for instance?

To find the equivalent interest rate, **r**, we transpose the equation for the future value of money to equal **r**. The equation for future value is:

Present Value × (1 + r)^{n }= Future Value

First, divide both sides by the Present Value:

(1 + r)^{n} = Future Payment/Present Value.

Take the **n ^{th}** root of both sides:

1 + r = (Future Payment/Present Value)^{1/n}.

Then subtract **1** from both sides, to arrive at **r**, the interest rate for the discount:

r = | ( | FV PV | ) | ^{1n} | - 1 | r = Interest Rate of Discount per time period n = number of time periods FV = Future Value PV = Present Value |

or |

### Example 4 — Calculating the Interest Rate of a Zero Coupon Bond

What interest rate is a zero coupon bond paying, that costs $600 and pays $1,000 in 10 years, assuming an interest rate that compounds annually?

**Solution:** Future Value = $1,000 par value, Present Value = $600 purchase price, n = 10 years

^{1/10}- 1 =

**5.24%**

Using a scientific calculator: **1000** ÷ **600** = ^{x}√y **10** - **1** = **.0524** = **5.24%**

Note that if the interest is compounded at different intervals, such as quarterly or daily, then the interest rate **r** and the number of compounding periods must be adjusted. But if compounding of interest is not specified, as with the zero coupon bond, what value do we use? *We use the value that allows us to compare it to another investment.* We can specify that the interest rate be compounded daily instead of annually, which will result in a lower interest rate. We would do this to compare the zero, for instance, to a savings account that is paying interest compounded daily. So how would our zero coupon bond example compare with a savings account that paid 5%, compounded daily? In other words, if we put $600 in the savings account instead of buying the bond, would we have more or less than $1,000 after 10 years?

We can solve this problem in 2 ways. We can solve the problem either by calculating the future value of $600 earning 5% compounded daily, or we can calculate the equivalent interest rate for the zero coupon bond when compounded daily. In either case, we need to know how many compounding periods there are. Since there is 365 days to a year, there is 3,650 compounding periods in 10 years. However, because the interest rate is listed as 5% per year, compounded daily, we need to find that .05/365 = .000136986 = the daily interest rate. Substituting these values into the IRD formula, the future value of the savings account is:

Future Value = 600 * (1 + .000136986)^{3650} = $989.20

We can already see that the zero coupon bond pays better, but let's see what the interest rate of the bond would be if compounded daily, like the savings account.

(1,000/600)^{1/3650} - 1 = .000139962 (daily interest rate) * 365 = .0511 = **5.11%** annual interest rate.

### Example 5 — Calculating the Interest Rate of a Fluorescent Bulb

What did you say? Fluorescent bulbs don't pay interest! Let's see.

You have a light bulb in your house, that's on quite a bit, and it's a 100 watt bulb. You typically go through 13 bulbs in 5 years, for a total of 10,000 hours of light. But, what if you buy a 23-watt fluorescent bulb, instead, which gives off almost the same amount of light, and it last 13 times longer than a typical 100-watt bulb. And let's say that your electric rate is 10¢ per kilowatt hour. Let's also assume that you pay about 33% of your income in federal, state, and local taxes, which may well be a conservative estimate.

100-watt bulbs generally cost about .25 per bulb, so 13 will cost $3.25. A 23-watt fluorescent bulb can be bought for nearly the same price as the 13 incandescent bulbs. Yes, fluorescent bulbs are more expensive than incandescents, but wait! The amount of electricity consumed by a series of 100-watt bulbs in 10,000 hours is 1,000 kilowatts (100 × 10,000 = 1,000,000 watts or 1,000 kilowatts, since a kilowatt is 1,000 watts). At 10¢ a kilowatt hour, that's $100 worth of electricity needed for incandescent bulbs.

The fluorescent bulb consumes 230 kilowatts of electricity for the same amount of light (23 × 10,000 = 230,000 watts = 230 kilowatts). At 10¢ a kilowatt hour, that's $23 worth of electricity for a savings of $77 over the incandescents. So, in 5 years, for this one light socket, you would save $77 over a period of 5 years. But, to save $77 is the same thing as earning $77 tax-free! So you paid $3.25 to earn $77 tax-free over a period of 5 years. So what is the equivalent interest rate that is compounded annually of this?

**Solution:** Future Value = $77 saved, Present Value = $3.25 purchase price, n = 5 years

^{1/5}- 1 = 88.33%

Using a calculator: **77** ÷ **3.25** = ^{x}√y **5** - **1** = **.883308...** = **88.33%**

That's a pretty impressive rate of return—a tax-free rate of return—and not only that, it's virtually guaranteed! Your only small risk is that the bulb breaks or turns out to be defective, in which case, you can probably return it to the store for another one. Note that the $77 is real money, even though you only saved it and didn't really earn it. It's real, however, because if you had bought the incandescent bulbs instead, that would be $77 less than you would have had over the 5 years. Granted, these savings aren't going to allow you to buy a Lamborghini for your next car, but it's something that not only saves you money, but saves you time as well, since you won't have to change the bulb 12 times during those 5 years! Multiply that by the number of bulbs in your house. Or if you have a business! Even a small business like a medical center could have a hundred or more light bulbs which are on all day during business hours. The savings could be quite substantial. Not only that, but you help to conserve energy and protect the world from global warming!

Since this money is earned free of all taxes, we can also calculate a taxable equivalent yield. The formula for this is:

Taxable Equivalent Yield = (Tax-free yield)/(100% - Tax%)

We assumed that you pay 33% of your income in federal, state, and local taxes. This yields:

88.33%/(100% - 33%) = 88.33%/67%= .8833/.67 = 1.318358... = **131.84%**

That's a taxable equivalent yield of 131.84%, compounded every year, for 5 years. In other words, you would have to earn that rate of return to equal the tax-free yield of 88.33%. Compare that to a savings account!

Obviously, this rate of return will vary depending on the actual value of the variables. For instance, if the light bulb was not on so much, and it lasted 10 years, then obviously this will lower the equivalent interest rate, but it will still be substantial.

Here we can see that, even though a zero and a fluorescent bulb pay no actual interest, we can find an equivalent interest rate that's compounded daily, weekly, quarterly, or whatever, so we can compare it to investments that do pay interest. This is the value of the formulas for the present value and the future value of money!

## Interest Rate Conversions

In investments, pricing and returns are often expressed in interest rates that are compounded in specific time intervals. The actual interest rate or yield will depend on the compounding period.

To find a compounded interest rate where the compounding period is shorter in duration that a given interest rate per period, simply take the appropriate n^{th} root of 1 plus the given interest rate, where n equals the number of compounding periods within the old time frame:

New Interest Rate = (1 + Old Interest Rate)^{1/n}

For instance, if a bank account pays a nominal interest of 6% per year, then the equivalent interest rate compounded every 6 months would equal the square root of 1.06%, or 1.02956%. If the new compounding period is longer than the one given, then:

New Interest Rate = (1 + Old Interest Rate)^{n}

where n = the number of compounding periods in the new period.

Thus, if a bank account paid 1.02956% interest semiannually, then that would be the equivalent of 6% per annum, since (1.02956%)^{2} = 6%.

In many financial calculations, continuous compounding is used, especially in pricing derivatives. Converting continuous rates to discrete rates, and vice versa, is only a little bit more complicated. If **c** is a continuously compounded rate and **r**_{n} is an equivalent rate that is compounded n times per year, then the following equations must be true:

PV | × | e^{c} | = | PV | × | (1+ | r_{n}n | )^{n} |

Divide both sides of the equation by PV:

e^{c} | = | (1+ | r_{n}n | )^{n} |

Take the natural logarithm (ln) of both sides to find the continuous rate that equals the discrete rate compounded n times:

c | = | n | × | ln (1+ | r_{n}n | ) | c = Continously compounded rate r = discrete interest rate n = number of compounding periods at rate r |

The formula for finding the discrete rate compounded n times that is equal to a given continuously compounded rate:

r^{n} | = | n | × | (e^{c/n} -1) | c = Continously compounded rate r = discrete interest rate n = number of compounding periods at rate r |

Math note: *ln* designates the natural logarithm of a number, such that if x = e^{y}, then y = ln x.

### Example 6: Converting a Discrete Rate into a Continuously Compounded Rate

If a bank account paid 5% per year, with biannual compounding, then the equivalent continuously compounded rate would be:

r = 2 × ln (1+ | 5% 2 | ) = 4.9385% |

### Example 7: Converting a Continuously Compounded Rate to a Discrete Rate

If a bank account pays a continuously compounded rate of 6% per year, but that is paid quarterly, then the equivalent rate compounded quarterly would be:

Quarterly Compounded Rate = 4 × (e^{(6%/4)} – 1) = 6.0452%

## Conclusion

The present value and future value of a dollar is a lump sum payment. A series of equal lump sum payments over equal periods of time is called an **annuity**. This is a more general concept than the insurance product that most people think of when they see the word *annuity*; it includes loans, interest payments from bonds, and so on—even the annuity insurance product. Like the present value and future value of a dollar, the present value and future value of an annuity allows you compare investments, or the costs of loans. For instance, you can answer questions such as, How much would my payments be on a $200,000 mortgage with a 6% interest rate? or How much of a mortgage could I get, if the interest rate is 5%, but I can only afford to pay $1,000 per month? next: The Present Value and Future Value of an Annuity