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Present Value

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How should you decide whether it is worth it to go to college or not? If the cost of college tuition is $50,000, and you can expect your annual income to increase by $10,000 after you get the degree, is that increase worth the cost? How do you compare the additional $10,000 year after year into the future with the one-time cost of $50,000? How do you even begin to think about weighing the financial costs and benefits of this decision?

You will be able to address this situation with ease once you read our explanation of present value.

Whenever undertaking important financial decisions, individuals and investors consider the benefits of the project and weigh the benefits against the opportunity cost of investing their money. Investors are highly cautious when making long-term investments. They carefully calculate the future investment income by translating it into an equivalent amount in today's money.

How do they equate future income with the value of that money in today's terms? They use a very important concept called **present value**.

**Present value** refers to the current value today of an amount of money, or stream of income, to be received at a particular future date.

Basically, it measures how much your future money is worth today.

Consider this scenario. You have $1,000 in a savings account at the bank, and your friend wants to borrow it. You know this friend to be very trustworthy, and they will pay you back in one year from now with interest--an extra $50! BUT, your bank is currently paying you 6% interest on your savings. If you lend the money to your friend, you will have $1,050 at the end of the year. However, your opportunity cost is the next best alternative, and if you leave your $1,000 in the bank, you will have 6% x $1,000 = $1,060. The risk-free interest rate from the bank is your opportunity cost of lending your friend the $1,000. After accounting for this opportunity cost, the net present value of lending the money to your friend is, roughly speaking, *negative *$10! This is the general idea underlying the time value of money, although the formula is a little more precise than this.

To account for the difference between today's money and future money, the calculation of present value makes use of a **discount rate**. The discount rate is the interest rate in the economy. It provides the rate of return an investor could be guaranteed to get by putting their money in a risk-free alternative, like depositing it in a bank. In the U.S. personal bank deposits are insured by the FDIC (up to a point).

The higher this discount rate, the lower the current value of that future income will be.

It helps to keep in mind that money loses value daily, monthly, and yearly. This percentage value loss over time compounds upon itself continuously, just as interest in your bank compounds. That means you earn interest not only on the principal amount that you originally invested, but also on the interest that you have been earning.

The purchasing power of $100 a year ago is not typically equivalent to the purchasing power of $100 now, and that's not typically equal to the purchasing power of $100 one year from now. This is because of inflation and other economic circumstances that contribute to the value loss of money, like increased uncertainty.

Present value is beneficial in accounting for inflation while calculating the current value of expected future income.

The formula for the present value of income received *n* periods in the future, with discount rate *r* is:

To understand this, consider how interest compounds over time. If you earn interest rate *r* on an investment of $100, then at the end of one year you haveIf *r* was 10%, then you now have $110. If you keep the money invested for another year, you are earning interest rate *r* on the total amountAt the end of the second year you have and each additional year that earns *r* multiplies the investment by *1+r*. If there are *n* periods total, then you earn .

Present value discounts future income in the same way, but reversed. If you know the income is arriving *n* periods in the future, then you divide the future amount by to get the equivalent amount in terms of present value. Let's discuss the individual components of the present value formula to understand it better.

Future value is just the nominal amount of money that you are expecting to receive at some time in the future. If you receive income of $100 in one year from now, then the future value of that income is $100. It is just the future amount of expected income or cash flow. This is the amount that needs to be discounted back to the present in order to account for the time value of money.

Is it better to receive $10,000 now or to receive $1,000 every year for the next ten years? Regardless of the interest rate, receiving money now is better than later, but how much better? It depends on how quickly money loses value over time. Your $10,000 could retain its purchasing power if it is invested in an asset that generates a return, or interest, without any risk of losing the principal amount.

The rate at which money loses value can be very subjective since it represents the projected rate of return you would earn if you had invested today's cash for an extended period. However, in many circumstances, a risk-free rate of return is used as a proxy for the discount rate. A risk-free rate of return means that it is guaranteed that you will have the return on your investment bank, and there will not be a default. If one was to hold cash instead of investing it in an asset, the discount rate is the opportunity cost of holding cash.

Because the government of the United States guarantees U.S. Treasury bonds and U.S. bank deposits, the interest rate on one of these assets is often used as the risk-free rate.

The number of periods refers to the time duration considered for the longevity of the investment. It could be daily, monthly, yearly. It basically shows how much money is growing throughout the considered period. If the number of a period is ten years, it shows how much worth $1000 is worth today vs. ten years from now.

A period can generally be a day, a month, a year, or almost any interval at all. The important factor is that the interest rate *r* must be the rate for one period of time using the same interval that is measured by *n*. For instance, if *r * is an **annual** rate of return on a U.S. Treasury bond, then *n* must count the number of **years **from the initial investment. If the investment pays an annual rate *r* in 18 months, then *n* would be 1.5, the number of intervals counted in years.

Similarly, if *r *is a **monthly** rate of return, then *n* must count the number of **months **instead. If the rate is 1% per month over the course of 3 years, then *n* would be 3x12 = 36. Then discount rate and the number of intervals must agree with each other.

Present value examples include assessing the current value of an investment based on its return.

Imagine you’re joining a company as a financial analyst at KKR, the leading global investment firm. As one of your first tasks, your manager asks whether you recommend purchasing a particular piece of real estate. You find out that it is a piece of land that costs $85,000, and after analyzing trends in the real estate market, you predict that the piece of land will increase in value to a price of $91,000 in the next year, a $6,0000 gain. If the interest rate earned on a certificate of deposit with the bank is currently 10%, should KKR invest in the land?

To decide whether to buy the land, the KKR investment team should calculate the present value of the future income, which is $91,000 one year from now, if the land is sold after one year. Considering that the interest rate on bank certificates of deposit is 10%, what does that $91,000 equate to in terms of present value? We use the present value formula for that.

Basically, if we invest $85,000 today, we would have $82,727.27 in income one year from now. We would lose money!

That is of course based on the fact that we can instead put the same $85,000 in the bank and earn 10% risk-free. Then, one year from now we would have $85,000 + $8,500 = $93,500. This is more than the $91,000 we would get for the land. At this discount rate, the land purchase does not measure up.

This means that it wouldn’t be a wise decision for KKR to invest the money in the piece of land.

Just as we calculate the present value of future income, we can also calculate the future value of current income.

When calculating future value, we consider the worth of a current asset at a particular time *n* periods into the future. With present value, we use a **discount **rate to account for money losing value of time--the further in the future the income arrives, the lower its present value. With future value, we **compound **interest over time that can be earned from the same guaranteed risk-free opportunity that we use as the discount rate when calculate present value.

The main difference between present value and future value is that the present value provides today's current worth of money received in the future, while future value starts with today's money and provides the worth of that money at some point in the future.

When calculating future value, we are starting with money today, which is most valuable because money today can be invested at the same guaranteed risk-free rate. The future value of that money is the amount we can expect to have from this baseline activity of depositing it in the bank at the guaranteed risk-free interest rate *r*.

From the formula for present value (PV), we can derive the equation for calculating the future value (FV) of a certain amount of dollars today projected *n* periods into the future at baseline rate *r*, as follows:

The FV equation is based on the assumption of a constant growth rate over time and a single initial amount of money today. This is the future value of today's money.

This equation can also be used to determine exactly how much money a given investment will provide. If we know that an investment returns, say, a 9% dividend in each year, then we can plug in 9% for the variable *r* in the equation for future value. When *n* *= 1*, this formula gives the income that will be received after one year. When *n = 2*, it gives the income that this investment will provide after two years, and so on.

investors may use this formula to forecast the amount of profit that different types of investment opportunities can earn with differing degrees of accuracy. In order to do that, investors use the concept of Net Present Value.

The **net present value** of an investment project is the present value of all current and future income minus the present value of all current and future costs of the project.

Hopefully, you are beginning to see how helpful the concepts of present value and future value can be!

One of the most important aspects of using present value is to account for inflation and loss in purchasing power. If an economy experiences a 10% increase in inflation, meaning that the price of goods and services increased by that particular percentage, the money in your pocket will also lose value. $1000 would buy you $900 worth of goods and services in the following year after the prices have increased by 10%. The present value captures the impact of the inflation rate and helps calculate the current value of your money as well as inform you how much your $1,000 should be worth next year to have the same purchasing power under the inflation rate.

Additionally, it is very important in valuing assets and bonds in the financial market. Present value helps inform investors about the value of an asset in today’s terms. Also, it helps investors navigate through the various assets and securities they can invest in, and make apples-to-apples comparisons between them.

Present value is based on certain key principles about the value of money over time and how to compare options. These principles, or assumptions, include:

**1. Money loses value over time. **That means that the value of today’s money is not the same as its value a year from now. This is due to both inflation as well as impatience. Prices tend to increase over time, and people tend to prefer consumption now to consumption at any point in the future.

**2. Investment options can be characterized by a constant rate of return.** This is the idea that there is a reliable average rate of return that is stable over time for any given investment option. Since many investment returns follow a random walk, their average rate of return may not be constant over time, or even follow a constant trend. Nevertheless, for simplicity, present value analysis assumes a constant rate of return.

**3. There is a guaranteed risk-free rate of return. **Another principle is that you can guarantee a certain rate of return. In the U.S., depositing money in a bank is generally considered to be risk-free because bank deposits are insured by a federal agency called the FDIC (although there is a per-person limit on the amount that is insured).

- Present value refers to the current value of the future amount of money or stream of income at a future date.
- The formula for the present value
*PV*of income*FV*to be received*n*periods in the future, using discount rate*r*, is: The formula for the future value

*FV*, after*n*periods, of an amount of money*PV*today, using discount rate*r*, is:One of the most important reasons to use present value is to account for inflation and loss in purchasing power.

Present value is based on the principles that money loses value over time, there is a constant rate of return on investments, and there is a discount rate that is guaranteed in some way.

The net present value of an investment project is the present value of current and future income from the project minus the present value of current and future costs of the project.

**Present value** is the value of money right now, today. $100 today has a present value of $100, but $100 one year from now is worth slightly less, because money loses value over time as prices go up. The present value of $100 one year from now is whatever amount right now, today, is exactly equivalent in value. It is the value in today's dollars of a stream of income in the future.

*Y* to be received *n* periods in the future, divide *Y* by *(1+r)** ^{n}*, where

*X* in the present, its value at *n* periods from now in the future is *X* multiplied by *(1+r) ^{n} *where

*n* periods starting *k* periods from now. Using the formula for present value, the first payment is valued at *W* divided by *(1+r) ^{k} *and the second payment is valued at

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