## Unit 2 Study Guide: Time Value Of Money: Future Value, Present Value, And Interest Rates

##### 2a. explain the time value of money
• Why does the value of money change over time?
• How can one calculate the value of a lump sum in the future?
• How do I find the current value of a future lump sum?

While the value of money is relatively stable in the American economy, time does have an effect on its value. Time is a measure of risk in finance. Economic conditions, inflation, interest rates, the money supply, and investor expectations all change over time, and therefore, have effects on the value of money. The saying is true that "a dollar today, is worth more than a dollar tomorrow". The reason this is true is because no one can accurately predict what will happen between today and tomorrow, or even between now and the next minute. There is risk when investing or when holding money in any form for any period of time. The more money or assets that are held in forms less liquid than cash, the greater the risks. Financial calculations take into account the relationship between time and monetary value. These calculations compute what is called the "time value of money". When computing the value of money over multiple time periods, finance uses formulas to reflect that the value of a lump sum in the future (future value) or the value of a future lump sum in the present (present value) differs based on the interest rate, whether the investment earns simple or compound interest, the number of periods for which interest is earned, and the total life of the investment. When we calculate values in the future, it is called compounding. Calculating values back to the present is called discounting. To find the present value, you discount to the present. To find the future value of a lump sum, you compound into the future.

##### 2b. compute present values and future values
• How do we calculate the future value of a lump sum?
• How do we calculate the present value of a future lump sum?

To account for changes in the time value of money, finance has two ways of computing the value of a lump sum of money over a period of time. There are two basic questions in money valuation that an investor needs to answer: 1) How much is a lump sum worth at n periods into the future? and 2) How much is a future lump sum worth today? These questions represent the future value and the present value, respectively. To answer the first question, you need to compound; to answer the second question, you need to discount. To compound a lump sum, you multiply it by the compound factor. To discount a lump sum, you multiply it by the discount factor. The valuation of every type of financial instrument or transaction is a variation on the present value or future value computation. To accurately calculate the present or future value you need some standard inputs, such as the interest rate (which becomes the discount or compound rate), the number of times in one year the investment compounds, the total number of years in the life of the investment, and the lump sum amount. The lump sum amount is multiplied by all of the other elements of the equation, which combine to form either a compound factor or a discount factor. The compound factor and the discount factor are inverses of each other. Future values and present values can be computed for lump sums or for multiple flows. The future value of multiple flows and the present value of multiple flows requires the same basic components.

The present value and future value equations are:

Present Value: $\frac{FV}{\left ( 1+r \right )^t}$

Future Value: $PV \left ( 1+r \right ) ^t$

##### 2c. compute rates of return and know their use in making financial decisions
• How do we compute the profit earned on an investment?
• Why is profit earned quoted as a percentage rate?

The ultimate goal of investing is to profit from the investment. In finance, the profit is called "return". The rate of return, expressed as a percent, is the ratio of the profit to the amount invested. Sometimes, firms or investors desire a certain rate of return prior to entering into a transaction. After calculating the potential value, the investor can make the decision not to enter into the transaction if the rate of return is not acceptable. A predetermined rate of return can also be used as the discount rate when valuing a transaction. A desired rate of return can be arbitrarily chosen, based on market conditions, or can be based on historical returns an investor has received.

##### 2d. explain when to apply a simple interest calculation versus a compound interest
• What is the difference between simple interest and compound interest?

Interest is the amount paid for or earned on a transaction, depending on which side of the transaction you sit. An interest rate is simply a price paid for money, stated as a percentage of the principle or some other specified amount. There are many interest rates in the economy and they differ based on the relevant product and market.

There are two main ways to calculate interest: simple interest or compound interest. With simple interest, you only earn interest on the principal amount. Simple interest is calculated by multiplying that stated interest rate by the principal amount. The product of that operation is the amount of interest for the given period. Repeat these steps for each period that interest is earned. For the final step, sum all of the interest amounts for each period to get the total interest earned on the transaction using simple interest.

Compound interest differs from simple interest because it allows one to earn interest on their interest. This means that in the equation, over multiple periods the interest calculation includes the principal and interest from all prior periods. This results in a final amount when using compound interest that is greater than simple interest would be using the same inputs for the equation.

##### 2e. calculate the future value and present value of an amount using one period and multiple periods
• How does the value of a lump sum when it is invested for more than one period?

Financial investments are typically held over a period of time or usually earn interest over period of time. When calculating future or present value, the investment can last for one period or multiple periods. Each period represents a period of compounding. When computing future value, the greater number of periods, the greater the final amount. The time horizon matters, as well. The longer amount of time over which the investment compounds, the greater the final future value after multiple periods of compounding. The opposite is true of calculating the present value for multiple periods. The more frequently the investment is discounted and the longer the life of the investment, typically, the smaller the present value. When multiple periods are used, the value of n in the equations below is greater than one. Compounding for one period usually uses the word "annual", whereas compounding for multiple periods uses words other than annual, such as "daily," "continuously," "bi-annually," semi-annually," "quarterly," etc. The value of n is always written in comparison to annual terms. For example, an investment that compounds every two years (bi-annually) would have an n of 0.5, whereas something compounding every six months (semi-annually) would have an n of 2.

Try these examples:

1. You invest $1,000 earning 5% interest compounded semi-annually. How much do you have total at the end of 3 years? 2. You invest$1,000 earning 5% interest compounded bi-annually. How much do you have total at the end of 3 years?
##### Unit 2 Vocabulary

This vocabulary list includes terms that might help you with the review items above and some terms you should be familiar with to be successful in completing the final exam for the course.

Try to think of the reason why each term is included.

• Time value
• Present value
• Future value
• Compound
• Discount
• Discount rate
• Discount factor
• Compound factor
• Interest rate
• Rate of return
• Life of the investment