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Investment- Time Value of Money
Valuing cash flows, which occur over multiple times, is an essential topic in finance. You may be concerned with how much money you will have in the future (the future value) as a result of saving or investing over time. You may wish to know how much you should save in a particular amount of time to amass a specified quantity in the future. You may want to know what your expected return is on an investment with defined cash flows at different periods in time.
These types of difficulties occur every day in investing (e.g., in buying a bond), personal finance (e.g., in arranging an automotive loan or a mortgage), and corporate finance (e.g., in determining whether to develop a factory). These problems are known as ‘time value of money’ problems because their solutions reflect the idea that the timing of a cash flow influences the cash flow’s worth.
Interest
Borrowing and lending are transactions with cash flow repercussions. Someone who wants money borrows it from someone who does not need it in the moment (a saver) and is ready to lend it. In the present, the borrower has gotten money and the lender has given up money. In the future, the borrower will give up money to pay back the lender; the lender will get money as repayment from the borrower in the form of interest, as indicated below.
The lender will also receive back the money originally lent to the borrower. The money originally borrowed, which interest is computed on, is termed the principle. Interest can be described as payment for the use of borrowed money.
Hence, interest is additional money paid on top of the original amount borrowed on a loan or received on top of a deposit in a savings or investment account. Interest is paid by a borrower and earned by the lender to compensate the lender for opportunity cost and risk. Opportunity cost, in general, is the worth of other alternatives that have been given up by the lender, including lending to others, investing elsewhere, or simply spending the money. The following display provides examples of borrowers and lenders.
Interest Payment for the usage of borrowed money
Borrower
Someone or an entity who needs money and borrows it from someone or an entity who can lend it
Lender Someone or an entity who lends money to someone or an entity who needs it
Opportunity Cost
The cost of any activity defined in terms of the value of the best alternative that is not chosen; the value that investors forfeit by adopting a certain course of action.
Money Money originally borrowed, on which interest is computed
Simple Interest
A simple interest rate is the cost to the borrower or the rate of return to the lender, per period, on the original principal (the amount borrowed). Conventionally, interest rates are presented as annual rates, hence the period is presumed to be one year unless stated otherwise. The cost or return is presented as a percentage rate of the initial principal, so the rates can then be compared, independent of the amount of principal they apply to.
For example, a loan with a 5% interest rate is more expensive to the borrower than a loan with a 3% interest rate. Similarly, a loan with a 5% interest rate delivers a higher projected return to the lender than a loan with a 3% interest rate.
Simple Interest Rate Calculations
The actual amount of interest earned or paid relies on the simple interest rate, the amount of principal lent or borrowed, and the number of periods over which it is lent or borrowed. We show this mathematically in the following situation.
Simple interest = Simple interest rate × Principal × Number of periods
If you deposit money in a bank account and the bank offers a simple interest rate of 10% per annum (or annually), then for every GBP100 you put in, you (as a lender to the bank) will earn GBP10 in the course of the year (assuming at year-end to simplify calculations):
Interest = 0.10 × £100 × 1 = £10
If your money is left in the bank for two years, the interest paid will be GBP20:
Interest = 0.10 × £100 × 2 = £20
Simple interest is not reinvested and is applied just to the original principle.
Compound Interest
Interest compounds when it is added to the initial principal. Compound interest is commonly referred to as ‘interest on interest’. As opposed to simple interest, interest is supposed to be reinvested so future interest is earned on principal and reinvested interest, not simply on the initial principal.
If a deposit of GBP100 is made and earns 10% and the money is reinvested (remains on deposit), then further interest is gained in the course of the second year on the GBP10 of interest obtained in the first year. The interest is being compounded. Total interest after two years will now be GBP21; GBP10 (= £100 × 0.10) for the first year, plus GBP11 (= £110 × 0.10) for the second year.
The second year’s interest is calculated on the initial GBP100 principal plus the first year’s interest of GBP10. As seen in the next display, the total interest after two years is GBP21 rather than GBP20 as in the case of simple interest indicated in the preceding example.
Calculating Compound Interest of 10% on GBP100 Original Principal :
The link between the original principal and its future value when interest is compounded can be defined as follows:
Future value = Original principal × (1 + Simple interest rate)Number of periods
In the deposit example, £100 × (1 + 0.10)2 = £100 × (1.10)2 = £121. With compounding, the value at the end of two years is GBP121.
Compound interest is incredibly strong for savers; reinvesting the interest collected on investments is a technique of boosting funds.
Annual Percentage Rate
The rate mentioned is frequently the annual percentage rate (APR), which is a simple interest rate that does not require compounding.
Effective Annual Rate
Another often used rate is the effective annual rate (EAR). This rate includes annualising, by compounding, a rate that is paid more than once a year — commonly monthly, quarterly, or semiannually.
Whenever an interest rate compounds more than annually, in general, the EAR is bigger than the APR. In other words, more frequent compounding leads to a greater EAR.
Valuing cash flows, which occur over multiple times, is an essential topic in finance. You may be concerned with how much money you will have in the future (the future value) as a result of saving or investing over time. You may wish to know how much you should save in a particular amount of time to amass a specified quantity in the future. You may want to know what your expected return is on an investment with defined cash flows at different periods in time.
These types of difficulties occur every day in investing (e.g., in buying a bond), personal finance (e.g., in arranging an automotive loan or a mortgage), and corporate finance (e.g., in determining whether to develop a factory). These problems are known as ‘time value of money’ problems because their solutions reflect the idea that the timing of a cash flow influences the cash flow’s worth.
Interest
Borrowing and lending are transactions with cash flow repercussions. Someone who wants money borrows it from someone who does not need it in the moment (a saver) and is ready to lend it. In the present, the borrower has gotten money and the lender has given up money. In the future, the borrower will give up money to pay back the lender; the lender will get money as repayment from the borrower in the form of interest, as indicated below.
The lender will also receive back the money originally lent to the borrower. The money originally borrowed, which interest is computed on, is termed the principle. Interest can be described as payment for the use of borrowed money.
Hence, interest is additional money paid on top of the original amount borrowed on a loan or received on top of a deposit in a savings or investment account. Interest is paid by a borrower and earned by the lender to compensate the lender for opportunity cost and risk. Opportunity cost, in general, is the worth of other alternatives that have been given up by the lender, including lending to others, investing elsewhere, or simply spending the money. The following display provides examples of borrowers and lenders.
Interest Payment for the usage of borrowed money
Borrower
Someone or an entity who needs money and borrows it from someone or an entity who can lend it
Lender Someone or an entity who lends money to someone or an entity who needs it
Opportunity Cost
The cost of any activity defined in terms of the value of the best alternative that is not chosen; the value that investors forfeit by adopting a certain course of action.
Money Money originally borrowed, on which interest is computed
Simple Interest
A simple interest rate is the cost to the borrower or the rate of return to the lender, per period, on the original principal (the amount borrowed). Conventionally, interest rates are presented as annual rates, hence the period is presumed to be one year unless stated otherwise. The cost or return is presented as a percentage rate of the initial principal, so the rates can then be compared, independent of the amount of principal they apply to.
For example, a loan with a 5% interest rate is more expensive to the borrower than a loan with a 3% interest rate. Similarly, a loan with a 5% interest rate delivers a higher projected return to the lender than a loan with a 3% interest rate.
Simple Interest Rate Calculations
The actual amount of interest earned or paid relies on the simple interest rate, the amount of principal lent or borrowed, and the number of periods over which it is lent or borrowed. We show this mathematically in the following situation.
Simple interest = Simple interest rate × Principal × Number of periods
If you deposit money in a bank account and the bank offers a simple interest rate of 10% per annum (or annually), then for every GBP100 you put in, you (as a lender to the bank) will earn GBP10 in the course of the year (assuming at year-end to simplify calculations):
Interest = 0.10 × £100 × 1 = £10
If your money is left in the bank for two years, the interest paid will be GBP20:
Interest = 0.10 × £100 × 2 = £20
Simple interest is not reinvested and is applied just to the original principle.
Compound Interest
Interest compounds when it is added to the initial principal. Compound interest is commonly referred to as ‘interest on interest’. As opposed to simple interest, interest is supposed to be reinvested so future interest is earned on principal and reinvested interest, not simply on the initial principal.
If a deposit of GBP100 is made and earns 10% and the money is reinvested (remains on deposit), then further interest is gained in the course of the second year on the GBP10 of interest obtained in the first year. The interest is being compounded. Total interest after two years will now be GBP21; GBP10 (= £100 × 0.10) for the first year, plus GBP11 (= £110 × 0.10) for the second year.
The second year’s interest is calculated on the initial GBP100 principal plus the first year’s interest of GBP10. As seen in the next display, the total interest after two years is GBP21 rather than GBP20 as in the case of simple interest indicated in the preceding example.
Calculating Compound Interest of 10% on GBP100 Original Principal :
The link between the original principal and its future value when interest is compounded can be defined as follows:
Future value = Original principal × (1 + Simple interest rate)Number of periods
In the deposit example, £100 × (1 + 0.10)2 = £100 × (1.10)2 = £121. With compounding, the value at the end of two years is GBP121.
Compound interest is incredibly strong for savers; reinvesting the interest collected on investments is a technique of boosting funds.
Annual Percentage Rate
The rate mentioned is frequently the annual percentage rate (APR), which is a simple interest rate that does not require compounding.
Effective Annual Rate
Another often used rate is the effective annual rate (EAR). This rate includes annualising, by compounding, a rate that is paid more than once a year — commonly monthly, quarterly, or semiannually.
Whenever an interest rate compounds more than annually, in general, the EAR is bigger than the APR. In other words, more frequent compounding leads to a greater EAR.
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