Module 9 Lesson 3 - Read
Read: Valuing Income Streams
Overview
The value of an asset is a stock concept that reflects the capitalized value of the flow, or stream, of net income associated with owning that asset. The capitalized value is the anticipated earnings over the usable lifetime of an asset, which takes into account that money received in the future is not as valuable as money in the present.
The value of an asset at a point in time is a stock concept, while the income the asset produces through time is a flow concept.
Simple Interest
Simply put, interest (I) is a levy or fee for the use of money, such as a loan. The actual amount of money borrowed is called the principal (P). Borrowed funds are paid back using simple interest. In a simple interest scenario, the amount of money that is paid back to the lender is an amount that includes the principal plus interest on the principal. The interest on the principal is calculated with the following formula:
Interest (I) = Principal (P) x Rate (R) x Time (T)
Note: R = the interest rate; T = interest time period.
The basic idea with simple interest is that the appropriate interest amount is determined using the principal after the first interest period; after the second interest period, the appropriate interest is determined using the original principal. This process is repeated and completed over the relevant number of time periods. Figure 1 illustrates this process for simple interest applied to a $1,000 loan (i.e., principal) over three periods at an interest rate of 12%.
Compound Interest
With a compound interest-bearing loan, the amount of money that is paid back to the lender is an amount that includes the principal plus all the interest that has been accrued at each relevant point in time.
For compound interest, the appropriate interest amount is determined after the first interest period using the original principal, and then added to the principal amount. Then, the interest in the next period (i.e., the second interest period) is computed on this higher principal. For the second interest period, the appropriate interest is determined using the higher principal (i.e., the principal at the end of the first period) and then added to the higher principal, to arrive at an even higher principal going into the next period. This process is repeated and completed over the relevant number of time periods. Figure 2 illustrates this process for compound interest applied to a $1,000 principal over three periods at an interest rate of 12%.
As the illustrations show, compound interest-bearing amounts grow faster than simple interest-bearing amounts. A borrower can calculate the total amount he would pay on a compound interest-bearing loan using the formula:
A = P * (1 + r/n)nt
Note: P = principal; n = number of times interest is compounded annually; t = time in years;
r = interest rate.
A = 1,000 * (1 + 0.12)3 = $1,404.9
Present Value
Present value is simply how much a sum of money is worth today. Discounting is the process of determining the present value of money that will be received in the future. Future income flows are discounted, at an appropriate discount rate, to determine their value today.
Suppose that a bank was expecting cash flows from the repayment of a loan, as illustrated in Figure 2. That is, at the end of year one, the bank expects an inflow of $1,120, and so on, until the end of year three. Figure 3 depicts those cash flows.
The present value of this loan is the sum of the discounted value of each of the cash flows. In other words, the present value analysis determines the present value of each of the cash flows in years one, two, and three, and adds them up to arrive at the present value of the loan to the bank. To get the present value of year one cash flows, discount that amount by one year. The present value of year three cash flows requires that they are discounted by three years. Each cash flow is discounted using a duration equal to the number of interest periods (in this case, the periods are annual) that it takes for that cash flow to be received in the future.
The formula for determining the present value of a cash flow is given by:
The present value of future cash flows is the sum of the present value of each individual cash flow, mathematically:
Assuming a discount rate of 15%:
The present value for year one cash flow is: 1,120 / (1 + 0.15) = $973.9
The present value for year two cash flow is: 1,254.4 / (1 + 0.15)2 = $948.5
The present value for year three cash flow is: 1,404.3 / (1 + 0.15)3 = $923.3
Thus, the present value of the loan is: 973.9 + 948.5 + 923.3 = $2,845.7.
Figure 4 illustrates the discounting of each cash flow to arrive at the present value of this particular cash flow stream.
Notice that the present value of the year three cash flow is lower than that of year two, which is lower than the year one cash flow. This is because the further into the future a cash flow is received, the less it is worth today.
Future Value
Future value is how much an amount of money will be worth in the future. The process of determining exactly how much an amount of money will be worth in the future is imputed via compounding.
Suppose a household deposited $2,000 into an interest-bearing bank account today. Future analysis would help that household determine how much it would have in a year, or in two years, or in n years from now. Using an interest rate, that household can make these determinations.
For example, if the household is facing an annual interest rate of 12%, then it will have:
$2,240 a year from now ($2,240 = 2000 + (2,000 * 0.12))
$2,508.8 two years from now ($2,508.8 = 2240 + (2,240 * 0.12))
$2,809.8 three years from now ($2,809.8 = 2508.8 + (2508.8 * 0.12))
The future value of a $2,000 deposit three years from now at an annual interest rate of 12% is $2,809.80. This represents the value of the deposit compounded over time.
The formula for determining the present value of a cash flow is given by:
Future value (FV) = Present value (PV) ∗ (1 + r)n
Note: n = number of times interest is compounded annually; r = interest rate.
Using this formula, the future value of a $2,000 deposit is: 2,000 * (1 + 0.12)3 = $2,809.80.
Expand: Present and Future Value
Discover
Many factors can affect the stream of income associated with a particular asset. With time value of money assessments, the duration and the interest rate are some of the main factors that affect income streams.
Time Value of Money
Suppose a household needs $15,000 in five years and the annual interest rate is 3.5%. How much would that household need to deposit in the bank now in order to meet its five-year financial goals? Using the formula for present value results in the following:
Present value = $15,000 / (1 + 0.035)5 = $12,629.5
Suppose instead that a household deposits $12,629.5 into an interest-bearing bank account today. Five years from now, given an annual interest rate of 3.5%, that deposit would be worth the following:
Future value = $12,629.5 * (1 + 0.035)5 = $15,000
Changes in Duration
For this future value example, suppose that instead of five years, the duration of the $12,629.50 deposit is seven years or just three years. How would that affect the value of the money in the bank?
Future value after seven years: $12,629.5 * (1 + 0.035)7 = $16,068
Future value after three years: $12,629.5 * (1 + 0.035)3 = $14,002.50
It’s easy to observe that for future value calculations, the longer the duration, the more compounding that occurs. Accordingly, the end amount, which includes principal plus all of the interest that has been accrued at each relevant interest bearing period, will be higher. Therefore, the shorter the duration, the lower the end amount.
Changes in the Interest Rate
For this present value example, suppose that instead of 3.5% per year, the interest is 1.5% or 5%. How would that affect the value of the money in the bank?
Present value (I=1.5%): $15,000 / (1 + 0.015)5 = $13,923.9
Present value (I=5%): $15,000 / (1 + 0.005)5 = $11,752.8
Here, it can be observed that higher interest rates need lower deposits today to attain the household’s five-year goal. This is because the higher interest rate implies that money is growing at a faster rate than in the case of a lower interest rate. As such, the household needs less money deposited in the bank today for it to grow to the $15,000 mark.
Interest-Bearing Periods
Suppose instead of the 3.5% interest being compounded annually, the interest rate is compounded semi-annually. How much would a deposit of $12,629.5 be worth one year from now? An annual interest rate means interest is applied once, typically at the end of the year. Semi-annual compounding means interest is applied twice. So, divide the interest rate into two equal halves, with the first half applied after six months and the next half applied at the end of the year, as in:
3.5% / 2 = 1.75%
FV First 6 months: $12,629.5 * (1 + 0.0175) = $12,850.50
FV Next 6 months: $12,850.5 * (1 + 0.0175) = $13,075.30
The above calculations can be re-written in one line as:
FV = $13,075.3 = $12,629.5 * (1 + 0.0175) * (1 + 0.0175) or $12,629.5 * (1 + 0.0175)2
Recall that in the future value formula, n is the number of times that interest is compounded. The value of $12,629.50 deposited in an account with 3.5% semi-annual compounding interest at the end of five years is:
FV = $12,629.5 * (1 + (0.035 / 2) )2*5 = $12,629.5 * (1 + 0.0175)10 = $15,022
Since interest is applied more frequently, the deposit grows faster than it would with annual compounding.
In order for some people to borrow, other people must save. Borrowing means being able to get an item of value today by paying in the future. Saving means delaying consumption until a future period. To delay consumption, the saver must be compensated and, because of this, the borrower must pay a premium. This is why interest rates essentially embody the opportunity cost of money. A person pays an extra cost to borrow money, while another person is compensated for saving money.
Large businesses go through a lot of trouble to figure out the right interest rates to evaluate the value of potential future income streams. It is important that an interest rate used reflects various risks that could impact the realization of the value of future income streams.
Module 9 Lesson 3 of 4