Lesson 4 – Time Value of Money (The Income Approach to Value)

Financial decision making involves the analysis of costs and benefits spread out over time. Time value of money concepts and techniques are used to calculate and to compare the values of sums of money at different points in time. The “time value of money” refers to the fact that a dollar today is worth more than a dollar in the future. The fundamental reason for this is that one can invest money in hand and end up with a greater amount of money in the future.

Discounting is a procedure used to convert periodic incomes, cash flows, and reversions into present value. It is a method of finding today's value for the right to receive future benefits (income) that a property may yield. Each future benefit (income payment) is discounted using one of the six functions summarized below. The present value is the sum of all the discounted payments. Discounting is based on the assumption that benefits received in the future are worth less than the same benefits received now.

A series of payments, usually made at equal intervals, is known as an annuity. The present value of an annuity is the sum of the several separate periodic incomes discounted to their respective present worth. Factors and rates used to convert annuities into value can be obtained from compound interest and annuity tables, personal computers, or financial calculators.

Compound interest and annuity tables generally contain six basic formulas that are used to find either a future worth of a present income or the present worth of a future income:

1. The future worth of the present value of money compounded by an interest rate: The amount to which money will grow in a given length of time (holding period) when invested or deposited at a given rate of interest. A holding period is a time span of, or term of, ownership of an investment.
2. The present value of future collections: The present value of money to be collected at a specific future time when discounted from that future time to the present date at a given rate of interest.

The Board publishes compound interest and annuity tables, both annual and monthly, for use by appraisers, in Assessors' Handbook Section 505, Capitalization Formulas, and Tables, (AH 505). The tables are arrayed by interest rates [i], ranging from 1 percent to 25 percent, in one-half percent increments. The tables are comprised of six different functions, and the numbers in each column represent a relationship between value and income (factor). The tables include:

Note: n = number of compounding periods, and i = effective interest rate for each

Future Worth of 1 (FW1) – Column 1

The future worth of one dollar is the amount to which an investment or deposit of one dollar will grow in a given number of periods including the accumulation of interest at the effective rate per period. For example, if you deposit $500 into a savings account, the future worth of one factor will tell you what that deposit will be worth in 30 years.

The future worth of one factor is based on the premise that one dollar deposited at the beginning of a period earns interest that accrues during the period and becomes part of the principal at the end of that period or the beginning of the second period. All other functions of a dollar assume the deposit(s) are at the end of the period.

DEMONSTRATION: Future Worth of 1

If you deposit $6,000 in your credit union today, how much will you have in 11 years if the effective interest rate is 6½ percent compounded monthly?

A Single Deposit × Future Worth of 1 = Future Value

Future Worth of 1 per Period (FW1/P) – Column 2

The future worth of one per period is the total accumulation of principal and interest of a series of equal deposits or installments of one dollar per period, for a given number of periods, with interest at the effective rate per period. The future worth of a dollar per period is applicable to ordinary annuity problems. An ordinary annuity may be defined as a series of periodic payments, usually equal in amount, and payable at the end of the period. For example, if you deposit $500 per month in a payroll savings plan, the future worth of one per period factor will tell you how much those deposits will be worth in 30 years.

DEMONSTRATION: Future Worth of 1 per Period

If you sign up for a savings plan with your credit union and they begin automatically deducting $375 per month beginning one month from today, how much will you be able to withdraw in 20 years if the effective interest rate remains at 7 percent compounded monthly?

A Series of Equal Deposits × Future Worth of 1 per Period = Future Value

Sinking Fund Factor (SFF) – Column 3

The sinking fund factor is a series of equal periodic investments or deposits required to accumulate one dollar in a given number of periods including the accumulation of interest at the effective rate. For example, if you need to replace a $10,000 roof in five years, the sinking fund factor will tell you how much money you need to save each month.

Present Worth of 1 (PW1) – Column 4

The present worth of one dollar is for the right to receive one dollar to be collected at a given future time when discounted at the effective interest rate (yield rate) for the number of periods from now to the date of collection. It is the discounted amount of a future value, so the factor is always less than one. It is based on the fact that money collectible in the future is always worth less than current money. For example, if you will receive a $10,000 settlement in five years, the present worth of a dollar will tell you much money you could get today if you sell the rights to that settlement.

DEMONSTRATION: Present Worth of 1

Your uncle, whom you did not know, recently passed away. He bequeathed you a lump sum of $2.2 million. You are entitled to receive this amount in seven years. The interest that is accrued from the inheritance goes to the SPCA. A tax loophole will be eliminated in one year that will cause the $2.2 million dollars to be heavily taxed unless it is invested soon. Your only hope is to cash in the trust and take a discount on the future value. What is the present value of the trust if you sell it today and the buyer discounts it at 15 percent per year compounded at an annual rate for the next seven years?

A Single Future Payment × Present Worth $1 = Present Value

Present Worth of 1 per Period (PW1/P) – Column 5

The present worth of one per period is used to compute the present value of a series of future equal installments or payments of one dollar per period for a given number of periods when discounted at the effective interest rate (yield rate). Make note of this sentence. It discounts an annuity to an indicator of value as of today (the present). For example, the present worth of one period will tell you the remaining balance on a loan.

DEMONSTRATION: Present Worth of 1 per period

Congratulations! You have just won the state lottery. When you cashed in your ticket, you received $750,000. The remaining amount will be available to you annually for the next 19 years in amounts equal to your original payment. You decide that you would like to have the money now so that you can enjoy it in your youth. The only way that you can do this is to sell the rights to collect the future payments. Assuming the best deal you can make is to sell the rights discounted at 14 percent, how much money will you receive?

A Series of Future Payments x Present Worth $1 per Period = Present Value

Periodic Repayment (PR) – Column 6

Periodic repayment is used to compute the amount of periodic installments that will pay interest and provide full recapture of an investment of one dollar in a given number of periods with interest at the given rate per period. For example, the periodic repayment factor is used to calculate the periodic amount necessary to amortize, or pay off, a loan of one dollar given a periodic interest rate and the number of repayment periods. Part of the periodic repayment is interest on the outstanding loan balance and part is repayment of loan principal.

It is composed of the sinking fund factor plus the effective interest rate. For example, the Sinking Fund Factor at six percent annual compounding for five years was given in Example 4-3 to be 0.177396. Add to that the six percent interest rate, 0.06, and the total is 0.237396, the Periodic Repayment factor that will be used in Example 4-6, below.

The sinking fund factor is the level periodic installment or deposit that will pay interest and provide full recapture of an investment of one dollar in a given number of periods with interest (yield) at a given interest rate.

DEMONSTRATION: Periodic Repayment

You are facing a dilemma. You can buy a fancy sports car that you have always wanted. The current price is $115,000. If you buy it in five years, the price is projected to be $145,000. Based on the following information, determine the monthly payments if you purchase the car now versus the amount you would need to deposit into an account monthly in order to buy the car five years from now.

1. Assuming you have $10,000 in your credit union account that can be used as a down payment, what would your monthly payments be if you bought it today with a 12 percent loan financed for five years with monthly payments?
2. What if you wait five years, and the amount you have today remains earning 6½ percent, compounded monthly, how much would you have to deposit monthly in order to buy the car for $145,000? Assume that your monthly savings will earn at the same rate as your other savings.

1. If you bought the car today, you calculate your monthly loan payments based on the following formula: Loan Amount × Periodic Repayment = Payment Amount

   Cells of note are highlighted. MONTHLY COMPOUND INTEREST TABLES

   Note this value.ANNUAL RATE 12.00%

   Years	Column 1Future Worth of 1	Column 2Future Worth of 1 per Period	Column 3Sinking Fund Factor	Column 4Present Worth of 1	Column 5Present Worth of 1 per Period	Column 6Periodic Repayment	Months	Column 7Mortgage Constant
   1	1.1268250	12.682503	0.0788488	0.8874492	11.255077	0.088849	12	1.066185
   2	1.2697346	26.973465	0.0370735	0.7874551	21.243387	0.047073	24	0.564882
   3	1.4307688	43.076878	0.0232143	0.6989249	30.107505	0.033214	36	0.398572
   4	1.6122261	61.222608	0.0163338	0.6202604	37.973959	0.026334	48	0.316006
   Note this value.5	1.8166967	81.669670	0.0122444	0.5504496	44.955038	Note this value.0.022244	60	0.266933

   $115,000

   Purchase Price Today

   −

   $10,000

   Cash Down Payment

   $105,000

   Loan Amount

   ×

   0.022244

   (PR, 12%, Monthly, 5 Years)

   $2,335.62

   Monthly Payment Required To Pay Off Loan
2. If you decide to wait to buy the car in five years when it will cost $145,000, you will need to perform 2 calculations to determine the amount you will need to deposit monthly in order to buy the car since you plan to leave the $10,000 in your credit union account earning interest. The first calculation determining the future amount, at the end of five years, of the $10,000 earning 6 ½ percent compounded monthly. The second calculation determining the future amount required after reducing the purchase price by the future amount of the $10,000 deposited in the credit union; then applying the sinking fund factor to determine the monthly deposit required to receive the required amount at the end of five years.

   A Single Deposit × Future of $1 = Future Value

   Cells of note are highlighted. MONTHLY COMPOUND INTEREST TABLES

   Note this value.ANNUAL RATE 6.50%

   Years	Column 1 - Note this value.Future Worth of 1	Column 2Future Worth of 1 per Period	Column 3Sinking Fund Factor	Column 4Present Worth of 1	Column 5Present Worth of 1 per Period	Column 6Periodic Repayment	Months	Column 7Mortgage Constant
   1	1.0669719	12.364034	0.0808798	0.9372318	11.587967	0.086296	12	1.035557
   2	1.1384289	25.556111	0.0391296	0.8784035	22.448578	0.044546	24	0.534555
   3	1.216716	39.631685	0.0252323	0.8232678	32.627489	0.030649	36	0.367788
   4	1.2960204	54.649927	0.0182983	0.7715928	42.167488	0.023715	48	0.284579
   Note this value.5	Note this value.1.3828173	70.673968	0.0141495	0.7231613	51.108680	0.019566	60	0.2534794

   $10,000

   Savings Account

   ×

   1.382817

   (FW1, 6.5%, Monthly, 5 Years)

   $13,828.17

   Future Value of Current Savings Account

   Amount Required (Including Earned Interest) x Sinking Fund Factor = Payment

   Cells of note are highlighted. MONTHLY COMPOUND INTEREST TABLES

   Note this value.ANNUAL RATE 6.50%

   Years	Column 1Future Worth of 1	Column 2Future Worth of 1 per Period	Column 3 - Note this value.Sinking Fund Factor	Column 4Present Worth of 1	Column 5Present Worth of 1 per Period	Column 6Periodic Repayment	Months	Column 7Mortgage Constant
   1	1.0669719	12.364034	0.0808798	0.9372318	11.587967	0.086296	12	1.035557
   2	1.1384289	25.556111	0.0391296	0.8784035	22.448578	0.044546	24	0.534555
   3	1.216716	39.631685	0.0252323	0.8232678	32.627489	0.030649	36	0.367788
   4	1.2960204	54.649927	0.0182983	0.7715928	42.167488	0.023715	48	0.284579
   Note this value.5	1.3828173	70.673968	Note this value.0.0141495	0.7231613	51.108680	0.019566	60	0.2534794

   $145,000.00

   Future Purchase Price

   −

   $13,828.17

   Future Value of Current Savings Account

   $131,171.83

   Amount To Be Saved Over The Next 5 Years

   ×

   0.014149

   (SFF, 6.5%, Monthly, 5 Years)

   $1,855.95

   Monthly Deposit Required To Save For New Car

Mortgage Constant (MC) – Column 7 (monthly tables only)

The monthly compound interest and annuity tables perform the same functions as the annual compound and annuity interest tables except that the interest is compounded monthly and the resulting answer is a monthly amount. The monthly compound interest and annuity tables also include a seventh column entitled mortgage constant (Rm). The mortgage constant annualizes the monthly periodic repayment factor. The mortgage constant is the ratio of annual debt service to the loan principal. It is used to find the annual debt service of a loan with interest at a given rate per period. Therefore, the mortgage constant is the sum of 12 equal monthly payments expressed as a factor to be applied to a remaining principal loan amount that is to be amortized over a certain term.

Earlier in this lesson we discussed the six functions of dollar and listed the formulas used to calculate these functions where i ≡ the effective interest rate and n ≡ the number of compounding periods. (Note: ≡ means "identical to" and "=" means "equal to") In summary:

The student may have noticed common elements in these formulas. The functions are, in fact, all related, and all start with the basic formula that one plus the effective growth rate (such as the interest rate), raised to the number of compounding periods, will give you what one will grow to at the end of those compounding periods. Here are some of the more common and easily understood relationships:

The Future Worth of One and the Present Worth of One are reciprocals of each other:

• FW1 ≡ Sⁿ FW1 ≡ Sⁿ = (1 +i) ⁿ (1+i)ⁿ = 1 ÷ PW1
• PW1 ≡ 1/Sⁿ FW1 ≡ Sⁿ = (1 +i) ⁿ i ÷ (1+i)ⁿ = 1 ÷ FW1

The Future Worth of One per Period and the Sinking Fund Factor are reciprocals of each other:

• FW1/P ≡ S_n FW1 ≡ Sⁿ = (1 +i) ⁿ {(1+i)ⁿ−1} ÷ i = 1 ÷ SFF
• SFF ≡ 1/S_n FW1 ≡ Sⁿ = (1 +i) ⁿ 1 ÷ {(1+i)ⁿ−1} = 1 ÷ FW1/P = PR−i

The Present Worth of One per Period and the Periodic Repayment factor are reciprocals of each other:

• PW1/P ≡ a_n FW1 ≡ Sⁿ = (1 +i) ⁿ [1 − {1 ÷ (1+i)ⁿ}] ÷ i = 1 ÷ PR
• PR ≡ 1/a_n FW1 ≡ Sⁿ = (1 +i) ⁿ i ÷ [1 − {1 ÷ (1+i)ⁿ}] = 1 ÷ PW1/P = SFF+i

The difference between the Sinking Fund Factor and the Periodic Repayment factor is the (effective) interest rate; that is, the Sinking Fund Factor, which shows the amount of principal to deposit, plus the interest rate, will equal the Periodic Repayment, which shows the amount of principal and interest to pay:

• SFF ≡ 1/S_n FW1 ≡ Sⁿ = (1 +i) ⁿ i ÷ {(1+i)ⁿ−1} = 1 ÷ FW1/P = PR − i
• PR ≡ 1/a_n FW1 ≡ Sⁿ = (1 +i) ⁿ i ÷ [1 − {1 ÷ (1+i)ⁿ}] = 1 ÷ PW1/P = SFF+i

The basic theoretical underpinning of the income approach to value uses the process of discounting a series of future payments. As such, a thorough understanding of the time value of money needs to be understood before continuing with the remaining lessons. If you have any unresolved issues with the concepts covered in this lesson, further review of compound interest and annuity concepts is indicated. Please return to the Self-Paced Online Learning Session on the Time Value of Money – Six Functions of a Dollar if you need to review this topic again before you continue.

In the next Lesson, we will explain the definition of the income approach to value, discuss what the income approach entails, and discuss the provisions and directives that Property Tax Rule 8 provides for using the income approach to derive an opinion of value.