The Time Value of MoneyThe Effects of Compound Interest

Concentration of WealthPresent Value

Robert M. Hayes

2005

Overview

Time Value of MoneyEffects of Compound InterestConcentration of WealthPresent Value

Why is there a time value for money?

A dollar in hand today is worth more than a dollar in hand tomorrow. Why is that?

I could buy something today and thus get the use today of what I buy.

I could invest today and gain the return from that investment. I could avoid the loss of value due to inflation in costs. I could lend the money today and gain the interest on that loan.

Why is there interest on a loan? There needs to be a return, given the value today vs. tomorrow. The loss of value from the other potential uses must be recognized. There are risks that the loan may not be repaid.

The Relevant Variables

There are therefore four relevant variables in dealing with the time value of money:

The initial amount lent, called the principal amount The time period of the loan The interest rate The time period to which the interest rate applies

Note that there are two separate and potentially different (in fact, usually different) time periods involved: (1) the time period of the loan, and (2) the time period to which the interest rate applies.

Methods of Calculating InterestSimple vs. Compound Interest

Simple interest is applied to the initial amount, called the principal, for a given time period for interest. If the period of the loan is greater than the time period for interest, the simple interest will be repeated, at the same amount, and accumulate during successive time periods for interest until the end of the time period of loan.

Compound interest is applied to the initial sum, plus any previous accumulated interest that has not been paid, for each successive time period for interest.

The rationale for compound interest is that the interest is in fact money that should be in hand at the end of the time period for interest, i.e., at the time it is due. Therefore, if that interest is not received, it is, in effect, also lent and therefore should also bear interest.

The Relevant Formulas

Let the four relevant variables be represented as follows: Principal amount, P Time period of loan, L Interest rate. I Time period for interest, T

Let C be the total amount due at the end of L, and let N be the ratio of the two time periods, L/T

For simple interest, the formula for total amount due, C, at the end of the time period of loan, L, is:

C = P* (1 + (L/T)*I) = P* (1 + N*I) For compound interest, the formula is:

C = P*(1 + I)(L/T) = P*(1 + I)N

Note that compound interest is exponential.

The Power of Compound Interest

Albert Einstein is reputed to have said, “Compound interest, not E = MC2, is the greatest mathematical discovery of all time"

He also is credited with discovering what is called “the compound interest Rule of 72”. The Rule of 72 says that the principal amount will double in 72/I years, where I is the rate of interest. For example, if the interest rate is 6%, the principal amount will double in 12 years.

To illustrate, suppose that in 1955, a person invested $5,000 in a mutual fund and, during the ensuing 50 years, all dividends were reinvested. Today, that fund is worth $160,000.

Let’s apply the Rule of 72: $160,000/$5,000 = 32. That is 25, so the original investment doubled five times. That means it doubled every 10 years, so the average interest rate was 7.2%.

Of course, during that 50 years, the inflation rate averaged about 4%, so the net gain was probably not that much.

The Effect on Concentration of Wealth

Interest, especially compound interest, plays a significant role in the concentration of wealth.

In a society, considered from an economic standpoint, there are two primary means for production: Labor and Capital. The former represents the results of the contributions of individuals as the agents of production; the latter, the results from investment of accumulated past savings in the tools for production.

The relationship between production, on the one hand, and labor and capital, as the means for production, on the other, is usually represented by a “production function”, a relatively simple example of which is the Cobb-Douglas production function.

In a very real sense, interest represents the societal income from the investment of the capital.

The question at hand here is the relationship between interest and the concentration of wealth.

To examine that relationship, let’s let the Capital owned by person k be C(k), the income from Capital be i*C(k) (so the interest rate, or “return on capital”, is i), the production generated by person k from Labor be L(k), and the total expenditures related to person k be E(k).

It is relevant and even important to note that the total expenditures, E(k), related to a person consist of three components: (1) personal expenditures (for self and dependents), (2) production expenditures (represented in part by “overhead”, which includes “management”, space, etc., and in part by materials), and (3) societal expenditures (represented primarily by government and, thus, taxes).

(The difference between the E(k) and the personal expenditures of person k, is what Karl Marx refers to as “surplus value”. That is, it is the excess of a person’s production over what is directly received for it.)

Normally, one would expect the total expenditures, over all persons, to be less than or at most equal to the total production over all persons (otherwise the accumulated social wealth of the past will be dissipated).

If L(k) – E(k) + i*C(k) > 0, there will be a net addition to societal capital (and, of course, if L(k) – E(k) + i*C(k) < 0, a net reduction to societal capital) from person k. Let’s suppose that person k is permitted to keep the increase (or lose the decrease) and add it to (or subtract it from) C(k).

Now, consider two persons, P1 and P2. Let C(k), k = 1, 2 be their respective ownership of the societal wealth. So their income from Capital will be i*C(k), respectively. Let their respective production from their Labor be the same, L, and let their respective consumption also be the same, E. Thus, in this context, they differ only in their relative wealth.

Then, their respective net savings will be S(k) = i*C(k) + L – E. The total societal net savings will be S(1) + S(2) = i*(C(1) + C(2)) + 2*(L – E).

The individual net savings result in a new distribution of capital wealth: C’(k) = C(k) + S(k) = C(k) + i*C(k) + L – E

Let C(1) = C(2) + X, so that, if X > 0, P1 has more wealth than P2. Then, C’(1) = C(2) + X + i*(C(2) + X) + L – E = (1 + i)*(C(2) + X) + L – E and C’(2) = (1 + i)*C(2)

+ L – E C’(1)/C’(2) = 1 + (1 + i)*X/C’(2) = 1 + (1 + i)*X/((1 + i)*C(2) + L – E) If L – E < 0, then (1 + i)*C(2) > (1 + i)*C(2) + L – E and therefore (1 + i)/((1 + i)*C(2) +

L – E) > 1/C(2) Therefore, C’(1)/C’(2) > 1 + X/C(2) = C(1)/C(2) Thus, if the expenditures related to a person are greater than the production related to that

person, that person’s relative share of the wealth will be reduced, even though his amount of wealth may increase.

Stages in Wealth Concentration

I think there is value in understanding the stages in wealth concentration, especially as represented by the effects of interest.

At the simplest level, such as a primitive agricultural society, every person is effectively at the level of subsistence, making just enough to meet the needs of themselves and their dependents.

At the next level, there is societal capital, as an investment in tools for production, that permits a more complex society, with greater production than mere subsistence. For a variety of reasons, there is almost certain to be some degree of concentration of wealth, with some persons in the society having more than others. And, as just shown, the degree of concentration is almost certain to increase over time.

At the next level, the degree of concentration reaches the point where those with the most wealth do not need to subsist on the results of their labors, but can do so solely on the income from their wealth.

Let’s suppose that subsistence requires an income of at least Z. In current economic terms, that might be the “federal poverty level”, which for a family of 4 is about $20,000. If the interest rate is, just for illustration, at 5%, a level of wealth of $400,000 would generate that level of income, without the need to work. Working would then, of course, provide the resources for life beyond the poverty level, or for increasing one’s wealth, or for some mix of the two.

At the next level, the degree of concentration reaches the point where those with the most wealth do not need to subsist on the results of their labors, but can do so solely on the income from the income from their wealth.

Continuing with the example of Z = $20,000 as the subsistence level, that means that the income from the wealth generates $400,000 per annum so, at 5%, the wealth must be $8,000,000.

The important point here is that the growth in wealth no longer depends at all upon labor, but can be generated solely from the interest. Indeed, if the person could subsist on the $20,000 per annum, the capital wealth would increase by nearly 5% per annum and thus would double in 15 years!

At this level or perhaps at the next one, the capital ceases to be money. It becomes power and control.

I want to examine one final level simply to show what happens. Let the wealth accumulated by a person be such that subsistence can be obtained from the income of the income on the income (three levels remove from the need for labor).

Continuing with Z = $20,000 as the subsistence level, the wealth would need to generate $8,000,000 in income so, at 5%, that implies wealth of $160,000,000. Clearly, this is at the level where money represent power.

And we have not gotten even close to Bill Gates!

Application to U.S. National Economy

Simply to illustrate some of the relationships among the things I have just discussed, let’s look at the U.S. national economy.

From the 1997 Input/Output Tables, we have the following data: Total Intermediate Input = $6.7 Trillion Product from Capital, Labor (Value Added) = $8.8 Trillion Government taxation = $2.7 Trillion Additions to capital = $1.3 Trillion Net for Capital and Labor = $4.8 Trillion

From the Cobb-Douglas production model: 8.8 = a*(L)b *(C)(1-b)

Let C = K*L. Then 8.8 = a*L*K(1-b)

Net for Capital and Labor = L + i*C = 4.8 If i = 5%, then L + .05*C = 4.8 If a* K(1-b) = 10, then L = .88 and C = 3.92

Present Value

The Role of Present Value of Money Calculating Present and Future Value of Money Using Net Present Value Analysis Selecting a Discount Rate Identifying Cash Flows to Consider Determining Cash Flow Timing Selecting the Best Alternative Identifying Issues and Concerns

The Role of Present Value of Money

Why is a dollar today worth more than a dollar a year from now? Investment Inflation Use and Enjoyment

The Role of the Discount Rate The bases for choice of the discount rate The role of risk assessment The role of capitalization rates The effect of the time period

Present and Future Value of Money

Present Value and Future Value Effects of inflation Present value of a cash stream. Present value of a cash stream in perpetuity

Calculating Present Value  t

P = Fy/(1 + i)y

y = 1 P = present dollar value

Fy = future dollar value in year y

i = annual rate of return (e.g., 0.05 is 5% per annum) y = the succession of yearst = number of years in the future 

If the future dollars are the same for each year, say Fy = F,   t 1 t (1 + r) t-1 1 let S = ——— so (1 + r)*S = ——— = ——— y=1 (1 + r) y y=1 (1 + r) y y=0 (1 + r) y    1 1 1 (1 + r)t - 1(1 + r)*S – S = ——— – ——— = 1 – ——— = ————— (1 + r)0 (1 + r)t (1 + r)t (1 + r)t

Hence:  (1 + r)t - 1P = F*S = F ————— r*(1 + r)t  

There are times when the present value analysis needs to consider a cash stream in perpetuity—for an infinite period of time. Consider the formula shown above

(1 + r)t - 1 P = F*S = F ———— r*(1 + r)t

but let t be infinity. Note that the second term in the expression on the right becomes zero and the first term, 1/r. The result is that

P = F/r

Using Net Present Value Analysis

Illustrative Contexts for use of Present Value Lease-purchase Different lease alternatives Life-cycle cost Trade-off of acquisition costs and costs of operation

Factors Affecting Net Present Value The timing of the cash flow The discount rate

Steps in Net Present Value Analysis

Step 1. Select the discount rate. Step 2. Identify the costs/benefits to be considered Step 3. Establish the timing of the costs/benefits. Step 4. Calculate net present value of alternatives Step 5. Select the option with best net present value.

Selecting A Discount Rate

Nominal Discount Rates Real Discount Rates Selecting the Rate for Analysis.

Nominal Discount Rates

Most benefit-cost analyses should use nominal discount rates (i.e., discount rates that include the effect of actual or expected inflation or deflation).

Maturity in Years

Discount Rate

3 years 5.60% 5-year 5.70% 7-year 5.80%

10-year 5.90% 30 years 6.10%

Real Discount Rates

For some projects, it may be more reasonable to assess in terms of constant dollars. The real discount rate is the nominal discount rate adjusted to eliminate the effect of anticipated inflation/deflation.

Maturity in Years

Discount Rate

< 3 years 3.40%5 year 3.50% 7 year 3.50%

10 year 3.60% > 30 years 3.80%

Determining the Discount Rate

Once the type of discount rate has been selected (whether nominal or real), the values to be used are then determined from the appropriate table (using linear interpolation to determine values for years between those in the table).

Identifying Cash Flows To Consider

Cash Flow Identify all relevant cash flows, both costs and benefits Alternatives should clearly identify the cash flows that are

specifically significant Points to Consider in Identifying Costs and Benefits

Include the same cash flows in all alternatives Include cash flows in which alternatives will differ Do not include cash flows that are identical for alternatives Do not include sunk costs or benefits

Analysis Period For leasing contexts, use the leasing period plus renewal For acquisition contexts, use life cycle period For equipment context, use amortization period

Representative Costs & Benefits Net Purchase Price Costs for Transportation, Installation, Site preparation Costs for Design, Training, and Management. Repair and improvement costs, including:

Estimated unplanned service calls Improvements required to assure continued operation.

Operation and maintenance, including: Operating labor and supply requirements; and Routine maintenance.

Disposal costs and salvage value, including: Cost of modifications to return equipment to original configuration Cost or modifications to return facilities to original configuration Salvage value at the end of the period for analysis

Determining Cash Flow Timing

Bases for determining cash flow timing Offer-Identified Cash Flows. Government-Identified Cash Flows.

End-of-year payment When to use End-of-Year Discount Factors End-of-Year Discount Factor Calculation. Repetitive End-of-Year Cash Flows.

Mid-Year Payment When to Use Mid-Year Discount Factors. Mid-Year Discount Factor Calculation. Repetitive Mid-Year Cash Flows.

Calculating Net Present Value to Select The Best Alternative

Lease-Purchase Decision, Example 1 Lease-Purchase Decision, Example 2

Lease-Purchase Decision, Example 1

Which of the following will result in the lowest total cost of acquisition?

A: Proposal to lease the asset for 3 years. The annual lease payments are $10,000 per year, the first payment due at the beginning of the lease and the remaining two payments due at the beginning of Years 2 and 3. 

B: Proposal to purchase the asset for $29,000. It has a 3-year useful life. Salvage value at the end of 3-year period will be $2,000.

Step 1. Select the discount rate. The term of the lease analysis is three years, so we will use the nominal discount rate for three years, 5.4 percent.

Steps 2 and 3. Identify and establish the timing of the costs/benefits to be considered in analysis. The expenditures and receipts associated with the two offers and their timing are delineated in the table below: (Parentheses indicate a cash outflow.)

y Offer A Offer B0 ($10,000) ($29,000)1 ($10,000) 02 ($10,000) 03 0 $2,000

Step 4. Calculate net present value. The table below summarizes for each alternative.

Step 5. Select the offer with the best net present value. In this example, it is Offer B, the offer with the smallest negative net present value.

y Cash Flow DF PV Cash Flow DF PV0 ($10,000) 1.000 ($10,000) ($29,000) 1.000 ($29,000)1 ($10,000) 0.947 ($9,470)2 ($10,000) 0.897 ($8,968) $2,000 0.849 $1,698

($28,438) ($27,302)

NET PRESENT VALUE OF OFFER A

NET PRESENT VALUE OF OFFER B

Net Present Value Net Present Value

Lease-Purchase Decision, Example 2

Which of the following will result in the lowest total cost of acquisition?

A: Proposal to lease the asset for 3 years. The monthly lease payments are $1,500; that is, the total amount for each year is $18,000. These payments are spaced evenly over the year, so the use of a MYDF would be appropriate.

B: Proposal to purchase the asset for $56,000. It has a 3-year useful life. At the end of the 3-year period it will have a $3,000 salvage value.

Step 1. Select the discount rate. The term of the lease analysis is three years, so we will use the nominal discount rate for three years, 5.4 percent.

Steps 2 and 3. Identify and establish the timing of the costs/benefits to be considered in analysis. The expenditures and receipts associated with the two offers and their timing are delineated in the table below: (Parentheses indicate a cash outflow.)

t Offer A Offer B0 0 ($56,000)1 ($18,000) 02 ($18,000) 03 ($18,000) $3,000

Step 4. Calculate net present value. The table below summarizes for each alternative.

Step 5. Select the offer with the best net present value. In this example, it is Offer A, the offer with the smallest negative net present value.

t Cash Flow

DF PV Cash Flow DF PV

1 ($18,000) 0.9731 ($17,516) ($56,000) 1 ($56,000)2 ($18,000) 0.9215 ($16,587)3 ($18,000) 0.8727 ($15,709) $3,000 0.849 $2,548

($49,812) ($53,452)

Net present value of Offer A Net present value of Offer B

Net Present Value Net Present Value

Identifying Issues and Concerns

Is net present value analysis used when appropriate? Are the dollar estimates for expenditures and receipts reasonable? Are the times projected for expenditures and receipts reasonable? Are the proper discount rates used in the net present value

calculations? Are the proper discount factors used in analysis? Are discount factors properly calculated from the discount rate? Have all cash flows been considered?

THE END