Teoría del cuidado humano de Jean Watson

2.2 Marco Teorico

2.2.3 Teoría De Enfermería

2.2.3.1 Teoría del cuidado humano de Jean Watson

The Income Approach to value is designed to compute the “present worth of future benefits.” One method of determining present worth is by the use of Compound Interest Tables. These tables are summaries of various compound interest factors using different interest rates, time periods, and payment schedules.

The tables can be found in textbooks (e.g., Property Assessment Valuation) or the factors can be calculated with financial calculators such as the HP-12C or within Microsoft Excel. A sample table based on 6% annual interest will be used in our examples and is presented on the next page.

The Compound Interest Tables are most commonly presented in a format with 6 columns in specific order. There are various names for the functions, but they are typically arrayed as follows:

Column 1: Amount of 1

Amount of 1

Future Worth of $1 at Compound Interest Future Value of $1

Future Worth of $1 (FW $1)

amount of one (Sn) - The compound interest factor that indicates the amount to which $1 (or other unit of currency) will grow with compound interest at a specified rate for a specified number of periods. The amount of one is one of the six functions of one found in standard financial tables; also called future value of one.

This series of factors shows the amount to which $1 will grow at a given interest rate in a given number of years. The column is constructed by adding compound interest to a one- time deposit at the beginning of the first year.

Amount of 1 Example 6-1

What is the value 5 years from now of a $1,000 deposit in your savings account if the interest rate is 6% compounded annually?

Answer: Look to the 6% Annual Table, Column 1 (Amount of 1) for 5 years shows a factor of 1.338226. Thus, a one-time deposit made today of $1,000 will grow to $1,338.25 in five years:

$1,000 x 1.338226 = $1,338.23 in 5 years at 6% interest when compounded annually We can also calculate the amount $1 will grow by using a financial calculator such as the HP-12C. The HP-12C keystrokes necessary to solve or example are provided below. (HP Keystrokes: “1000 CHS PV”, “6 i”, “5 n”, “FV”) display 1,338.23

The formula used to derive the Amount of 1 (compounded annually) factor is: FW $1 = (1 + i)n

Where FW $1 is the Amount of 1, i is the interest rate expressed as a decimal, and n is the number of years in the compounding period. In the above example, the Amount of 1 Factor can be calculated as follows:

FW $1= (1 + .06)5

FW $1 = (1.06) x (1.06) x (1.06) x (1.06) x (1.06) FW $1 = 1.338226

(Try it on your calculator.)

Column 2: Amount of 1 per Period

Amount of 1 Per Period

Future Worth of $1 Per Period w/Interest Future Worth of $1 Per Period (FW $1/P)

amount of one per period (Sn¬) - The compound interest factor that indicates the amount to which $1 (or other unit of currency) per period will grow with compound interest at a specified rate for a specified number of periods. The amount of one per period is one of the six functions of one found in standard financial tables; also called sinking fund accumulation factor or future value of one per period.

Amount of 1 per Period Example 6-2

What is the value 5 years from now of a $1,000 deposit in your savings account each year (at the end of the year) if the interest rate is 6% compounded annually?

Answer: Looking to the 6% Annual Table, Column 2 (Amount of 1 Per Period) for 5 years shows a factor of 5.637093. Thus, a deposit of $1,000 each year (at the end of the year) for 5 years will grow to $5,637.09.

$1,000 x 5.637093 = $5,637.09

(HP Keystrokes: “1000 CHS PMT”, “6 i”, “5 n”, “FV”) display 5,637.09

The formula used to derive the Amount of 1 Per Period Factor is:

FW $1/P =

( )

i 1

1+ −

Where FW $1/P is the Amount of 1 Per Period, i is the interest rate expressed as a decimal, and n is the number of years in the compounding period.

In the above example, the Amount of 1 Per Period Factor can be calculated as follows:

FW $1/P =

(

)

06 . 1 06 . 1+ 5 − =

( ) ( ) ( ) ( ) ( )

06 . 1 06 . 1 06 . 1 06 . 1 06 . 1 06 . 1 × × × × −

This one is a little trickier to try on your calculator but it is simplified as:

FW $1/P = 1.338226 – 1 or 0.338226 = 5.63709

Column 3: Sinking Fund Factor

Sinking-Fund Factor (SFF) Accumulation to $1

Periodic Payment to Grow to $1

sinking fund factor (1/Sn¬) - The compound interest factor that indicates the amount per period that will grow, with compound interest, to $1 (or other unit of currency). The sinking fund factor is one of the six functions of one found in standard financial tables.

This series of factors shows the annual deposit required to accumulate $1 at a given rate of interest in a given number of years.

Sinking-Fund Factor Example 6-3

An investor wants the sum of $1,000 to be available in 5 years. The interest rate is 6% compounded annually. What amount must he deposit (at the end of the year) annually to reach the goal?

Answer: Looking to the 6% Annual Table, Column 3 (Sinking-Fund Factor) for 5 years shows a factor of 0.177396. Thus, in order to accumulate $1,000 at the end of 5 years, a deposit at the end of each year in the amount of $177.40 is made.

$1,000 x 0.177396 = $177.40

(HP Keystrokes: “1000 FV”, “6 i”, “5 n”, “PMT”) display -177.40

Note: When you solve for PMT, the answer is negative indicating that it is money paid out at the end of each year.

The formula used to derive the Sinking-Fund Factor is:

SFF =

( )

1+ −1 n i

Where SFF is the Sinking-Fund Factor, i is the interest rate expressed as a decimal, and n is the number of years in the compounding period.

In the example on the previous page, the Sinking-Fund Factor can be calculated as follows: SFF =

(

1 .06

)

1 06 . 5 − + ₌1.338226 1 06 . − = 0.338226 06 . = 0.177396

Column 4: Present Worth of 1:

Present Worth of 1 Present Value of $1

Present Worth of $1 (PW $1) Reversion Factor

present value of one (1/Sn) - A compound interest factor that indicates how

much $1 (or other unit of currency) due in the future is worth today. The present value of one is one of the six functions of one found in standard financial tables; also called present worth of one.

This series of factors shows the present worth of a single amount of money to be collected after a given number of years at a given interest rate (the interest rate is also known as the “discount rate”).

Present Worth of 1 Example 6-4

What is the present value of a $1,000 received 5 years from now if the interest rate is 6% compounded annually?

Answer: Looking to the 6% Annual Table, Column 4 (Present Worth of 1) for 5 years shows a factor of 0.747258. Thus, a one-time payment of $1,000 received 5 years from now is worth $747.26 today.

$1,000 x 0.747258 = $747.26

(HP Keystrokes: “1000 FV”, “6 i”, “5 n”, “PV”) = -747.26

Note: When you solve for PV, the answer is negative indicating that it is money paid out today.

The formula used to derive the Present Worth of 1 (compounded annually) Factor is:

PW $1 =

( )

n i + 1 1

Where PW $1 is the Present Worth of 1, i is the interest rate expressed as a decimal, and n is the number of years in the compounding period.

In the above example, the Present Worth of 1 Factor can be calculated as follows:

PW $1 =

(

)

5 06 . 1 1 + ₌1.338226 1 = 0.747258

Column 5: Present Worth of 1 per Period:

Present Worth of 1 Per Period (PW $1/P) Present Value of 1 Per Period

Ordinary Level Annuity

present value of one per period (an¬) - A compound interest factor that

indicates how much $1 (or other unit of currency) paid periodically is worth today. The present value of one per period is one of the six functions of one found in standard financial tables; also called present worth of one per period or ordinary level annuity factor.

This series of factors shows the present value of the right to receive $1 deposited each year (at the end of the year) at a given rate of interest for a given number of years.

Present Worth of 1 Per Period Example 6-5

What is the present value of $1,000 received each year (at the end of the year) for 5 years if the interest rate is 6% compounded annually?

Answer: Looking to the 6% Annual Table, Column 5 (Present Worth of 1 Per Period) for 5 years shows a factor of 4.212364. Thus, an income stream of $1,000 each year (at the end of the year) for 5 years is worth $4,212.36 today.

$1,000 x 4.212364 = $4,212.36

(HP Keystrokes: “1000 CHS PMT”, “6 i”, “5 n”, “PV”) display 4212.36 The formula used to derive the Present Worth of 1 Per Period Factor is:

( )

+i n −1 1 1

Where PW $1/P is the Present Worth of 1 Per Period, i is the interest rate expressed as a decimal, and n is the number of years in the compounding period.

In the above example, the Present Worth of 1 Per Period Factor can be calculated as follows: PW $1/P =

(

)

06 . 06 . 1 1 1− + 5 = 06 . 338226 . 1 1 1− = 06 . 747258 . 0 1− PW $1/P = 06 . 252742 . 0 = 4.212364

Column 6: Partial Payment Factor

Partial Payment

Amount to Amortize $1

Installment to Amortize $1 (ITAO) Direct Reduction Loan Factors

installment to amortize one - The compound interest factor that represents the

installment needed to repay one unit of currency with interest at a specified rate for a specific number of periods; the reciprocal of the level annuity factor. Sometimes abbreviated ITAO; also called the partial payment factor or amortization factor. When expressed annually, it may be referred to as the mortgage constant, loan constant, annual constant, or mortgage capitalization rate.

This series of factors shows the amount required to amortize principal and interest on an investment or loan at a given rate of interest in a given number of years.

Partial Payment Example 6-6

An investor wants to know what to pay per year (at the end of the year) to pay off a $1,000 loan in 5 years at an interest rate of 6% compounded annually.

Answer: Looking to the 6% Annual Table, Column 6 (Partial Payment) for 5 years shows a factor of 0.237396. Thus, in order to pay off a loan of $1,000 in 5 years at 6% interest the annual payments are $237.40.

(HP Keystrokes: “1000 PV”, “6 i”, “5 n”, “PMT”) display -237.40

Note: When you solve for PMT, the answer is negative indicating that it is money paid out. The formula used to derive the Partial Payment Factor or Installment to Amortize 1 is:

ITAO =

( )

n i i + −1 1 1

Where ITAO is the Partial Payment Factor, i is the interest rate expressed as a decimal, and n is the number of years in the compounding period.

In the above example, the Partial Payment Factor can be calculated as follows:

ITAO =

(

)

5 06 . 1 1 1 06 . + − ₌1 11.338226 06 . − ₌₁₋₀_..₇₄₇₂₅₈06 ₌₀_.₂₅₂₇₄₂.06 _{= 0.237396}

Interrelationships Among the Tables – The Reciprocal

A reciprocal is defined in Merriam-Webster’s Collegiate Dictionary as:

“either of a pair of numbers (as 2/3 and 3/2 or 9 and 1/9) whose product is one”

In other words, reciprocals are numbers divided into 1. For example, the reciprocal of 10 is 1/10 (10 x 1/10 = 1). A look at the compounded interest tables shows us that the functions of $1 are reciprocals.

Using the 6% Annual Table for 5 Periods results in the following factors:

FW of $1 1.338226

FW of $1 Per Period 5.637093

SFF 0.177396

PW of $1 0.747258

PW of $1 Per Period 4.212364 Partial Payment Factor 0.237396

If you have been paying close attention, you may have noticed that many of the numbers appear over and over again. This is because some of the factors are reciprocals of the others.

The proof is in the math presented below:

The PW of $1 is the reciprocal of the FW of $1 (0.747258 x 1.338226 = 1.0).

The Partial Payment Factor is the reciprocal of the PW of $1 Per Period (0.237396 x 4.212364 = 1.0).

The SFF is the reciprocal of FW of $1 Per Period (0.177396 x 5.637093 = 1.0).

Another important relationship is between the sinking fund factor and the partial payment factor. The partial payment factor considers both a return on and return of the investment. Adding the interest rate or return on of 0.0600 to the sinking fund factor or return of 0.177396 results in the partial payment factor 0.237396. This relationship will be revisited in the recapture provision methodologies detailed later in the course.

Important Notes:

All of the previous discussions and examples were based on the following two assumptions:

Payments were made annually.

Payments were made at the end of the year.

If the payments are made at the beginning of the period, the factors identified in the tables cannot be used without modification.

If you have a HP-12C calculator, the window shows nothing when the payments are made at the end of the period. To calculate the factors for payments made at the beginning of the period, touch “g” then “BEG” in blue (the number 7 key). To get it back to the end of the period, touch “g” and “END” in blue (the number 8 key).

For the purpose of this course, always assume that the payments are made at the end of the period.

Compounding Periods

So far we have looked at annual compounding (or when the interest is calculated only once a year). This is not the only compounding method available. The following table summarizes some of the compounding options and the periods per year:

Compounding Periods Per Year Annual 1 Semi-Annual 2 Quarterly 4 Monthly 12 Daily 365

Let’s take on a problem to show the differences in investment growth amounts depending on the type of compounding. We have $100 to put into a savings account and five different savings accounts (A–E) choices. Each pays a nominal rate of 6% interest. However, the way they compound the interest differs.

Savings Account Deposit Compounding Nominal Interest Rate

A $100.00 Annual 6.00%

B $100.00 Semi-Annual 6.00%

C $100.00 Quarterly 6.00%

D $100.00 Monthly 6.00%

E $100.00 Daily 6.00%

nominal interest rate (I) - A stated or contract rate; an interest rate, usually

annual, that does not necessarily correspond to the effective or compound interest rate.

In which account should we invest our $100?

Savings

Account Deposit Compounding

Nominal Interest Rate Periods Per Year Interest Rate Per Period FV Factor 1 Year Future Value A $100.00 Annual 6.00% 1 6.0000% 1.0600 $106.00 B $100.00 Semi-Annual 6.00% 2 3.0000% 1.0609 $106.09 C $100.00 Quarterly 6.00% 4 1.5000% 1.0614 $106.14 D $100.00 Monthly 6.00% 12 0.5000% 1.0617 $106.17 E $100.00 Daily 6.00% 365 0.0164% 1.0618 $106.18

Be aware there are also Monthly Tables, Quarterly Tables, and Semi-Annual Tables. Make sure you are looking at the correct table!

Chapter 6 Quiz: Compound Interest Tables

The 11% annual interest rate table is provided on the page following the quiz questions. 1.) An investor purchased a parcel of land 10 years ago for $25,000. Ignoring holding

costs, how much must the investor sell the subject property for to have earned 11% on the investment over the 10-year period?