# Lesson 10 – Rates and Factors Revisited (The Income Approach to Value)

## Appraisal Training: Self-Paced Online Learning Session

Open All Close AllThe investor decides to purchase property based on their anticipated income, expenses, and recapture. Appraisers use this direct relationship to derive multipliers and rates, and value property using multipliers and rates, and using factors that include rate information. This lesson serves as a summary of previous chapters and discusses the following:

- What is a "Rate"
- What is the difference between rates of return, recapture and recovery rates, and yield rates, and capitalization rates
- What are some of the other rates and capitalization methods used by appraisers – why are different methods used, and which ones are used by property tax appraisers

### Rates

A rate, in appraising, expresses the relationship between income and value. Using an example from finance, consider a bank deposit of 100 dollars ($100 is the value) that earns ten percent interest (calculated annually) per year – the bank pays the depositor ten dollars of interest at the end of the year ($10 is the income). Ten percent (10%) is the rate.

In Lesson 9, we discussed the derivation and use of multipliers – multipliers that could be based on monthly rent or annual incomes – both gross incomes and effective gross incomes. In the following chapters we will look at rates. In real estate appraising, rates, like multipliers, provide us with a ratio between income and value. Although they are applied to different income levels, and therefore cannot be used interchangeably, they can be thought of as reciprocals of each other. That is, we derived multipliers by dividing a sales price – the dividend or numerator – by the income anticipated by the buyer – the income was the divisor or denominator; the result – the quotient – was the multiplier indicated by the sale.

**M**=

**SP**

**Ant I**

Where "**M**" is the **M**ultiplier, "**SP**" is the **S**ales **P**rice of the property, and "**Ant I**" is the **I**ncome **A**nticipated by the buyer.

A rate can be thought of as the reciprocal of a multiplier. That is, where "**R**" is a **R**ate:

**R**=

**1**

**M**

**1**

**SP ÷ Ant I**

**1**

**SP**

**Ant I**

**Ant I**

**SP**

The student is cautioned that **this is a theoretical conclusion**; this relationship between multipliers and rates is **not used** in appraisal practice, because the anticipated income that the appraiser uses in deriving a multiplier from the sales price of a closely comparable property is **not** the same anticipated income that the appraiser uses in deriving a rate from the sales prices of a recently sold comparable property. In Lesson 9 we derived multipliers using gross income, gross rents, and effective gross income.

There is nothing magical about a rate – it's just a number that expresses the relationship between two (or more) other items. As a simple example, consider depositing $100,000 in a financial institution (a bank, say) for a year; at the end of the year your $100,000 will be available to you, along with $3,000 the bank will pay your for the use of your money for a year – they are, in effect, paying you a rent of $3,000 for the use of your money for a year. The interest rate – the relationship between the $100,000 value of your deposit and the $3,000 income the deposit earns – is three percent. R = I ÷ V = $3,000 ÷ $100,000 = 3%.

The incomes that the appraiser uses in deriving rates have usually been processed by deducting expenses and other items. For instance, in Lessons 11 and 12 we will work with OverAll Rates [OAR]; the income used in deriving an OverAll Rate is the anticipated Net Operating Income – the difference between gross return (gross income) and gross outgo (including vacancy and collection losses and current expenses, capital expenses [or annualized allowances for their expenditure], and other outlay required to develop and maintain the estimated gross income stream).

#### Various Rates

We previously mentioned and discussed multipliers, rates and factors; Lesson 9 developed and utilized multipliers, and starting in Lesson 11 we will develop and use rates. Before we continue, it might be helpful to define some terms – ones we have used, and some we will be using.

**Amortization Factor**: The Periodic Repayment factor [PR]; "Column 6" in the Assessors' Handbook section AH 505 – the equal periodic payment that has a present value of $1, for a specified number of periods and at a specified discount rate. It is also known as the Partial Payment factor.

**Annuity**:

- The right to receive money, or its equivalent, usually in fixed equal amounts at regular intervals, and usually for a definite term.
- A level, increasing, or decreasing, stream of scheduled and predictable income or payment amounts. If the amounts are increasing or decreasing, it is usually at a constant rate.

**Annuity Factor**: The Present Worth of 1 per Period [PW1/P]; "Column 5" in the Assessors' Handbook section AH 505. The present value of a series of amounts of one (1), receivable at the end of each period, for a specified number of periods and at a specified growth or interest rate. It is also known as the "Inwood coefficient".

**Building Capitalization Rate**: We will discuss building capitalization rates in Lesson 16, when we introduce the residual techniques. A building capitalization rate is the sum of the recapture and return rates on an income-producing property; the rate applies only to the improved portion of a property.

**Capitalization Rate**: Capitalization was discussed in Lesson 8. A capitalization rate is any rate used to convert an estimate of future income to an estimate of market value. Commonly called the "cap rate", the capitalization rate is a divisor used to determine value. It reflects the ratio of income (frequently net operating income) to market value. The net income from an investment divided by the cap rate will equal the value. When appraising for *ad valorem* property taxation purposes, the cap rate is a combination of the effective tax rate [ETR], a return (recapture or recovery) **of** the investment [CRR] and a return **on** the investment [Y].

**Cash-Flow Rate [R _{e}]**: The ratio of annual cash flow to the investment; a rate that reflects the relationship between the equity dividend (the annual pre-tax* cash flow to equity) and the equity investment (such as the down payment). That is, a single year's Net Income Before Recapture [NIBR], less the Annual Debt Service [ADS], with the difference divided by the equity investment. The Cash-Flow Rate is used to develop an OverAll Rate [OAR] by the band-of-investment method, and sometimes used to convert the equity dividend into an indicator of equity value. It is also known as the "cash-on-cash rate", the "equity capitalization rate", and the "equity dividend rate".

**Debt Capitalization Rate**: The debt component of an overall direct capitalization rate; it can be computed by dividing annual interest payments by the market value of debt.

**Debt Rate**: The interest rate on borrowed money.

**Discount Rate**: One of the components of a capitalization rate, the discount rate, in appraisal terminology, is the rate of return **on** investment; the rate an investor requires to discount future income to its present worth. The discount rate consists of an interest rate and an equity yield rate. Theoretical factors considered in setting a discount rate are the safe rate earned from a completely riskless investment (this rate may reflect anticipated loss of purchasing power due to inflation) and compensation for risk, lack of liquidity, and investment management expenses. The property tax appraiser develops a discount rate by sales comparison analysis or by band-of-investment analysis. Sometimes referred to as a "Rate of Return". For comparison, see Recapture Rate.

**Effective Tax Rate**:

- The tax rate expressed as a percentage of market value. Beginning in 1981 California law [Government Code § 16101.5 and R&T Code § 135] prescribed the assessed value would be 100 percent of the full value; prior to that time the effective tax rate would have been different from the nominal tax rate, as the assessment ratio was not equal to one (1).
- The relationship between dollars of tax and dollars of market value of a property. The rate may be calculated either by dividing the property tax by value, or by multiplying a property's assessment level by its nominal tax rate.

**Equity Capitalization Rate**: Equity component of an overall direct capitalization rate. Computed by dividing equity earnings by the market value of equity.

**Equity Dividend [d _{e}]**: The annual pre-tax cash flow to equity — a single year's Net Income Before deducting for Recapture [NIBR], less the Annual Debt Service [ADS]. It is also known as "cash-on-cash."

**Equity Yield Rate**:

- The required rate of return
**on**equity capital. A component of the capitalization rate (or discount rate or mortgage-equity overall rate) that must be separately specified in band-of-investment analysis (and in mortgage-equity analysis). - The equity yield rate reflects all yield to equity, whether from annual cash flows or future reversions; it considers the effect of debt financing on the cash flow of the equity investor.

**Interest, or Interest Rate**: The premium paid for the use of money; a (rate of) return on capital. The interest rate usually incorporates a risk factor, an illiquidity factor, a time-preference factor, an inflation factor, and potentially, other factors. See also discount rate.

**Internal Rate of Return**:

- The annualized yield rate on capital that is (or could be) generated by an asset or within a group of capital assets over a period of ownership.
- The rate that discounts all future cash flows to a net present worth equal to the original investment. The internal rate of return is calculated, usually by trial and error, from a knowledge of the relevant cash flows.

**Inwood Coefficient**: A factor (the Present Worth of 1 per Period [PW1/P]) used to estimate the present value of a level stream of income. It is also known as the Annuity factor.

**Land Capitalization Rate**:

We will discuss land capitalization rates, along with building capitalization rates, in Lesson 16, when we introduce the residual techniques. A land capitalization rate provides for a return on an investment in land; it does not include any provision for recapture of the investment, as the appraisal models we are discussing in these online learning sessions assume land maintains its value, and income-producing ability, into perpetuity, that is, land is a non-wasting asset.

**Management rate:**: A rate that reflects the costs of managing the property. This management rate is added to the risk rate, the safe rate, and the non-liquidity rate, to yield the return rate in the summation method.

**Market Rate of Return**: The typical return on an investment in a given type of property in a given market. (Not necessarily the actual rate of return indicated by a property's actual income.)

**Mortgage Constant [ R _{M} | ƒ ]**: "Column 7" of the Monthly Compound Interest Tables in the Assessors' Handbook section AH 505. Annual debt service, including any amortization and interest, expressed as a percentage of the original (principal) amount of the loan. It is also known as the "loan constant" and the "mortgage capitalization rate." It is also known as the "Constant Annual Percent".

**Nominal Tax Rate**: The stated tax rate, which does not necessarily correspond to the effective tax rate. In 1981 California law made the effective tax rate the same as the nominal tax rate when the 100 percent assessment ratio was legislated. For comparison, see Effective Tax Rate.

**Non-liquidity Rate**: In building a rate of return by the summation approach, an appraiser must add in a factor or rate for non-liquidity. This rate compensates the investor for his or her inability to make immediate use of the investment amount. Generally, the longer the holding period for the investment, the higher the non-liquidity rate will be.

**OverAll Rate [OAR]**: A rate that blends all requirements of discount and recapture for both land and improvements; it is used to convert annual net operating income into an indicated overall property value. When used in property tax appraising the effective tax rate is added to the overall rate, to develop the capitalization rate used by the appraiser.

**Recapture:** A portion of the overall capitalization rate in an income approach representing the return of an owner's investment in property; it is usually expressed as the current year's percentage of the remaining economic life.

**Recapture Rate:** The return of an investment; the annual amount that can be recaptured from an investment divided by the original investment; primarily used in reference to wasting assets (improvements).

**Required rate of Return on Equity:** A component of the discount rate, as it is understood from the point of view of band-of-investment analysis, and a component of the overall rate developed according to mortgage-equity analysis.

**Reversion Factor**: : The Present Worth of 1 [PW1]; "Column 4" in the Assessors' Handbook section AH 505 – the present value of one (1) receivable at the end of a specified number of periods and discounted at a specified rate. Looking at it alternatively, it is the lump sum amount that would have to be set aside to accumulate to one (1), with periodic compounding at the end of a specified number of periods and at a specified rate of growth or interest.

**Sinking Fund Factor [SFF]**: The Sinking Fund Factor indicates the amount of periodic deposit required which will grow at compound interest to a specified future amount of one (1) at the end of a specified number of periods; deposits are made at the end of the period. It is also known as the "amortization rate–sinking fund" and as "a periodic deposit that will grow to one at a future date"; in conjunction with annuity capitalization it is known as the "capital recovery rate".

**Straight-Line Recapture**: The recovery of capital invested in a wasting asset in equal periodic amounts over the remaining economic life of the asset.

**Tax Rate**:

- The amount of tax stated in terms of a unit of the tax base, for example, 2 percent, $250 per parcel, and 2 cents per gallon.
- For ad valorem property taxes, the percentage of assessed value at which each property is taxed in a given district, local government jurisdiction, or tax area. Distinguish between effective tax rate and nominal tax rate.

**Yield Rate**:

- The return on investment applicable to a series of incomes that results in the present worth of each. Examples of yield rates are the interest rate, discount rate, equity yield rate, and internal rate of return.
- The required rate of return on equity capital; a component of the capitalization rate (or discount rate or mortgage-equity overall rate) that must be separately specified in band-of investment analysis and mortgage equity analysis.

### Differences between Rates and Factors

#### Factors

When we use the term, "factor", we are usually referring to a number that was looked up in a Compound Interest Table, such as the AH 505, *Capitalization Formulas and Tables*. You have already used these tables, first when you took the Self-Paced Online Learning Session, "*Time Value of Money - Six Functions of a Dollar* (Use of AH 505)", and then when you worked the exercises in Lesson 4. Towards the end of Lesson 4 was a discussion, "Relationships between the Functions.' The functions in the AH 505 are all related, 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 equal what one will grow to at the end of those compounding periods**.

The discussion noted:

- The
**F**uture**W**orth of**One**and the**P**resent**W**orth of**One**are reciprocals of each other. - The
**F**uture**W**orth of**One****per P**eriod and the**S**inking**F**und**F**actor are reciprocals of each other. - The
**P**resent**W**orth of**One****per****P**eriod and the**P**eriodic**R**epayment factor are reciprocals of each other. - The difference between the
**S**inking**F**und**F**actor and the**P**eriodic**R**epayment factor is the (effective)nterest rate.*i*

The appraiser, like the investor, is primarily interested in today's worth of future payments. Let's review the six compound interest and annuity factors previously discussed in Lesson 4

- The
shows the amount to which a single initial deposit of one will grow with compound**F**uture**W**orth of**One**[**FW1**]nterest at a specified rate for a specified*i***n**umber of periods.**FW1**, @interest rate, for*i***n**periods, ≡**(1+**.*i*)^{n} - The
shows the amount to which a series of deposits of one per period will grow with compound**F**uture**Worth of****One****per****P**eriod [**FW1/P**]nterest at a specified rate for a specified*i***n**umber of periods. It is the sum of all the**FW1**factors for the specified**n**umber of periods.**FW1/P**≡ (**FW1**-1) ÷. It is also the reciprocal of the*i***SFF**, which will be discussed next; that is,**FW1/P**≡ 1 ÷**SFF**. - The
is the level periodic payment or investment required to accumulate an amount of "one" in a given**S**inking**F**und**F**actor [**SFF**]**n**umber of periods, including the accumulation ofnterest. It is the reciprocal of the*i***FW1/P**factor.**SFF**≡ 1 ÷**FW1/P**≡÷ (*i***FW1**-1) - The
shows how much an amount of one due in the future is worth today. It is the reciprocal of the FW1 factor.**P**resent**W**orth of**One**[**PW1**]**PW1**≡ 1 ÷**FW1**. - The
shows how much an amount of one per period due periodically in the future is worth today. It is the sum of all the**P**resent**W**orth of**One****per****P**eriod [**PW1/P**]**PW1**factors for the specified**n**umber of periods.**PW1/P**≡ (1 –**PW1**) ÷. It is also the reciprocal of the PR factor, which will be discussed next; that is,*i***PW1/P**≡ 1 ÷**PR**. - The
is the direct reduction of loan factor for a loan given the**P**eriodic**R**epayment [**PR**]nterest rate and amortization term. It is the reciprocal of the*i***PW1/P**factor.**PR**≡ 1 ÷**PW1/P**≡÷ (1 –*i***PW1**) - The difference between the
and the**S**inking**F**und**F**actor [**SFF**]factor is the (effective)**P**eriodic**R**epayment [**PR**]nterest rate; that is, the**i****SFF**, which shows the amount of**principal**to deposit,*plus*thenterest rate, will equal the**i****PR**, which shows the amount of**principal**to pay.*and*interest**SFF**+=*i***PR**.

Let's see why these relationships hold true. Given an interest of 10 percent, annual compounding, let's first construct a Future Worth of 1 [FW1] table for five years. ("BoY" is the amount deposited, or on hand, at the beginning of the year; "EoY" is the amount deposited at the end of the year.)

[**F**uture**W**orth of**1****FW1**] FW1, 10%, Ann, 5yrs

If we deposit one dollar ($1.00) at the start of year one, it will earn 10 cents (10¢) that first year, and there will be $1.10 at the end of the first year, which is the beginning of the second year. That $1.10 will earn 11¢ during the second year, growing to $1.21. This continues until the end of the fifth year, when, with the accumulated compound interest, there will be $**1.61** on deposit. This corresponds with the FW1 factor, at 10 percent, annual compounding, for five years, which is **1.61**0510.

FW1 | Yr | BoY | interest |
EoY |
---|---|---|---|---|

≡ (1 + i)^{n} |
1 | $1.00 | + $0.10 | = $1.10 |

= (1 + 10%)^{5 yrs} |
2 | $1.10 | + $0.11 | = $1.21 |

= (1 + 0.10)^{5} |
3 | $1.21 | + $0.12 | = $1.33 |

= (1.10)^{5} |
4 | $1.33 | + $0.13 | = $1.46 |

= 1.61051 |
5 | $1.46 | + $0.15 | = $1.61 |

**S**inking**F**und**F**actor**[SFF]**SFF, 10%, Ann, 5yrs

The Sinking Fund Factor [SFF] shows what periodic equal amount you need to deposit to grow to "one". The SFF can be calculated from the FW1, or it can be looked up in compound interest tables – the SFF below was calculated, from the FW1 of **1.61**0510 above, to be **0.16**3797, and the same number will be found in the tables, at 10 percent, annual compounding, for five years. If we deposit 16 cents per year for five years, and that deposit will earn 10 percent, at the end of five years there will be one dollar on deposit.

SFF | Yr | BoY | interest |
deposit | EoY |
---|---|---|---|---|---|

≡ i ÷ (FW1 − 1) |
1 | $0.00 | + $0.00 | + $0.16 |
= $0.16 |

= 10% ÷ (1.610510 − 1) |
2 | $0.16 | + $0.02 | + $0.16 |
= $0.34 |

= 0.10 ÷ (0.610510) |
3 | $0.34 | + $0.03 | + $0.16 |
= $0.54 |

= 0.163797 |
4 | $0.54 | + $0.05 | + $0.16 |
= $0.76 |

5 | $0.76 | + $0.08 | + $0.16 |
= $1.00 |

#### EXAMPLE 10-1: Using the Sinking Fund Factor

The owner of a 15 year old duplex knows that the roof will need to be replaced in five years; a new roof is estimated to cost $5,000, material and labor. The owner would like to set aside money at the end of each year until then, so there will be enough money on deposit to replace the roof. The owner’s credit union is currently paying 10 percent interest on periodic deposits such as the ones envisioned.

How much does the owner have to deposit at the end of each year?

- The Sinking Fund Factor at 10 percent, annual compounding, for five years, is 0.163797.
- $5,000 × 0.163797 (SFF, 10%, Ann, 5yrs) = $819.

The owner needs to deposit $819 annually, and that, with the interest paid by the credit union, will grow to $5,000 in five years.

**P**resent**W**orth of**O**ne**[PW1]**PW1, 10%, Ann, 5yrs

The Present Worth of 1 [PW1] is related to the FW1; the latter tells you what an amount of one will grow to, and the PW1 tells you what single deposit today will grow to one, given periodic compound *i*nterest for **n** periods. The PW1 is the reciprocal of the FW1, and therefore, using the **1.61**0510 calculated above, 10 percent, annual compounding, for five years, the PW1 is 0.620921. The next table shows how a single deposit of 62 cents today, earning 10 percent interest, will grow to one dollar in five years.

PW1 | Yr | BoY | interest |
FoY |
---|---|---|---|---|

≡ 1 ÷ FW1 |
1 | $0.62 |
+ $0.06 | = $0.68 |

= 1 ÷ 1.610510 |
2 | $0.68 | + $0.07 | = $0.75 |

= 0.620921 |
3 | $0.75 | + $0.08 | = $0.83 |

4 | $0.83 | + $0.08 | = $0.91 | |

5 | $0.91 | + $0.09 | = $1.00 |

#### EXAMPLE 10-2: Using the Present Worth of One

An investor has the opportunity to purchase a bond that pays no interest or dividend, but is guaranteed to pay $5,000 to the bearer in five years.

Given a 10% interest rate (annual compounding), what is the bond worth today

- The Present Worth of One, at 10 percent, annual compounding, for five years, is 0.620921.
- $5,000 × 0.620921 (PW1, Ann, 10%, 5yrs) = $3,104.61

If the investor pays $3,104.61 for the bond, the rate of return is 10 percent.

**P**resent**W**orth of**O**ne**p**er**P**eriod**[PW1/P]**PW1/P, 10%, Ann, 5yrs

Frequently real estate transactions involve periodic payments or deposits, not a single payment; for example, loans may be paid off monthly, and rent may be paid monthly. The Present Worth of 1 per Period [PW1/P] is the sum of the PW1 factors. For instance, we could use the 10 percent compound interest tables to look of the PW1 factors for five years:

Yr | PW1 (from tables) |
Sum of current plus previous year(s) PW1s | Calculated PW1/P |
---|---|---|---|

1 | 0.909091 | 0.909091 | ≡ PW1/P, 10%, Ann, 1 yr |

2 | 0.826446 | 1.735537 | ≡ PW1/P, 10%, Ann, 2 yrs |

3 | 0.751315 | 2.486852 | ≡ PW1/P, 10%, Ann, 3 yrs |

4 | 0.683013 | 3.169865 | ≡ PW1/P, 10%, Ann, 4 yrs |

5 | 0.620921 | 3.790787 | ≡ PW1/P, 10%, Ann, 5 yrs |

The PW1/P factor is sometimes called an annuity factor, since it represents a level series of periodic payment. We can, of course, also calculate PW1/P using the formula given earlier,

**PW1**)

*i*

**0.620921**

*10%*

*10*

**3.79**0787

You can see, in the table below, that if you have $**3.79** deposited today, and it pays 10 percent interest annually, you can draw out $1.00 each year, and at the end of five years you will have drawn out $5.00, and you will have nothing left in the deposit account.

PW1/P | Yr | BoY | interest |
payment | FoY |
---|---|---|---|---|---|

≡ (1 − PW1) ÷ i |
1 | $3.79 |
$0.38 | ($1.00) | $3.17 |

= (1 − 0.620921) ÷ 10% |
2 | $3.17 | $0.32 | ($1.00) | $2.49 |

= (0.379079) ÷ 0.10 |
3 | $2.49 | $0.25 | ($1.00) | $1.74 |

= 3.790787 |
4 | $1.74 | $0.17 | ($1.00) | $0.91 |

5 | $0.91 | $0.09 | ($1.00) | $.00 | |

sum of payments: ($5.00) |

#### EXAMPLE 10-3: Using the Present Worth of One per Period

A retiree wants to purchase a five-year annuity from an insurance company that will pay $5,000 at the end of each year, for the next five years. The insurance company is willing to sell annuities of this sort based on a 10 percent rate of return.

What is the present worth of this annuity; that is, given the 10 percent rate, what should the retiree pay of this annuity … what is the insurance company's price for this annuity?

- The Present Worth of One per Period, at 10 percent, annual compounding, for five years, is 3.790787.
- $5,000 × 3.790787 (PW1/P, Ann, 10%, 5yrs) = $18,953.94.

The fair price for the annuity, given the interest rate and term, is $18,953.94.

**P**eriodic**R**epayment**[PR]**PR, 10%, Ann, 5yrs

The Periodic Repayment is the reciprocal of the PW1/P. For instance, the PW1/P is the present value of a series of one dollar ($1.00) payments, as was illustrated above, whereas the PR is the series of equal periodic payments that will pay off a loan on one dollar made today. This is illustrated in the next table.

PR | Yr | BoY | interest |
payment | FoY |
---|---|---|---|---|---|

≡ i ÷ (1 − PW1) |
1 | $1.00 | $0.10 | ($0.26) | $0.84 |

= 10% ÷ (1 − 0.620921) |
2 | $0.84 | $0.08 | ($0.26) | $0.66 |

= 0.10 ÷ (0.379079) |
3 | $0.66 | $0.07 | ($0.26) | $0.46 |

= 0.263797 | 4 | $0.46 | $0.05 | ($0.26) | $0.24 |

5 | $0.24 | $0.02 | ($0.26) | $0.00 |

#### EXAMPLE 10-4: Calculating the Loan Payments on a Fully-Amortizing Loan

A newly married couple wants to buy a home, which is selling for $250,000. They don't want to be saddled with debt for 30 years, and plan to pay off the 10 percent loan in five years, making annual payments at the end of each year. They will make a 20 percent down payment. What will their payments be?

- $250,000 less 20 percent down = $250,000 less $50,000 down = $200,000 loan.
- $200,000 × 0.263797 (PR, 10%, Ann, 5yrs) = $52,759.40 payments each year.

Let's see if this is true.

BoY balance | 10% interest | EoY payment | FoY balance | |
---|---|---|---|---|

purchase price: | $250,000 | |||

down payment: | ($ 50,000) | |||

1st year | $200,000 | $20,000 | ($52,759.40) | $167,241 |

2nd year | $167,241 | $16,724 | ($52,759.40) | $131,205 |

3rd year | $131,205 | $13,121 | ($52,759.40) | $ 91,566 |

4th year | $ 91,566 | $ 9,157 | ($52,759.40) | $ 47,964 |

5th (last) year | $ 47,964 | $ 4,796 | ($52,759.40) | $ 0.59 |

Yes, the loan will be paid off in five years. (The 59 cents showed owing at the end of the fifth year is due to rounding errors. If we used a PR factor carried to seven decimal places the answer would be off by two cents, and with eight places, zero.)

The PR is the number that shows how much it costs to periodically pay off a loan of one dollar; from the lender's point of view, it's the amount of principal and interest that they receive from the borrower that has a present value of one. The SFF factor, remember, is the number that shows how much you have to put aside periodically to grow to an amount of one; the depositor pays the principal, and the bank pays the interest. With both factors we're taking about a series of deposits or payment made to the financial institution; the difference: who pays the interest. The borrower or depositor pays the principal in either case. The borrow pays the interest on the loan; the bank pays the interest to the sinking fund.

Let's check this, where the interest rate is still 10 percent, and the term is still five years:

**–**0.163797

**=**0.100000

The 10 percent interest rate is the rate of return ** on** investment. The other component of the PR factor is the return

**the investment, the**

*of**recapture*

**the investment.**

*of*In Lesson 6 we discussed five income streams; two of the streams described a terminating series of periodic payments – you might want to refer back to the graphs in Lesson 6.

- The straight-line declining terminal income stream described a series of payments that declined, at constant rate, over the term of the income stream,
, the recapture was at a constant, level, amount over the life of the income.*and* - The level terminal income stream described a series of equal payment that remained constant (level) over the term of the income stream; the recapture amount increased over the term of life of the income.

#### Rates of Recapture

Earlier we discussed recapture – we recognized that an owner will want more than just a return on investment – a return of the investment is also required. There may be a return of investment because the property's value does not change, and the investor can sell it for the original purchase price. For instance, if you deposit $100,000 in the bank, the bank will pay you interest (the return *on* investment), and you expect that the $100,000 will still be there when you want to withdraw it (the return *of* the investment).

There may be an actual increase of value – investors often buy real estate anticipating appreciation in the property's value.

The return of an investment, whether calculated as a dollar amount, or as a percentage when used in a capitalization formula, is dependent upon the shape of the income stream. In either case, the amount to be recovered is the amount that was invested and is wasting away – it's the investment in the wasting assets (improvements).

**Straight-Line Declining Terminal Income Stream:** The amount to be recaptured is the investment in the wasting asset; the annual amount to be recaptured can be found taking the total amount to recapture over the term of the investment, and dividing that amount by the remaining life of the income stream – the (remaining) economic life of the improvement or wasting asset. For instance, given a Building Value [BV] of $100,000, and a **Remaining Economic Life [REL]** of five years:

The rate of recapture is called the Capital Recovery Rate [CRR]. For straight line declining terminal income streams, the CRR is the reciprocal of the REL.

In addition to dividing the amount to be recaptured by the REL, the annual recapture amount can also be found by multiplying the amount to be recaptured by the CRR. Therefore, given the same value and life used above,

Annual Recapture Amount = BV × CRR = $100,000 × 20% = $20,000 per year.

**Level Terminal Income Stream:** The amount to be recaptured is the investment in the wasting asset; the annual amount to be recaptured, for the first year can be found taking the total amount to recaptured over the term of the investment, and multiplying that amount by the SFF for the (remaining) economic life of the improvement or wasting asset. For instance, given a BV of $100,000, and a REL of five years, assuming a 10 percent rate of return:

#### Capitalization Rates

In Lesson 8 we discussed capitalization rates. We noted that Income × Rate = Value. Capitalization rates include provision for (1) return ** on** investment, (2) recapture

**investment in wasting assets, (3) property taxes, where the appraiser is doing an appraisal for**

*of**ad valorem*property tax purposes.

The return ** on** investment may be provided for with a Yield rate. The recapture

**investment is provided by using a Capital Recovery Rate or a Sinking Fund Factor.. In some instance, an OverAll Rate [[OAR] is used, that provides for both return**

*of***and recapture**

*on***investment.**

*of*The tax component of a capitalization rate is provided by using the Effective Tax Rate [ETR] for the property.

### The Relationship between Rates and Value

Where the risk is greater, investors will seek a higher rate of return. U S Savings Bonds may be paying ¼-to-½ percent at the same time that savings accounts are paying 1 percent. Lenders are getting 3-to-4 percent on vehicle loans and real estate mortgages, but will be requiring 12 percent on computer loans and 15 percent or more on unsecured personal loans.

Where the time to recover an investment in a wasting asset is short, the investor will seek a greater rate of recapture. For instance, where there is an investment of $100,000 in a building with a remaining life of five years, the investor might seek a return that includes investment recapture of 20 percent a year, in order to have earned enough additional income to recover the $100,000 investment in five years — $20,000 per year. However, if the building had a remaining live of 25 years, a lower rate of recapture would be appropriate – 4 percent a year — $4,000 per year – would recover the $100,000 investment in 25 years.

Where the risk is greater, the investment in a wasting asset is larger, and/or the wasting assets have a shorter remaining economic life, the investor will require a higher rate a return, and the appraiser will use a higher capitalization rate to value the property.

### Summary

The lesson you just read summarized what you have learned about rates and factors in past lessons of this self-paced online learning session. It introduced a glossary of terms that were in previous lessons, and terms that will appear in future lessons. This lesson also discussed the relationship between rates and factors – a continuation of the discussion from Lesson 8 – and delved into the important relationships between the functions in compound interest and annuity tables, such as those found in the Assessors' Handbook section *AH 505, Capitalization Formulas and Tables*, (06-93). In the next lesson, Lesson 11, you will learn how to derive an OverAll Rate followed by Lesson 12, which discusses valuation using overall rates.

Note: Before proceeding on to the next lesson, be sure to complete the exercises for this lesson.