Yield Capitalization — Theory and Basic Applications

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The Student Handbook toTHE APPRAISAL OF REAL ESTATE

Chapter 23

Yield Capitalization — Theory and Basic Applications

Chapter 23 2Student Handbook to THE APPRAISAL OF REAL ESTATE

Modern computers allow appraisers to easily discount future cash flows to present value. In the past this process was difficult.

Any investment can be considered the present worth of future benefits.

A dollar today will grow to $X over a period of time, and $X to be received in the future is worth a dollar today if discounted at the same rate.


Discounting is the process in which future cash flows are converted into present value.

Discounting requires many calculations that are tedious for any human, but easy for many modern computers.

The sum of the present values of the future cash flows is the present value of the investment. This includes periodic cash flows and the reversion.


The time value of money is represented in the “six functions of a dollar” tables. These tables show how the value of a dollar can

be impacted with compound interest. They reflect the impact on the dollar from

interest on interest. There are six tables to follow. Each one shows a

different impact of compound interest.


What will $1 today grow to with compound interest?

1 + interest x Previous Balance = New BalanceEnd of year 1 1.10 x $1.0000 = $1.10

End of year 2 1.10 x $1.1000 = $1.21 End of year 3 1.10 x $1.2100 = $1.33 End of year 4 1.10 x $1.3310 = $1.46 End of year 5 1.10 x $1.4641 = $1.61 End of year 6 1.10 x $1.6105 = $1.77 End of year 7 1.10 x $1.7716 = $1.95 End of year 8 1.10 x $1.9487 = $2.14 End of year 9 1.10 x $2.1436 = $2.36 End of year 10 1.10 x $2.3579 = $2.59

Amount of original deposit = $1.00Interest rate = 10%


Table 23.2 Present Value of $1

What is $1 in the future worth today?Year 10% Present worth

1 $1.00 / 110.00% = $0.9091

2 $0.91 / 110.00% = $0.8273

3 $0.83 / 110.00% = $0.7545

4 $0.75 / 110.00% = $0.6818

5 $0.68 / 110.00% = $0.6182

6 $0.62 / 110.00% = $0.5636

7 $0.56 / 110.00% = $0.5091

8 $0.51 / 110.00% = $0.4636

9 $0.47 / 110.00% = $0.4273

10 $0.42 / 110.00% = $0.3818

$1 that is paid in one year is only worth $0.9091 today

$1 that is paid in three years is only worth $0.754545 today


Table 23.3 Discounted Cash Flow—Level Annuity

What is $1 per period worth today ($100,000 per period and $1,000,000 at the end @10%)?

10%Year Cash flow Discount. rate Present Value

1 100,000 0.9091 90,9092 100,000 0.8264 82,6453 100,000 0.7513 75,1314 100,000 0.6830 68,3015 100,000 0.6209 62,0926 100,000 0.5645 56,4477 100,000 0.5132 51,3168 100,000 0.4665 46,6519 100,000 0.4241 42,41010 1,100,000 0.3855 424,098

Total 1,000,000


Table 23.4 Investment Analysis of Drugstore

Year Cash flow x Discount Factor= Present Value

1 $190,674 x 0.92593 = 176,550$

2 $190,674 x 0.85734 = 163,472$

3 $190,674 x 0.79383 = 151,363$

4 $190,674 x 0.73503 = 140,151$

5 $190,674 x 0.68058 = 129,770$

6 $190,674 x 0.63017 = 120,157$

7 $190,674 x 0.58349 = 111,256$

8 $190,674 x 0.54027 = 103,015$

9 $190,674 x 0.50025 = 95,384$

10 $190,674 x 0.46319 = 88,319$

1,279,438$

10 0.26974 = 555,672$

1,835,110$ Total

$2,060,000

Discount rate for the periodic cash flows = 8%

Present value of the periodic cash flows only

Discount rate for the reversion @ 14%


Table 23.5 Straight-Line Change

$ - 1 $1.00 x 0.9174 = 0.91743$ 2 $1.05 x 0.8417 = 0.88376$ 3 $1.10 x 0.7722 = 0.84940$ 4 $1.15 x 0.7084 = 0.81469$ 5 $1.20 x 0.6499 = 0.77992$ 6 $1.25 x 0.5963 = 0.74533$ 7 $1.30 x 0.5470 = 0.71114$ 8 $1.35 x 0.5019 = 0.67752$ 9 $1.40 x 0.4604 = 0.64460$ 10 $1.45 x 0.4224 = 0.61250$ 11 $1.50 x 0.3875 = 0.58130$ 12 $1.55 x 0.3555 = 0.55108$ 13 $1.60 x 0.3262 = 0.52189$ 14 $1.65 x 0.2992 = 0.49376$ 15 $1.70 x 0.2745 = 0.46671$ 16 $1.75 x 0.2519 = 0.44077$ 17 $1.80 x 0.2311 = 0.41593$ 18 $1.85 x 0.2120 = 0.39219$ 19 $1.90 x 0.1945 = 0.36953$ 20 $1.95 x 0.1784 = 0.34794$ 21 $2.00 x 0.1637 = 0.32740$ 22 $2.05 x 0.1502 = 0.30787$ 23 $2.10 x 0.1378 = 0.28934$ 24 $2.15 x 0.1264 = 0.27177$ 25 $2.20 x 0.1160 = 0.25513$ 26 $2.25 x 0.1064 = 0.23938$ 27 $2.30 x 0.0976 = 0.22450$

14.13278$ Total

$0.05 change in income each yearDiscount rate of 9%


Table 23.6 Exponential-Curve Change

Year Cash Flow Discount Factor Present Value1 $1.00 x 0.9174 = $0.917432 $1.04 x 0.8417 = $0.875353 $1.08 x 0.7722 = $0.835194 $1.12 x 0.7084 = $0.796885 $1.17 x 0.6499 = $0.760336 $1.22 x 0.5963 = $0.725457 $1.27 x 0.5470 = $0.692178 $1.32 x 0.5019 = $0.660429 $1.37 x 0.4604 = $0.6301310 $1.42 x 0.4224 = $0.6012211 $1.48 x 0.3875 = $0.5736412 $1.54 x 0.3555 = $0.5473313 $1.60 x 0.3262 = $0.5222214 $1.67 x 0.2992 = $0.4982715 $1.73 x 0.2745 = $0.4754116 $1.80 x 0.2519 = $0.4536017 $1.87 x 0.2311 = $0.4328018 $1.95 x 0.2120 = $0.4129419 $2.03 x 0.1945 = $0.3940020 $2.11 x 0.1784 = $0.3759321 $2.19 x 0.1637 = $0.3586822 $2.28 x 0.1502 = $0.3422323 $2.37 x 0.1378 = $0.3265324 $2.46 x 0.1264 = $0.3115525 $2.56 x 0.1160 = $0.2972626 $2.67 x 0.1064 = $0.2836327 $2.77 x 0.0976 = $0.27061

$14.37121 Total

$0.04 change in income each yearDiscount rate of 9%


Table 23.7 Level Income in Perpetuity

Year Income Discount Rate Value1 $25,000 x 0.925926 = $23,1482 $25,000 x 0.857339 = $21,4333 $25,000 x 0.793832 = $19,8464 $25,000 x 0.735030 = $18,3765 $25,000 x 0.680583 = $17,0156 $25,000 x 0.630170 = $15,7547 $25,000 x 0.583490 = $14,5878 $25,000 x 0.540269 = $13,5079 $25,000 x 0.500249 = $12,50610 $25,000 x 0.463193 = $11,58011 $25,000 x 0.428883 = $10,72212 $25,000 x 0.397114 = $9,92813 $25,000 x 0.367698 = $9,19213 $312,500 x 0.367698 = $114,906

$312,500

Analysis of property with level income and no change in value at 8%

Net present value

Direct Capitalization = $25,000 / 0.08 =$312,500


Table 23.8 Analysis of Property with Level Income and Change in Value

Year Cash Flow Discount Factor Present Value0.1000

1 $25,000 x 0.9090909 = $22,7272 $25,000 x 0.8264463 = $20,6613 $25,000 x 0.7513148 = $18,7834 $25,000 x 0.6830135 = $17,0755 $25,000 x 0.6209213 = $15,5235 $328,869 x 0.6209213 = $204,202

$298,972Net present value* The reversion was calculated as 125% of the present value, or $298,972 x 1.10 = $328,869

R = 0.10 – (0.10 x 0.16380)R = 0.083620Direct capitalization: $25,000/0.083620 = $298,972

Δ = 10%Y = 0.10N = 5 years

R = Y – (Δ 1/x S n ⎤)


Table 23.9 Constant Change in Income and Value

Year Adj Income Disc. Rate = Present Value0.08

1 $35,000 $35,000 x 0.925926 = $32,4072 $35,000 + $2,178 = $37,178 x 0.857339 = $31,8743 $35,000 + $4,356 = $39,356 x 0.793832 = $31,2424 $35,000 + $6,533 = $41,533 x 0.735030 = $30,5285 $35,000 + $8,711 = $43,711 x 0.680583 = $29,7496 $35,000 + $10,889 = $45,889 x 0.630170 = $28,9187 $35,000 + $13,067 = $48,067 x 0.583490 = $28,0468 $35,000 + $15,244 = $50,244 x 0.540269 = $27,1469 $35,000 + $17,422 = $52,422 x 0.500249 = $26,22410 $35,000 + $19,600 = $54,600 x 0.463193 = $25,29010 $1,050,000 0.463193 = $486,353

= $777,778

R = 0.08 - 0.035R = 0.045

Net present value

777,778 x (0.35 x 0.10) x 0.08 = the increase in cash flows777,778 x 0.035 x 0.08 = 2,177.78

Initial Income Income Change

Direct Capitalization: $35,000 / 0.045 = $777,778To calculate the amount of cash flow change each period, the following formula applies:V x (Δ 1/ ) = n x Y the increase in cash flows

= 0.08Y = - (R Y Δ 1/ )n = 0.08 - (0.35 * 0.10)R

= 35.00%Increase in income and value = 35,000Income $ = 10 Holding period years


Table 23.10 Constant-Ratio Change in Income and Value

R = Y - CRR = 0.08-0.03R = 0.05

Direct Capitalization: $35,000 /0.05 = $700,000

Year Income Reversion Cash Flows Discount Factor Net Present Value3.00%

1 $35,000.00 $35,000 0.925925926 $32,407.412 $36,050.00 $36,050 0.85733882 $30,907.063 $37,131.50 $37,132 0.793832241 $29,476.184 $38,245.45 $38,245 0.735029853 $28,111.545 $39,392.81 $39,393 0.680583197 $26,810.086 $40,574.59 $40,575 0.630169627 $25,568.887 $41,791.83 $41,792 0.583490395 $24,385.138 $43,045.59 $43,046 0.540268885 $23,256.199 $44,336.95 $44,337 0.500248967 $22,179.5110 $45,667.06 $45,667 0.463193488 $21,152.6911 $47,037.07 $47,037 0.428882859 $20,173.3912 $48,448.19 $48,448 0.397113759 $19,239.4413 $49,901.63 $49,902 0.367697925 $18,348.7314 $51,398.68 $51,399 0.340461041 $17,499.2515 $52,940.64 $52,941 0.315241705 $16,689.1016 $54,528.86 $1,123,295 $1,177,823 0.291890468 $343,795.41

$700,000.00Total =

Assume the property is in an area where 8% yields are required. Properties are appreciating 3% per year.Income is $35,000/year and it is increasing at 3% per year also.


Table 23.11 Constant-Ratio Change in Income and Value

Income is $35,000/year and it is increasing at 3% per year also. R = Y - CRR = 0.08-0.03R = 0.05

Direct Capitalization: $35,000 /0.05 = $700,000

Year Income Reversion Cash Flows Discount Factor Net Present Value3.00%

1 $35,000.00 $35,000 0.925926 $32,407.412 $36,050.00 $36,050 0.857339 $30,907.063 $37,131.50 $37,132 0.793832 $29,476.184 $38,245.45 $38,245 0.735030 $28,111.545 $39,392.81 $39,393 0.680583 $26,810.086 $40,574.59 $40,575 0.630170 $25,568.887 $41,791.83 $41,792 0.583490 $24,385.138 $43,045.59 $43,046 0.540269 $23,256.199 $44,336.95 $44,337 0.500249 $22,179.5110 $45,667.06 $45,667 0.463193 $21,152.6910 * $904,741 0.463193 $435,745.32

$700,000.00

Assume the property is in an area where 8% yields are required. Properties are appreciating 3% per year.

Total =* $700,000 x 1.0310 = $940,744.47


Problems

Suggested solutions begin on page 365.


Problem 1D

RO = YO - CR 0.02 0 CF

RO = 0.10 - 0.02 100,000$ x 1.02 = 102,000 CF1

RO = 0.08 102,000$ x 1.02 = 104,040 CF2

104,040$ x 1.02 = 106,121 CF3

100,000 / 0.08 = 1,250,000 106,121$ x 1.02 = 108,243 CF4

108,243$ x 1.02 = 110,408 CF5

110,408$ x 1.02 = 112,616 CF6

112,616$ x 1.02 = 114,869 CF7

114,869$ x 1.02 = 117,166 CF8

117,166$ x 1.02 = 119,509 CF9

119,509$ + 1,523,743$ = 1,643,252 CF10

Solve for NPV @ 10% = 1,249,999


Problem 2B

RO = = I/V Prove RO = $7,000/$150,000 N 5RO = 0.04667 I 111/S n = 0.16057 PV -150,000Δ = ( - )/1/ )Y R Sn = (0.11 - 0.04667)/0.16057 PMT 7,000Δ = = 0.06333/0.16057 Solve FV 209,164Δ = = 0.39441F V = 150,000 1.39441$ x = 209,162 $


Problem 3B

Clear registers1 P/yr.0 CFO

5,000 CFJ

3 NJ

6,000 CFJ

5 NJ

9,000 CFJ

4 NJ

1,209,000 CFJ

9 I/yr.Solve for NPV 439,660


Problem 4B

Clear registers1 P/yr.

5,000 CFO

5,000 CFJ

2 NJ

6,000 CFJ

5 NJ

9,000 CFJ

5 NJ

1,200,000 CFJ

9 I/yrSolve for NPV 444,003


Problem 5B

RO = YO - Δ 1/Sn Prove

0.08 – (0.10 0.170456)x N 5

0.062954 I 8

V =O /I V PV - 476,538$

30,000/0.062954 PMT 30,000 $

476,538

476,538 1.10 = 524,192x Solve FV 524,189


Problem 6C

9 N

-750,000 PV

59,000 PMT

750,000 x 1.25 = 937,500 FV

Solve for I = 9.7288


Problem 7A

RO = 100,000 /1,250,000

YO = Ro + CR

YO = 0.08 + 0.01

YO = 0.09


Problem 8B

-175,300 PV

25,000 PMT

25 N

0 FV

Calculate I 13.68


Problem 9C

100,000 PMT

11 N

100,000 FV

9 I

Solve for PV 719,272

Yield Capitalization — Theory and Basic Applications

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