1. BASIC MICRO-LEVEL VALUATION: "DCF" & “NPV”
Chapter 10:
BASIC MICRO-LEVEL VALUATION:
"DCF" & “NPV”
Note: Yellow slides are covered in Geltner-Miller text.
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2. THE famous DCF VALUATION PROCEDURE...
FORECAST THE EXPECTED FUTURE CASH FLOWS;
ASCERTAIN THE REQUIRED TOTAL RETURN;
DISCOUNT THE CASH FLOWS TO PRESENT VALUE AT THE REQUIRED RATE OF RETURN.
THE VALUE YOU GET TELLS YOU WHAT YOU MUST PAY SO THAT YOUR EXPECTED RETURN WILL EQUAL THE "REQUIRED RETURN" AT WHICH YOU DISCOUNTED THE EXPECTED CASH FLOWS.
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where:
CFt = Net cash flow generated by the property in period “t”;   Vt = Property value at the end of period “t”;   E0[r] = Expected average multi-period return (per period) as of time “zero” (the present), also known as the “going-in IRR”;   T = The terminal period in the expected investment holding period, such that CFT would include the re-sale value of the property at that time (VT), in addition to normal operating cash flow.
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Numerical example...
·        Single-tenant office bldg
·        6-year “net” lease with a “step-up”...
·        Expected sale price year 6 = $15,000,000
·        Required rate of return (“going-in IRR”) = 10%...
·        DCF valuation of property is $15,098,000:
Lease:
Year: CF:
2001 $1,000,000
2002 $1,000,000
2003 $1,000,000
2004 $1,500,000
2005 $1,500,000
2006 $1,500,000
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Why is the DCF procedure important?
Recognizes asset valuation fundamentally depends upon future net cash flow generation potential of the asset.
Takes long-term perspective appropriate for investment decision-making in illiquid markets (multi-period, typically 10 yrs in R.E. applications).
Takes the total return perspective necessary for successful investment.
Due to the above, the exercise of going through the DCF procedure, if taken seriously, can help to protect the investor from being swept up by an asset market “bubble” (either a positive or negative bubble – when asset prices are not related to cash flow generation potential).
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Remember:
Investment returns are inversely related to the price paid going in for the asset.
e.g., in the previous example, if we could get the asset for $14,000,000 (instead of $15,098,000), then our going-in return would be 9.6% (instead of 8%):
vs.
What is the fundamental economic reason for this inverse relationship? [Hint: What determines fut. CFs?]
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7. Match the discount rate to the risk. . .
r = rf + RP
 Disc.Rate = Riskfree Rate + Risk Premium
(Riskfree Rate = US T-Bill Yield.)
Note: Underlying (implicit) assumption of a “twin” or “replicating” investment in the market, that the subject investment could be replicated by an alternative investment with expected return “r”. (This is the same assumption that we will see later in this course will be used to evaluate “real options”.)
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8. 10.2.1 Match the Discount Rate to the Risk:
Intralease & Interlease Discount Rates
Projected operating CFs will be contractual (covered by leases). 1st 6 yrs in current lease, remainder in a subsequent lease. Prior to signing, lease CFs are more risky (interlease disc rate), once signed, less risky (intralease disc rate). DCF Valuation:
Here we have estimated the discount rate at 7% for the relatively low-risk lease CFs (e.g., if T-Bond Yld = 5%, then RP=2%), and at 9% for the relatively high-risk later CFs ( 4% risk premium).  Implied property value = $18,325,000.
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Blended IRR:
A single Discount Rate
Current practice usually is not this sophisticated for typical properties. If the lease expiration pattern is typical, a single “blended” discount rate is typically used.
Thus, for this building, we would typically observe a “going-in IRR” of 8.57%, applied to all the expected future CFs…
In principle, this can allow “arbitrage” opportunities if investors are not careful (especially as the ABS mkt develops…)
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Example:
Suppose property with 7 yrs (instead of 6 yrs) remaining on vintage lease but same expected CFs as before.
Purchase for $18,325,000, then sell lease for $7,083,000 and property residual for $11,319,000, for a profit of $77,000:
1st 6 CFs in Lease  $18.325M Value
1st 7 CFs in Lease  $18.402M Value
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11. Valuation shortcuts: “Ratio valuation”...
10.3 Ratio Valuation Procedures
Valuation shortcuts: “Ratio valuation”...
1) DIRECT CAPITALIZATION:
A WIDELY-USED SHORTCUT VALUATION PROCEDURE:
·       SKIP THE MULTI-YEAR CF FORECAST
·       DIVIDE CURRENT (UPCOMING YEAR) NET OPERATING INCOME (NOI) BY CURRENT MARKET CAP RATE (YIELD, NOT THE TOTAL RETURN USED IN DCF)
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12. (FORMALLY, NOT CAUSALLY)
The idea behind direct capitalization…
IF "CAP RATE" = NOI / V ,
THEN:
V = NOI / CAP RATE
(FORMALLY, NOT CAUSALLY)
MOST APPROPRIATE FOR BLDGS W SHORT-TERM LEASES IN LESS CYCLICAL MARKETS, LIKE APARTMENTS.
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EXAMPLE:
250 UNIT APARTMENT COMPLEX
AVG RENT = $15,000/unit/yr
5% VACANCY
ANNUAL OPER. EXPENSES = $6000 / unit
8.82% CAP RATE (KORPACZ SURVEY)
VALUATION BY DIRECT CAPITALIZATION:
POTENTIAL GROSS INCOME (PGI) = 250*15000 = $3,750,000
- VACANCY ALLOWANCE (5%) = 0.5* = ,500
- OPERATING EXPENSES = 250*6000 = 1,500,000
NET OPER.INCOME (NOI) = $2,062,500
V = 2,062,500 / = $23,384,354, say approx. $23,400,000
Cap rate is from Korpacz Survey cerca See how different the property value would be at the current Korpacz-reported cap rate for apartments, holding all else the same…
Expenses might be a bit less than $6000 for unoccupied units. A more refined analysis would account for that. Analysts often try to distinguish between “fixed” vs “variable” operating expenses. The latter are proportional to occupancy (or stated per occupied unit). Argus easily allows for this. Parts C & D of PS1 will show you how Argus does this (or you can do yourself it in Excel).
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14. 2) GROSS INCOME MULTIPLIER (GIM):
GIM = V / GROSS REVENUE
COMMONLY USED FOR SMALL APARTMENTS.
(OWNER'S MAY NOT RELIABLY REVEAL GOOD EXPENSE RECORDS, SO YOU CAN'T COMPUTE NOI (= Rev - Expense), BUT RENTS CAN BE OBSERVED INDEPENDENTLY IN THE RENTAL MARKET.)
IN PREVIOUS APT EXAMPLE THE GIM IS:
23,400,000 / 3,750,000 = 6.2.
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15. Empirical cap rates and market values. . .
Cap rates are a way of quoting observed market prices for property assets (like bond “yields” are the way bond prices are reported).
E.g., PwC Survey 
Note: Cap rates are not the causal structure determining property values. Investors don’t buy property just to get one year’s income. Investors are interested in the total return over the lifetime of the investment.
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17. DANGERS IN MKT-BASED RATIO VALUATION. . .
1) DIRECT CAPITALIZATION CAN BE MISLEADING FOR MARKET VALUE IF PROPERTY DOES NOT HAVE CASH FLOW GROWTH AND RISK PATTERN TYPICAL OF OTHER PROPERTIES FROM WHICH CAP RATE WAS OBTAINED. (WITH GIM IT’S EVEN MORE DANGEROUS: OPERATING EXPENSES MUST ALSO BE TYPICAL.)
2) Market-based ratio valuation won’t protect you from “bubbles”!
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18. 10.4 Typical mistakes in DCF application to commercial property...
CAVEAT!
BEWARE OF “G.I.G.O.”
===> Forecasted Cash Flows:
Must Be REALISTIC Expectations
(Neither Optimistic, Nor Pessimistic)
===> Discount Rate should be OCC:
Based on Ex Ante Total Returns in Capital Market
(Including REALISTIC Property Market Expectations)
·      Read the “fine print”.
·      Look for “hidden assumptions”.
·      Check realism of assumptions.
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10.4. Typical Mistakes in DCF Application to Commercial Property
If your case lacks merit, dazzle them with numbers:
Read the “fine print” What are the assumptions behind the numbers in the DCF?...
In reality rents in a given building don’t often grow as fast as inflation over the long run
In reality leases don’t renew with 100% probability, optimal vacancy is greater than zero on average
Etc…
Excessive laziness:
The “GIGO” problem (Garbage In, Garbate Out)…
Assumptions should be consistent with basic urban economics (see Chs.3-6)
Watch out for the cycle:
For rent forecast, where is the space market in the rental/occupancy cycle?
For cap rate forecast, where is the asset market in the capital cycle?
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10.5. Underwriting Haircuts
Cash flow pro-formas and DCF analyses based on them in the real world sometime deliberately use biased numbers, in particular
They presumably try to err on the conservative side (they give a “haircut” to cash flow projections or future cap rate projections, so as to reduce what the PV of the asset would otherwise appear to be.
This is supposed to help avoid paying too much for the property, or making too large a loan on it. It is referred to as: “underwriting assumptions”.
This can be a useful exercise, but…
It violates the basic spirit or purpose of the DCF analysis based in economic and statistical theory, which is to employ realistic (unbiased) expectations focused on realistic (unbiased) implications about asset value.
It puts the analyst out on “soft ground,” removing the analysis from objective reality
This opens the door for abuse. Puts analysts and decision makers under pressure to make numbers that appear more conservative than the actual reality, yet employ subtle (or not so subtle!) numerical assumptions and devices (such as those in 10.4 on the previous slide) to support “doing the deal” (whether it be buying, or lending to support buying).
The best practice for DCF is probably transparency and realism based on sound empirics and theory.
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Three most common mistakes in R.E. DCF practice:
Rent & income growth assumption is too high—
aka: “We all know rents grow with inflation, don’t we!”?...
Remember: Properties tend to depreciate over time in real terms (net of inflation).  Usually, rents & income within a given building do not keep pace with inflation, long run.
Capital improvement expenditure projection, &/or terminal cap rate projection, are too low –
Remember: Capital improvement expenditures typically average at least 10%-20% of the NOI (1%-2% of the property value) over the long run.
Going-out cap rate is typically at least as high as the going-in cap rate (older properties are more risky and have less growth potential).
Discount rate (expected return) is too high –
This third mistake may offset the first two, resulting in a realistic estimate of property current value, thereby hiding all three mistakes!
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22. Numerical example: Two cash flow streams . . .
Year: 1 2 3 4 5 6 7 8 9 10
$1,000,000
$1,050,000
$1,102,500
$1,157,625
$1,215,506
$1,276,282
$1,340,096
$1,407,100
$1,477,455
$17,840,274
$1,010,000
$1,020,100
$1,030,301
$1,040,604
$1,051,010
$1,061,520
$1,072,135
$1,082,857
$12,139,907
First has 5%/yr growth,
Second has 1%/yr growth.
Both have same initial cash flow level ($1,000,000).
Both have PV = $10,000,000: First 15%, Second 11%.
As both streams have same starting value & same PV, both may appear consistent with observable current information in the space and property markets. (e.g., rents are typically $1,000,000, and property values are typically $10,000,000 for properties like this.)
Suppose realistic growth rate is 1%, not 5%. Then the first CF projection gives investors an unrealistic return expectation of 15%.
“Unfair” comparisons (e.g., bond returns cannot be “fudged” like this).
Investor is “set up” to be disappointed in long run.
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23. Exhibit 11-2: NCREIF Same-Property NOI Growth vs Inflation: 1979-2011
Gray shade indicates GDP recession
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24. © 2014 OnCourse Learning. All Rights Reserved.
Going-in IRR used as discount rate in DCF tends to be too high (above historically achieved avg total return, even though infla was higher in history)
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25. Going-in IRR used as discount rate in DCF tends to be too high above the cap rate* (implying growth equal to infla instead of < infla)
*Cap rate can be more accurately objectively observed than IRR.
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26. Results of these types of mistakes:
Unrealistic expectations
Long-run undermining credibility of the DCF valuation procedure
Wasted time (why spend time on the exercise if you’re not going to try to make it realistic?...)
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27. 10.6 DCF and Investment Decision Rules: the NPV Rule...
DCF  Property value (“V”) . . .
But how do we know whether an investment is a “good deal” or not?...
How should we decide whether or not to make a given investment decision?
NPV = PV(Benefit) – PV(Cost)
i.e.: NPV = Value of what you get – Value of what you give up to get it,
All measured in equivalent “apples-to-apples” dollars, because we have discounted all the values to present using discount rates reflecting risk.
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28. “THE NPV INVESTMENT DECISION RULE”:
1) MAXIMIZE THE NPV ACROSS ALL MUTUALLY-EXCLUSIVE ALTERNATIVES; AND
2) NEVER CHOOSE AN ALTERNATIVE THAT HAS: NPV < 0.
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29. The NPV Investment Decision Rule
IF BUYING: NPV = V – P
IF SELLING: NPV = P – V
Where:
V = Value of property at time-zero (e.g., based on DCF)
P = Selling price of property (in time-zero equivalent $)
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Example:
DCF  V = $13,000,000
You can P = $10,000,000.
NPV = V-P = $13M - $10M = +$3M.
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Note: NPV Rule is based directly on the “Wealth Maximization Principle”. . .
WEALTH MAXIMIZATION  The NPV Rule   Maximize the current value of the investor’s net wealth. Otherwise, you’re “leaving money on the table”.
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32. Proof that Wealth Maximization implies the NPV Decision Rule:
Developers don’t always explicitly apply the NPV Rule.
But remember . . .
Proof that Wealth Maximization implies the NPV Decision Rule:
Suppose not.
Then I could maximize my wealth and still contradict NPV Rule.
I could choose a project with NPV < 0, or with NPV less than that of another mutually-exclusive feasible alternative.
But if I did that I would be “leaving money (i.e. “wealth”) on the table”, taking less wealth when I could have more.
This would not be wealth-maximization.
Hence: Contradiction.
QED (“Proof by Contradiction”).
Thus, if our definition of a “successful” developer is one who maximizes wealth, then successful developers must be applying the NPV Rule (implicitly if not explicitly). (Dvlprs who max wealth must ultimately supplant those who don’t…)
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NPV Rule Corollary:
"Zero NPV deals are OK!“
Why? . . .
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Zero NPV deals are not zero profit.
(They only lack “super-normal” profit.)
A zero NPV deal is only “bad” if it is prevents the investor from undertaking a positive NPV deal.
In fact, on the basis of “market value” (MV)…
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35. NPV(Buyer) = V-P = MV-P NPV = 0.
Based on “market value” (MV),
NPV(Buyer) = V-P = MV-P
NPV(Seller) = P-V = P-MV = -NPV(Buyer)
Therefore, if both the buyer and seller apply the NPV Rule (NPV0), then:
(i) NPV(Buyer)  0  -NPV(Seller)  0  NPV(Seller)  0;
(ii) NPV(Seller)  0  -NPV(Buyer)  0  NPV(Buyer)  0;
(i)&(ii) together  NPV(Buyer) = NPV(Seller) = 0.
Thus, measured on the basis of MV, we actually expect that:
NPV = 0.
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36. Sources of “illusions” of big positive NPVs
1)    OCC (discount rate) is not the cost of borrowed funds (e.g., mortgage interest rate).
2)    Land value? (not just historical cost)
3)    Search & Management Costs?
4)    “Private Info”? (But MV is based on public info.)
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37. © 2014 OnCourse Learning. All Rights Reserved.
However, in Real Estate it is possible to occasionally find deals with substantially positive, or negative, NPV, even based on MV.
Real estate asset markets not informationally efficient: - People make “pricing mistakes” (they can’t observe MV for sure for a given property) - Your own research may uncover “news” relevant to value (just before the market knows it)
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38. What about unique circumstances or abilities? . . .
Generally, real uniqueness does not affect MV.
Precisely because you are unique, you can’t expect someone else to be willing to pay what you could, or be willing to sell for what you would. (May affect “investment value” – IV, not MV.)
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39. 10.6.2 Choosing Among Alternative Zero-NPV Investments
Alternatives may have different NPVs based on “investment value”, even though they all have equal (zero) NPV based on “market value”. (See Sect in Ch.12.)
One alternative may be preferable for macro or strategic reasons (portfolio target, administrative efficiency, property size preference, etc.).
Alternatives may present different expected return “attributes” (initial yield, cash flow change, yield change). (See Appendix 10A & Sect )
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40. 10.6.3. Hurdle Rate Version of the Investment Rule: IRR vs. NPV
SOMETIMES IT IS USEFUL (anyway, it is very common in the real world) TO "INVERT" THE DCF PROCEDURE...
INSTEAD OF CALCULATING THE VALUE ASSOCIATED WITH A GIVEN EXPECTED RETURN,
CALCULATE THE EXPECTED RETURN (IRR) ASSOCIATED WITH A GIVEN PRICE FOR THE PROPERTY.
I.E., WHAT DISCOUNT RATE WILL CAUSE THE EXPECTED FUTURE CASH FLOWS TO BE WORTH THE GIVEN PRICE?...
THEN THE DECISION RULE IS:
1) MAXIMIZE DIFFERENCE BETWEEN:
IRR AND REQUIRED RETURN (ACROSS MUT.EXCLU ALTS)
2) NEVER DO A DEAL WITH:
IRR < REQ'D RETURN
REQ’D RETURN = “HURDLE RATE” = rf + RP
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When using the IRR (hurdle) version of the basic investment decision rule: Watch out for mutually exclusive alternatives of different scales.
e.g., $15M 15% is better than $5M if cost of capital (hurdle) in both is 10%.
e.g., for a 1-yr project…
NPV = ((1.15)$15 million)/ $15 million = ($17.25 million)/ $15 million = $ million - $15 million = +$682,000;
Versus:
NPV = ((1.20)$5 million)/ $5 million = ($6 million)/ $5 million = $5.455 million - $5 million = +$455,000.
As $682,000 > $455,000 and the two are mutually exclusive, of course you would rather have the $682,000, the larger project with the smaller IRR.
e.g., for a 1-yr project…
NPV = ((1.15)$15 million)/ $15 million = ($17.25 million)/ $15 million = $ million - $15 million = +$682,000;
Versus:
NPV = ((1.20)$5 million)/ $5 million = ($6 million)/ $5 million = $5.455 million - $5 million = +$455,000.
As $682,000 > $455,000 and the two are mutually exclusive, of course you would rather have the $682,000, the larger project with the smaller IRR.
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42. © 2014 OnCourse Learning. All Rights Reserved.
Appendix 10A (& Ch.26 Sect ) Property-Level Investment Performance Attribution
Pink slides: Fall course: Focus only on slides 57 & 61.
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43. Real Estate Investment “Performance Attribution”
DEFINITION:
The decomposition (or “breaking down”, or “parsing”) of the total investment return of a subject property or portfolio of properties (or an investment manager).
PURPOSE:
To assist with the diagnosis and understanding of what caused the given investment performance.
USAGE:
By investment managers (agents) and their investor clients (principals).
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44. at which performance attribution is performed:
Two levels
at which performance attribution is performed:
Property level
Pertains to individual properties or static portfolios of multiple properties.
Portfolio level
Pertains to dynamic portfolios or investment manager (or fund) level.
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45. Major attributes (return components): At the PORTFOLIO LEVEL:
At the PROPERTY LEVEL:
Initial Cash Yield
Cash Flow Change
Yield Change
At the PORTFOLIO LEVEL:
Allocation
Selection
Interaction
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46. “Performance Attribution”:
Often useful for diagnostic purposes to compare subject portfolio or mgr with an appropriate benchmark . . .
“Performance Attribution”:
Portf Tot.Return – Bnchmk Tot.Return
Allocation
Selection
Interaction
Portfolio Level:
Prop.IRR – Bnchmk Cohort IRR
Init.Yield
CF Growth
Yield Chge
Property Level:
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47. © 2014 OnCourse Learning. All Rights Reserved.
Property-Level Performance Attribution . . .
Property level performance attribution focuses on “property level” investment performance, i.e., the total return achieved within a given property or a static (fixed) portfolio of properties (that is, apart from the effect of investment allocation decisions, as if holding allocation among categories constant).
Property level attribution should be designed to break out the property level total return performance in a manner useful for shedding light on the four major property level investment management functions:
Property selection (picking “good” properties as found);
Acquisition transaction execution;
Operational management during the holding period (e.g., marketing, leasing, expense mgt, capital expenditure mgt);
Disposition transaction execution.
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48. Property-Level Performance Attribution . . .
These property-level management functions are related generally to three attributes (components) of the property-level since-acquisition IRR, essentially as indicated below…
Initial Yield
(IY)
Cash Flow Change
(CFC)
Yield Change
(YC)
Property Selection
Acquisition Transaction Execution
Operational Management
Disposition Transaction Execution
The above refers to use of property-level performance attribution in an ex post (“look back”) approach, to help diagnose investment management performance. But alternatively the performance attribution can be used in an ex ante (“going-in” or forward-looking) perspective based on the projected investment pro-forma. When used in this ex ante way, performance attribution is usually less about diagnosing management strengths than about understanding where is the projected return expected to come from. This can be helpful in at least two ways: (i) to help judge the reliability in the expected return (e.g., CFC or YC components may be viewed as less reliable than the IY component in many cases); (ii) to help relate the investment to the investor’s return component preferences (e.g., does the investor prefer short-term and continuing income yield as presented by the IY attribute, or longer-term growth in the income yield as presented by the CFC component, or even longer-term growth in capital value without intermediate payback as presented by the YC component?). This can help to match investor preferences to investment alternatives, when all investment alternatives appear to present an equal NPV (given that typically NPV=0 measured on a market value basis).
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49. © 2014 OnCourse Learning. All Rights Reserved.
Property-Level Performance Attribution . . .
Conventional property level performance attribution is based on periodic returns, or on time-weighted multi-period returns (TWRRs, e.g., as implemented by IPD in England and PCA in Australia).
But IRR-based performance attribution is arguably more useful for property level management diagnostic purposes, because:
At the property level, the investment manager is typically responsible for the major cash flow timing decisions that can significantly effect property level (static portfolio) returns, e.g., leasing decisions, capital expenditure decisions.
The IRR is sensitive to the effect of cash flow timing, the TWRR is not.
The IRR is cash flow based (net of capital improvement expenditures), therefore, more accurately reflecting the investment return effect of capital improvement decisions.
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50. Since-acquisition IRR
Property-Level Performance Attribution . . .
Useful IRR-Based property level performance attribution benchmarking requires the use of:
Since-acquisition IRR
IRR is computed since acquisition of property (or portfolio):
In order to reflect investment operational performance during entire holding period since acquisition;
Property investment holding periods are typically multi-year (single period or periodic returns do not reflect effective investment management holding period).
IRR is computed for appropriate benchmark cohort, defined as universe of similar investments by competing managers, measured from same inception date (equal to property acquisition date).
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51. © 2014 OnCourse Learning. All Rights Reserved.
Property-Level Performance Attribution . . .
Simple Numerical Example:
Property (or static portfolio) bought at initial cash yield of 9%.
Net cash flow grew at 2% per year.
Property (or properties) sold (or appraised) after 10 years at a terminal yield of 10%, based on yr.11 projected cash flow (also 2% more than yr.10).
IRR is 10.30%.
How much of this IRR is due to 3 components: Initial Yield (IY), Cash Flow Change (CFC), and Yield Change (YC)?…
This example is taken from Geltner-Miller, Chapter 10 Appendix (pp ). There is also a Study Question (10.24) at the end of that chapter, with solution on p.858.
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52. © 2014 OnCourse Learning. All Rights Reserved.
There are several ways one might answer this question. The approach that seems most intuitively related to the 4 basic mgt fcns is presented here…
Initial yield = 9.00%, computed from line (6) IRR.
Cash flow change component = 2.00% = 11%-9%, computed as the line (8) IRR less the line (6) IRR: = IRR with actual CF – IRR with no CF growth, (with constant yld at initial rate).
Yield change component = -0.68% = 8.32%-9.00%, computed as the line (10) IRR less the line (6) IRR: = IRR with actual yld chg – IRR with no yld chg, (with constant CF at initial level).
Interraction effect = -0.02%, the difference bertween the line (3) overall IRR and the sum of the three other attributes [10.3%-(9%+2%-0.68%)].
Other ways to define the break-down of the three components would be, for example, to define the components on the basis of the actual or the terminal cash flows rather than the initial cash flow as done here. No one method of basing the break-down is theoretically uniquely correct. I think basing the break-down on the initial cash flow as done here gives results that are the most intuitive for identifying the substantive property-level management functions of acquisition and operational management, and potentially on the disposition function as well (this last if the terminal value is based on a disposition price) .
You could try to apply this performance attribution technique in an “ex ante” perspective in the “Private Practice” case, to help compare the investment alternatives.
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53. © 2014 OnCourse Learning. All Rights Reserved.
Property-Level Performance Attribution . . .
Here is a graphical presentation of the IRR-Based property-level performance attribution we just performed:
Suppose we computed the same type of IRR component breakdown for an appropriate benchmark, that is, a NCREIF sub-index cohort spanning the same period of time…
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Property-Level Performance Attribution . . .
We could compare our subject performance with that achieved by a peer universe of managers, for similar properties (e.g., Calif. Industr. bldgs):
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Property-Level Performance Attribution . . .
Here is the relative performance, the difference between our subject property and its benchmark, by attribute:
The above pattern could be plausibly interpreted as tentative evidence for the following hypothesis: Subject performed relatively poorly due largely to some combination of poor selection, acquisition, and operational mgt, partially offset by some combination of good disposition execution (or optimistic terminal appraisal), future-oriented capital improvements, &/or market movements during the holding period.
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Property-Level Performance Attribution . . .
Here is the relative performance, the difference between our subject property and its benchmark, by attribute:
Now suppose we computed these relative performance differentials across a number of different properties (or portfolios) we have invested in…
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Property-Level Performance Attribution . . .
We might gain some insights about our property-level investment and management performance:
Such comparisons of relative performance across properties can sometimes be useful even if the properties have different acquisition dates. (Of course, the IRR-cohort benchmark against which the relative performance is computed must be “custom” for each property, corresponding to the holding period for each property.) All such formal quantitative exercises must be “taken with a grain of salt”, applied in an “investigative” spirit. They often bring insights by the questions they raise rather than by providing any definitive answers. They get property and asset managers thinking about how they did and what they did in specific cases, and what impact their actions had in the overall performance. These types of results should never be used mechanistically, without broader and more qualitative analysis, and they should generally not be used for evaluative purposes (in my opinion), but rather for diagnostic (I.e., “learning”) purposes.
In this case Subject Properties #1 & 3 have similarly poor performance (rel to benchmk), due to poor initial yield & poor CF change, suggesting poor acquisition & poor operational mgt. Property #2 did better, with good InitYld & CFchg, but poor YldChg (suggesting good acquisition, but poor disposition or mgt actions that hurt future outlook (e.g., inadequate Cap.Improvement). Mkt movements can also affect these results (less so the longer the holding period).
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58. A Method to Estimate Interlease Discount Rates
Appendix 10B:
A Method to Estimate Interlease Discount Rates
Suppose in a certain property market the typical:
Lease term is 5 years;
Cap rate (cash yield) is 7%;
Long term property value & rental growth rate is 1%/yr (typically equals inflation minus real depreciation rate);
Leases provide rent step-ups of 1%/yr (per above);
Tenant borrowing rate (intralease disc rate) is 6%...
Then (assuming pmts in arrears), PV of a lease, per dollar of initial net rent, is:
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A stylized space in this market has PV equal to S, as follows (ignoring vacancy between leases), assuming interlease discount rate is r:
This is a constant-growth perpetuity, so (using geometric series formula from Chapter 8) we can shortcut this as:
From the market’s prevailing cap rate we also know that:
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Putting these two together, we have:
The implied interlease discount rate is 8.48%.
This is almost 250 bps above the intralease rate of 6%.
But it is only about 50 bps above the blended going-in IRR of 8% (determined based on the same constant-growth perpetuity model, as the cash yield plus the growth rate: 7% + 1% = 8%).
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61. Certainty Equivalence Discounting
Appendix 10C:
Certainty Equivalence Discounting
Review of traditional RADR DCF
Introduction to Certainty-Equivalence DCF
Example
Reference: Geltner-Miller et al 3e, Chapter 10 Appendix C (on the CD)
Note: Yellow slides are covered in Geltner-Miller text.
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62. THE CLASSICAL (and most widely used) RADR METHOD OF DCF
Chapter 10: “Discounted Cash Flow Valuation”…
THE CLASSICAL (and most widely used) RADR METHOD OF DCF
Forecast the expected future cash flows;
Ascertain the required total return;
Discount the cash flows to present value at the required rate of return.
The value you get tells you what you must pay so that your expected return will equal the “required return” at which you discounted the expected cash flows.
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Chapter 10: “Discounted Cash Flow Valuation”…
where:
CFt = Net cash flow generated by the property in period “t”;   Vt = Property value at the end of period “t”;   E0[r] = Expected average multi-period return (per period) as of time “zero” (the present), also known as the “going-in IRR”;   T = The terminal period in the expected investment holding period, such that CFT would include the re-sale value of the property at that time (VT), in addition to normal operating cash flow.
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64. Match the discount rate to the risk. . .
Chapter 10: “Discounted Cash Flow Valuation”…
Match the discount rate to the risk. . .
r = rf + RP
 Disc.Rate = Riskfree Rate + Risk Premium
(Riskfree Rate = US T-Bill Yield.)
Note: Underlying (implicit) assumption of a “twin” or “replicating” investment in the market, that the subject investment could be replicated by an alternative investment with expected return “r”. (This is the same assumption that we will see later in this course will be used to evaluate “real options”.)
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65. Consider the fundamental element of our valuation model:
Appendix 10C: “Certainty-Equivalent Valuation”…
Consider the fundamental element of our valuation model:
Accounts for both time and risk in the discount rate in the denominator, as , where rf is riskless and accounts for the time value of money, and RPV is the market’s required risk premium in the expected return for the investment (ex ante).
Note: The green (or blue?) slides here (and next…) are from the forthcoming 2nd Edition of the Geltner-Miller text.
Manipulate this formula so that the denominator purely reflects the time value of money (the discounting is done risklessly) and the risk is completely and purely accounted for in the numerator, by reducing the cash flow amount in the numerator to a “Certainty Equivalent Value” . . .
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Appendix 10C: “Certainty-Equivalent Valuation”…
A little algebra . . .
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Appendix 10C: “Certainty-Equivalent Valuation”…
is the “Certainty Equivalent” value of
The risk-adjusted discount rate formulation accounts for both time and risk simultaneously in the denominator:
The certainty-equivalence formulation accounts for time only in the denominator, and risk only in the numerator, discounting the certainty-equivalent cash flow amount risklessly back to the present:
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68. © 2014 OnCourse Learning. All Rights Reserved.
Appendix 10C: “Certainty-Equivalent Valuation”…
is the “Certainty Equivalent” value of
The risk-adjusted discount rate formulation accounts for both time and risk simultaneously in the denominator:
Risk premium (in %)
The certainty-equivalence formulation accounts for time only in the denominator, and risk only in the numerator, discounting the certainty-equivalent cash flow amount risklessly back to the present:
Risk discount (in $), aka “Haircut”
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Appendix 10C: “Certainty-Equivalent Valuation”…
The investment market is indifferent between a claim on the actual risky amount of V1 one period in the future (with the actual amount of risk involved in that), versus a claim on the lesser amount of:
one period in the future with no risk at all.
The “haircutted” value is called the “certainty-equivalence value”, labeled “CEQ”:
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Appendix 10C: “Certainty-Equivalent Valuation”…
Why do we care about this? . . .
The certainty-equivalent approach is more general, can be applied in situations where RADR discounting can’t be applied, including options valuation and other derivatives valuations (e.g., swaps, forwards, futures).
Typically, CEQ discounting useful if:
Risk does not accumulate at a constant rate over time (i.e., there is no single correct OCC).
PV=0 even though future cash flow expectations are not zero (as with futures contracts)
Also useful for valuing real estate index futures contracts, that have zero cash exchange up front. (See Chapter 26 CD files.)
Consider a simple example . . .
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Appendix 10C: “Certainty-Equivalent Valuation”…
Suppose:
Riskfree interest rate = 3%
Commercial property market total return risk premium = 6%
Property market expected total return is:
E[rV] = rf + E[RPV] = 3% + 6% = 9%
“Binomial World”…
An office building that is worth $100M today (if it already exists, earning income) will next year have a value of either:
$113M (with 70% probability), or
$79M (with 30% probability).
Therefore, expected value of office bldg next yr is:
E[V1] = (0.70)113 + (0.30)79 = $103M
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72. Real probabilities (not “risk-neutral”) Real expected values
Appendix 10C: “Certainty-Equivalent Valuation”…
Next Year
Here is the situation …
V1 = $113 M
Prob = 70%
Today
E[V1] = $103 M
PV[V1]
Prob = 30%
V1 = $79 M
Real probabilities (not “risk-neutral”)
Real expected values
(not “risk-neutral”)
True expected value next year (nominal) = $103 M.
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73. How much is the forward claim on the office building worth today? . . .
Risk-adjusted discount rate valuation:
Certainty-equivalent valuation:
$97 is the certainty-equivalent value of V1 whose expected value is $103.
If we know rf and either E[RPV] or PV[V1] then we can derive above.
Thus, easy to apply RADR discounting to underlying asset (due to transparency in market for trading such built properties).
The risk-adjusted discount rate applicable to the building is the same as that of the stock market in this case, because the building has the same risk as the stock market (same volatility as the stock market, and perfectly correlated with the stock market, hence, the building’s expected total return must have the same expected risk premium as that of the stock market, 6%).
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74. How much is the forward claim on the office building worth today? . . .
If we know rf and either E[RPV] or PV[V1] then we can derive above.
Even if we just know V0 (current value of pre-existing “twin” building) and yV (cash payout rate, or income yield, of such building)…
PV[V1] = (1 + gV)V0 / (1 + rV) = V0 / (1 + yV),
e.g., with 6% “caprate”, $94 = $103 / 1.09 = $100 / 1.06.
This is due to relationship between rV, gV, & yV:
(1 + rV) / (1 + yV) = (1 + gV)  rV ≈ yV + gV
(This is an “accounting relationship”: Holds by definition.)
Thus, easy to apply RADR discounting to underlying asset (due to transparency in market for trading such built properties).
The risk-adjusted discount rate applicable to the building is the same as that of the stock market in this case, because the building has the same risk as the stock market (same volatility as the stock market, and perfectly correlated with the stock market, hence, the building’s expected total return must have the same expected risk premium as that of the stock market, 6%).
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75. How much is this option worth?
Appendix 10C: “Certainty-Equivalent Valuation”…
Now suppose the office building does not exist yet,
but we could build it in 1 year
by paying $90M of construction cost next year. (Construction is instantaneous: We can decide next year whether to build or not.)
Suppose our option to build this project expires in 1 year: We either build then or we lose the right to ever build.
Suppose market for such options is thin, lacks transparency, so we cannot directly observe value of option or of relevant E[RP] in discount rate…
How much is this option worth?
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Appendix 10C: “Certainty-Equivalent Valuation”…
Next Year
Label the value of the option today PV(C1).
Here is the situation we face…
Vup = $113 M
K = $90 M
Do dvlpt, get:
$113 - $90 = $23 M = Cup
Prob = 70%
Today
Vdown = $79 M
K = $90 M
Don’t build, get: $0 = Cdown
PV[C1]
Prob = 30%
In general: Cup = MAX[Vup-K, 0], Cdown = MAX[Vdown-K, 0].
We can’t value the option using RADR discounting (which works to value the forward claim on the building), because we don’t know what E[RPC ] should be. (For built property we could observe or derive RP in the mkt, but dvlpt option market is likely much thinner, less transparent, less homogeneous.)
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77. Built Property 37% range Devlpt Option 192% range
Invoke “linear pricing” principle (aka “Law of One Price”):
Same “Price of Risk” must apply to all assets at a given time.
Same expected return risk premium per unit of risk (ex ante).
Otherwise, “not fair” (people will trade until “One Price” holds).
Among “derivative” assets, relative risk can be measured by volatility (or return outcome spread range) ratio between the derivative & the underlying asset. “Derivatives” are assets that are perfectly correlated: derivative outcome depends entirely on underlying outcome.
Risk
E[r]
rf = 3%
34% 9%
Built Property
37% range
Devlpt Option
192% range
E[RP]
(We don’t know this yet.)
(We don’t know this yet.)

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Appendix 10C: “Certainty-Equivalent Valuation”…
Price of risk in the “underlying asset” (the completed built office bldg – what you can get by exercising the development option) = Risk Premium / Risk (spread)…
Therefore, price of risk must be the same in the derivative asset, the development option:
Slight discrepancies in the numbers are due to round-off errors. The methodology produces an exact equality if you keep all the digits.
Now we have all we need to apply the certainty-equivalence discounting formula to value the option…
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Appendix 10C: “Certainty-Equivalent Valuation”…
Now we have all we need to apply the certainty-equivalence discounting formula to value the option:
Risk
E[r]
rf = 3%
34% 9%
Built Property
37% range
Devlpt Option
192% range
E[RP]
Slight discrepancies in the numbers are due to round-off errors. The methodology produces an exact equality if you keep all the digits.
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80. E[RPC] = 31% = (5.2)*6% = (5.2)*E[RPV].
Appendix 10C: “Certainty-Equivalent Valuation”…
Having computed (derived ) PV[C1], we can now back out the implied E[RPC] and RADR for the option…
Suppose the construction cost today (at Time 0) would be $89 if we built the project immediately, and the completed building today would be worth $100. Then if we did the development today (at Time 0), we would net $100 (today’s value of the completed building) minus the $89 construction cost, or $11. But we would lose the option to wait and decide next year. Thus, we would gain $11 but lose something also worth $12. (NPV = $11 - $12 = -$1 < 0.) So, obviously, it would be better to wait, not build the project today, hold onto the option, and decide next year what to do. Thus, the binomial certainty-equivalent model allows us not only to evaluate the real option that we have, but also to know how to make an optimal decision as to whether (when) to build or not.
Note also that we did not use (we did not need to use) “risk-neutral” probabilities or “risk-neutral dynamics”. This procedure uses “real” (“actual”) probabilities and expected future values and distributions. In other words, by knowing the actual volatility of the completed office building (which gave us the range between Vup and Vdown) and the actual expected return on that building (which gave us the probability of the “up” jump given the “up” and “down” future values), we could completely specify the model as we did here. (See: T.Arnold & T.Crack, “Option Pricing in the Real World”, Univ. of Richmond, 2003.)
Which implies that the option has 31/6 = 5.2 times the investment risk that the built office property has, as:
E[RPC] = 31% = (5.2)*6% = (5.2)*E[RPV].
However, if we didn’t already know the option’s risk premium (31%), or hadn’t already done the certainty-equivalent valuation, we wouldn’t be able to use the RADR method of DCF valuation.
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Invoke “linear pricing” principle (aka “Law of One Price”):
Same “Price of Risk” must apply to all assets at a given time.
Same expected return risk premium per unit of risk (ex ante).
Otherwise, “not fair” (people will trade until “One Price” holds).
Among “derivative” assets, relative risk can be measured by volatility (or return outcome spread range) ratio between the derivative & the underlying asset…
Risk
E[r]
rf = 3%
34% 9%
Built Property
37% range
Devlpt Option
192% range
E[RP]
(Now we know this.)
(Now we can compute this.)
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82. © 2014 OnCourse Learning. All Rights Reserved.
Appendix 10C: “Certainty-Equivalent Valuation”…
Underlying asset outcome spread (Vup – Vdown) / V(0):
Up outcome: ($113 – $94) / $94 = 19/94 = +20%
Down outcome: ($79 – $94) / $94 = -15/94 = -17%
Spread = (+20% – (-17%)) = 37%.
Derivative asset outcome spread (Cup – Cdown) / C(0):
Up outcome: ($23 – $12) / $12 = 11/12 = %
Down outcome: ($0 – $12) / $12 = -12/12 = -100%
Spread = (+92% – (-100%)) = 192%.
Suppose the construction cost today (at Time 0) would be $89 if we built the project immediately, and the completed building today would be worth $100. Then if we did the development today (at Time 0), we would net $100 (today’s value of the completed building) minus the $89 construction cost, or $11. But we would lose the option to wait and decide next year. Thus, we would gain $11 but lose something also worth $12. (NPV = $11 - $12 = -$1 < 0.) So, obviously, it would be better to wait, not build the project today, hold onto the option, and decide next year what to do. Thus, the binomial certainty-equivalent model allows us not only to evaluate the real option that we have, but also to know how to make an optimal decision as to whether (when) to build or not.
Note also that we did not use (we did not need to use) “risk-neutral” probabilities or “risk-neutral dynamics”. This procedure uses “real” (“actual”) probabilities and expected future values and distributions. In other words, by knowing the actual volatility of the completed office building (which gave us the range between Vup and Vdown) and the actual expected return on that building (which gave us the probability of the “up” jump given the “up” and “down” future values), we could completely specify the model as we did here. (See: T.Arnold & T.Crack, “Option Pricing in the Real World”, Univ. of Richmond, 2003.)
Which implies that the option has 192/37 = 5.2 times the investment risk that the built office property has, the same as the risk premium ratio we just computed:
E[RPC] / E[RPV] = 31% / 6% = 5.2
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The “Law of One Price”…
Risk
E[r]
rf = 3%
34% 9%
Built Property
37% range
Devlpt Option
192% range
E[RP]
If this relationship does not hold, then there are “super-normal” (disequilibrium) profits (expected returns) to be made somewhere, and correspondingly “sub-normal” profits elsewhere, across the markets for: Land, Built Property, and Bonds (“riskless” CFs).
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Appendix 10C: “Certainty-Equivalent Valuation”…
Fundamentally, what we are doing here is making sure that the expected return risk premium per unit of risk is the same between an investment in the development option, and an investment in already-build (stabilized) property. For example, for stabilized property:
For the development option:
Thus, this procedure for the valuation of the development option is equivalent to a cross-market equilibrium condition: that the expected return risk premium per unit of risk (or the “risk-adjusted return”) presented by the investments should be the same between the market for development options and the market for stabilized built property. Otherwise, investors will buy one type of asset and sell the other type until prices equilibrate the risk-adjusted returns across the two markets.
Slight discrepancies in the numbers are due to round-off errors. The methodology produces an exact equality if you keep all the digits.
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Appendix 10C: “Certainty-Equivalent Valuation”…
General formula to value derivative “C” based on underlying asset “V”…
If the underlying asset (the stabilized built property, “V1” ) is not equal volatility and perfectly correlated with the “Mkt” (i.e., equal “risk” as the “Mkt” ), then you would use (E[rV]-rf ) / (Vup% – Vdown%) in the above formula, in place of (E[rMkt]-rf ) / (Mktup% – Mktdown%) in the numerator.
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86. RPC = 31% = 5.2*6% = 5.2*RPV (= 5.2*RPS).
Appendix 10C: “Certainty-Equivalent Valuation”…
The equivalent valuation using the risk-adjusted discount rate approach is:
Which implies that the option has 31/6 = 5.2 times the investment risk that the office building has (and that the stock market has), as:
RPC = 31% = 5.2*6% = 5.2*RPV (= 5.2*RPS).
However, if we didn’t already know the option’s risk premium (31%), or hadn’t already done the certainty-equivalent approach, we wouldn’t be able to use the risk-adjusted discount rate approach.
The certainty-equivalent approach does not require prior knowledge of RPC. It only requires knowledge of Cup , Cdown , and the “up” probability (70% in our example).
This makes the certainty-equivalent approach very useful for option valuation. Most development projects are essentially “options”.
Suppose the construction cost today (at Time 0) would be $89 if we built the project immediately, and the completed building today would be worth $100. Then if we did the development today (at Time 0), we would net $100 (today’s value of the completed building) minus the $89 construction cost, or $11. But we would lose the option to wait and decide next year. Thus, we would gain $11 but lose something also worth $12. (NPV = $11 - $12 = -$1 < 0.) So, obviously, it would be better to wait, not build the project today, hold onto the option, and decide next year what to do. Thus, the binomial certainty-equivalent model allows us not only to evaluate the real option that we have, but also to know how to make an optimal decision as to whether (when) to build or not.
Note also that we did not use (we did not need to use) “risk-neutral” probabilities or “risk-neutral dynamics”. This procedure uses “real” (“actual”) probabilities and expected future values and distributions. In other words, by knowing the actual volatility of the completed office building (which gave us the range between Vup and Vdown) and the actual expected return on that building (which gave us the probability of the “up” jump given the “up” and “down” future values), we could completely specify the model as we did here. (See: T.Arnold & T.Crack, “Option Pricing in the Real World”, Univ. of Richmond, 2003.)
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Appendix 10C: “Certainty-Equivalent Valuation”…
Formula for derivative asset valuation with Binomial outcome distribution (as before):
Formula for derivative asset valuation with general outcome distribution (continuous or discrete):
If the underlying asset (the stabilized built property, “V1” ) is not equal volatility and perfectly correlated with the “Mkt” (i.e., equal “risk” as the “Mkt” ), then you would use (E[rV]-rf ) / (Vup% – Vdown%) in the above formula, in place of (E[rMkt]-rf ) / (Mktup% – Mktdown%) in the numerator.
Formula for underlying asset valuation with general outcome distribution (based on CAPM):
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Appendix 10C: “Certainty-Equivalent Valuation”…
The “binomial world” is more realistic and useful than you might think, because we can define the temporal length of a period to be as short as we want, and we can link and branch individual binomial elements together to make a “tree” of asset value possibilities over time.
More generally for a cash flow T periods in the future with expected value E0[VT], define the certainty-equivalent amount CEQ[VT]. We have:
The first “aside” here allows the binomial model to approach a continuous-time, continuous future value distribution (log-normal probability distribution of future values of V, or normal distribution of instantaneous return on V) world as the time increment between periods in each binomial element approaches zero (i.e., as the number of periods of binomial “jumps” in value in any given chronological time span approaches infinity).
Regarding the third “aside” here, note that in the most important use of the certainty-equivalent approach (in real options valuation), we usually only apply the approach one period at a time.
The second “aside” here shows how the certainty-equivalent valuation model is consistent with the most widely-employed normative model of risk and return equilibrium in the capital markets, the so-called “Capital Asset Pricing Model” (CAPM) of Sharp & Lintner.
According to the CAPM, the “risk that matters” in any asset (i.e., the risk that is priced in equilibrium in the capital market, i.e., the risk that affects the market’s required expected total return on that asset) equals that asset’s marginal contribution to the risk (variance) in the “market portfolio” (the portfolio consisting of ALL assets). This marginal risk contribution is only the asset’s covariance with the market portfolio. In other words, “idiosyncratic risk”, the component of an asset’s return variance that can be diversified away in an investor’s portfolio and hence that is not reflected in the market’s return variance, does not matter and is not priced (hence, not reflected in the market’s required expected return for that asset).
Note that by the definition of “return” and of “covariance”: COV[rV,rMkt]V0 = COV[V1,rMkt]. This is because: rV = V1/V0 – 1 in which the only stochastic (uncertain) component as of Time 0 is V1.
To relate the CAPM formulation to the Binomial formulation, note that “Beta” in CAPM can also be equivalently expressed as: CORREL[rV,rMkt](STD[rV]/STD[rMkt]). Thus, the ratio of risk between any two assets C and V is the ratio of their correlations with the market times their ratio of volatilities. In the case where C is a derivative of V, the two assets are perfectly correlated, which gives them the same correlation coefficient with the market portfolio: CORREL[rC,rMkt] = CORREL[rV,rMkt]. Thus, relative risk (the ratio of the betas), simply equals the ratio of the two assets’ volatilities. In the binomial world, this ratio of volatilities exactly equals the ratio of outcome percentage spreads.
For underlying assets, a binomial model consistent with the CAPM gives “Beta” equal to the ratio of the subject asset’s “up” – “down” percentage movements divided by the Market’s “up” minus “down” percentage movements, the same as the certainty-equivalent formula presented on the previous slides (multiplied by V0 to get the risk premium in dollars instead of percents). In a CAPM binomial world, the only component of an asset’s “up” and “down” movement that would matter in the pricing of that asset’s risk (the only component that would affect its risk-adjusted discount rate or its certainty-equivalence discount from expected value) would be that component that is perfectly correlated with the Market Portfolio. On the other hand, a derivative asset can be priced relative to its underlying asset merely using the ratio of the two assets’ betas (ratio of their percentage outcome spreads in the binomial world).
It is important to note that a CAPM-based certainty-equivalence valuation formula such as that presented in the middle panel of this slide is valid for evaluating “underlying assets” to the extent that the CAPM model is true (i.e., to the extent that the CAPM is a complete and accurate model of equilibrium asset pricing in the market, the formula will be a complete and accurate model for underlying asset valuation). To the extent that the CAPM model is not complete, we would need to adjust the certainty-equivalence valuation formula to get an accurate formula so as to reflect what is left out of the CAPM (e.g., perhaps “Fama-French factors”, or whatever), in order to evaluate an “underlying asset”.
On the other hand, we can price derivative assets (such as options) using the certainty equivalence formula here, whether or not the CAPM is a true and complete model of the market’s underlying asset risk pricing. While we apparently need to know or assume a value for E[rV] in order to implement this formula and price the option (and E[rV] presumably reflects whatever is the “true” model of asset market equilibrium risk pricing for the underlying asset), it turns out that in fact the value we obtain for the option will be independent of the value we assume for E[rV] in the formula once we embed the formula in an appropriate model of underlying asset value dynamics (such as the binomial model we will present in a later lecture). We do, however, need to know the magnitudes of the Vup% – Vdown% and Cup – Cdown spreads, which requires knowledge of the current value and volatility of the underlying asset. The current value of the underlying asset is itself determined by an equilibrium asset pricing model.
“Derivative assets” are by definition assets whose returns can be perfectly replicated by some combination of other assets (as for example in our replicating portfolio we described for the call option, with a long and short position in the underlying asset and bonds), and are distinguished from “underlying assets” in this regard. Because derivative assets are perfectly correlated with underlying assets, it does not matter what is the market’s “true” risk valuation model (whether this is the CAPM or something else). We can price the derivative asset using the type of replicating portfolio and arbitrage argument described in the preceding slides. The result is a riskless hedge. As the hedge is riskless, the market’s risk model does not matter. Whatever is the market’s true model of risk valuation, this will be reflected in the current prices of the underlying assets, and the derivative asset can be priced based on, and given, the underlying assets’ current prices. In this sense, our price model of the derivative asset is only relative to the observed or presumed current price of the relevant underlying assets (such as the built property and the bonds described in our replicating portfolio).
Given the ability to construct a riskless hedge (a perfectly replicating portfolio), the risk-neutral binomial option valuation formula typically presented in MBA finance courses follows directly. The generic certainty-equivalence valuation formula is mathematically equivalent to this risk-neutral valuation in the binomial world. (The risk-neutral binomial model is mathematically equivalent to the certainty-equivalence model presented here, and is found simply by replacing rV with rf everywhere both in the valuation formula and in the computation of the binomial jump probabilities (“p ”) that will be described in a subsequent lecture.)
The certainty equivalence formula for option (“derivative asset”) valuation can be derived as follows (see Arnold & Crack, op.cit.). Start with the basic risk-adjusted discount rate valuation formula for the option:
Ct = E[Ct+1] / (1 + E[rC]) = E[Ct+1] / (1 + rf + E[RPC]) (1)
Algebraic manipulation puts this in certainty-equivalence form, to separate the risk valuation from the time value of money:
Ct = (E[Ct+1] – CtE[RPC] ) / (1 + rf ) (2)
Now define a concept known as the “option elasticity” that we will label η (“eta”), which is the percentage change in option value C , associated with a one percent change in the value of the underlying asset V : η = (dC/C) / (dV/V) = (dC/dV)/(C/V) = VdC/(CdV).
For example, if the underlying asset experiences a 1% increase in value, and η = 5, then the option will experience a 5% increase in value. If the underlying asset experiences a 2% decline in value and η = 3, then the option will experience a 6% decline in value.
Because the option, as a derivative asset , is perfectly correlated with the underlying asset, the risk in the option’s return (as the capital market values that risk, no matter which model the capital market uses to value risk in underlying assets as long as it is based on return variability) will equal exactly η times the risk in the underlying asset’s return. Thus:
E[RPC] = ηE[RPV] (3)
Of course, by definition: E[RPV] = E[rV] – rf (4)
Substituting (3) and (4) into (2) we have:
Ct = (E[Ct+1] – Ct η(E[rV] – rf ) ) / (1 + rf ) (5)
In a binomial world, the option elasticity is defined relative to the single-period “up” and “down” movements in the option value and in the underlying asset value. In particular, by definition:
η = (Cup% – Cdown% ) / (Vup% – Vdown% ), where, as we have defined these before, (Cup% – Cdown% ) and (Vup% – Vdown% ) are the one period “up” – “down” range in percentage movements in the option and in the underlying asset , respectively (relative to their previous values). For example, for the numerical example we will use on subsequent slides: (Cup% – Cdown% ) = ( 23/12 – 0/12 ) = 190%; and (Vup% – Vdown% ) = ( 113/94 – 79/94 ) = 120% - 84% = 36%. Thus: η = 190/36 = 5.3.
Of course, when we multiply the percentage change in Ct by Ct , we get the dollar change in Ct : Ct (Cup% – Cdown% ) = Cup – Cdown . Thus, by definition:
Ct η = Ct (Cup% – Cdown% ) / (Vup% – Vdown% ) = (Cup – Cdown ) / (Vup% – Vdown% ) (6)
Substituting (6) into (5) we obtain the generic certainty-equivalence valuation formula for the option presented as “Formula (2)” on the earlier slide:
Ct = (E[Ct+1] – Ct η(E[rV] – rf ) ) / (1 + rf ) = (E[Ct+1] – ((Cup – Cdown ) / (Vup% – Vdown% ))(E[rV] – rf ) ) / (1 + rf )
Ct = [E[Ct+1] – (Cup – Cdown )(E[rV] – rf )/(Vup% – Vdown% ) ] / (1 + rf ) = “Formula (2)”.
Expressing this as:
Ct = CEQ[Ct+1] / (1 + rf ), we see that the certainty-equivalent value of the option next period is:
CEQ[Ct+1] equals E[Ct+1] minus a “risk premium” in dollars.
This risk premium in dollars equals the range (or risk) in the option value in dollars, (Cup – Cdown ), times the market’s “price of risk” in the option’s underlying asset: (E[rV] – rf )/(Vup% – Vdown% ), which in turn is seen to equal the market’s percentage risk premium in the underlying asset’s return: (E[rV] – rf ) per unit of range (or risk) in the underlying asset’s value measured in percent (of the asset’s previous value): (Vup% – Vdown% ).
As noted, this valuation formula in the context of the binomial model to be presented in a subsequent lecture equates to the risk-neutral binomial model that follows directly from the replicating portfolio argument, with rV being replaced by rf in both the certainty-equivalence valuation formula and in the binomial jump probability formula.
Provided the risk accumulates continuously through the entire time to T.
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89. Intuition in the CEQ formula:
The certainty equivalent value next year is the downward adjusted value of the risky expected value for which the investment market would be indifferent between that value and a riskfree bond value of the same amount…
The certainty equivalent value next year is the expected value minus a risk discount.
The risk discount consists of the amount of risk in the next year’s value as indicated by the range in the possible outcomes times… 
times the market price of risk (for same “type” of risk, e.g., office building*)
*dvlpt option is perfectly correlated with building
The market price of risk is the market expected return risk premium per unit of return risk for the same asset, the ratio of…
the market expected return risk premium divided by the range in the corresponding return possible outcomes.
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