Transcript CHAPTER 13 LECTURE: LEVERAGE.

Chapter 13
LEVERAGE
(The use of debt)
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The analogy of physical leverage & financial leverage...
“Give me a place to stand, and I will move the earth.”
- Archimedes (287-212 BC)
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Terminology...
“Leverage”
“Debt Value”, “Loan Value” (L) (or “D”).
“Equity Value” (E)
“Underlying Asset Value” (V = E+L):
"Leverage Ratio“ = LR = V / E = V / (V-L) = 1/(1-L/V)
(Not the same as the “Loan/Value Ratio”: L / V,or “LTV” .)
“Risk”
The RISK that matters to investors is the risk in their total
return, related to the standard deviation (or range or spread)
in that return.
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L e v e ra g e R a tio & L o a n -to -V a lu e R a tio
25
LR 
20
1
1  LTV
LR
15
10
5
0
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
LTV
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L e v e ra g e R a tio & L o a n -to -V a lu e R a tio
100%
90%
80%
LTV  1 
70%
LTV
60%
1
LR
50%
40%
30%
20%
10%
0%
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
LR
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Effect of Leverage on Risk & Return
(Numerical Example)…
Example Property & Scenario Characteristics:
Current (t=0) values (known for certain):
E0[CF1] = $800,000
V0 = $10,000,000
Possible Future Outcomes are risky (next year, t=1):
"Pessimistic" scenario (1/2 chance):
CF1 = $700,000; V1 = $9,200,000.
"Optimistic" scenario (1/2 chance):
CF1 = $900,000; V1 = $11,200,000.
$11.2M
+ 0.9M
Property:
50%
$10.0M
50%
$9.2M
+0.7M
Loan:
$6.0M
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100
%
$6.0M
+0.48M
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Case I: All-Equity (No Debt: Leverage Ratio=1, L/V=0)...
Item
Pessimistic
700/10000= 7%
Inc. Ret. (y):
Ex Ante:
(1/2)7% + (1/2)9% = 8%
1%
RISK:
App. Ret. (g):
(9.2-10)/10 = -8%
Ex Ante:
(1/2)(-8) + (1/2)(12) = +2%
10%
RISK:
Optimistic
900/10000= 9%
(11.2-10)/10=+12%
Case II: Borrow $6 M @ 8%, with DS=$480,000/yr
(Leverage Ratio=2.5, L/V=60%)...
Item
Pessimistic
Optimistic
(0.7-0.48)/4.0= 5.5%
(0.9-0.48)/4.0= 10.5%
Inc. Ret.:
Ex Ante:
(1/2)5.5 + (1/2)10.5 = 8%
2.5%
RISK:
App. Ret.:
(3.2-4.0)/4.0 = -20%
(5.2-4.0)/4.0 = +30%
Ex Ante:
(1/2)(-20) + (1/2)(30) = +5%
25%
RISK:
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Exhibit 13-2: Typical Effect of Leverage on Expected Investment
Returns
Property
Levered Equity
Debt
Initial Value
$10,000,000
$4,000,000
$6,000,000
Cash Flow
$800,000
$320,000
$480,000
Ending Value
$10,200,000
$4,200,000
$6,000,000
8%
2%
10%
Income Return
Apprec.Return
Total Return
8%
5%
13%
8%
0%
8%
E xh ib it 1 3 -3 : S e ns itivity A n a lys is of E ffec t o f L e ve ra g e o n R isk in E qu ity R e tu rn C om p o n e n ts, as M e a su re d b y P e rc e n ta g e
R a n g e in P os sib le R e tu rn O u tc om e s . ($ V a lu es in m illio ns )
OPT
In itia l V a lu e
C a s h F lo w
E n d in g V alu e
P ro p e rty (L R = 1 )
PES
RANGE
L e ve re d E q u ity (L R = 2 .5 )
OPT
PES
RANGE
OPT
D e b t (L R = 0 )
PES
RANGE
$ 1 0 .0 0
$ 1 0 .0 0
NA
$ 4 .0
$ 4 .0
NA
$ 6 .0
$ 6 .0
NA
$ 0 .9
$ 0 .7
 $ 0 .1
$ 0 .4 2
$ 0 .2 2
 $ 0 .1
$ 0 .4 8
$ 0 .4 8
0
$ 1 1 .2
$ 9 .2
 $ 1 .0
$ 5 .2
$ 3 .2
 $ 1 .0
$ 6 .0
$ 6 .0
0
In c om e R e tu rn
9%
7%
1%
1 0 .5 %
5 .5 %
 2 .5 %
8%
8%
0
A p p re c .R e tu rn
12%
-8 %
10%
30%
-2 0 %
25%
0%
0%
0
T o ta l R etu rn
21%
-1 %
11%
4 0 .5 %
-1 4 .5 %
 2 7 .5 %
8%
8%
0
O P T = O u tc om e if "O p tim istic " S ce n a rio o c c u rs .
P E S = O u tc om e if "P e ss im istic " S c e n a rio o cc u rs.
R A N G E = H a lf th e d iffe re n c e b etw e e n "O p tim is tic " S c e n a rio o u tc om e a n d "P ess im istic " S c e n a rio o u tco m e.
N o te : In itia l va lu e s a re k n o w n d e term inistic a lly, as th e y a re in p re s e n t, n o t futu re , tim e, s o th e re is n o ran g e .
 Return risk (y,g,r) directly proportional to Levg Ratio (not L/V).
 E[g] directly proportional to Leverage Ratio.
 E[r] increases with Leverage, but not proportionately.
 E[y] does not increase with leverage (here).
 E[RP] = E[r]-rf is directly proportional to Leverage Ratio (here)…
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Exhibit 13-4: Effect of Leverage on Investment Risk and Return:
The Case of Riskless Debt...
E xp ected
T o tal
R eturn
13%
RP
5%
10%
RP
2%
8%
rf
0
R isk less
M ortgage
1
U n lev ered
E qu ity:
U n d erlyin g
P rop erty
8%
2 .5
R isk
F acto r
L e verage
R atio (L R )
L evered
E qu ity:
6 0% LT V
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Exhibit 13-5: Effect of Leverage on Investment Risk and Return:
The Case of Risky Debt...
E xp ected
T o tal
R eturn
13%
RP
10%
8%
RP
RP
7%
4%
2%
r f= 6 %
=6%
rf
6%
R isk
F actor
0
0
R isk y
M ortgage
1
2.5
U n lev ered
E qu ity:
P rop erty
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L e verage
R atio (L R )
L evered
E qu ity:
6 0% LT V
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Section 13.3
A really useful formula. . .
The "Weighted Average Cost of Capital" (WACC) Formula . . .
rP = (L/V)rD + [1-(L/V)]rE
Derivation of the WACC Formula:

V
V

V
V

E

V
D
V

E E
V

E
D D
V
D
V = E+D

E E
V
D  E
D D

 1 


V  E
V D

E

D D
V
 V  D  E D D



D
V D
 V  E
Where: rE = Levered Equity Return,
rP = Property Return,
 WACC :
rD = Debt Return,
LTV=Loan-to-Value Ratio (D/V).
rP  (1  LTV ) rE  ( LTV ) rD
Invert for equity formula:
rE 
rP  ( LTV ) rD
(1  LTV )
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Or, equivalently, if you prefer . . .
E = V-D

E

E

E
E

V V
E V
V

E

D

E
(V  E )  D
E
D
V V
E V

V V
E V

D D

E D
V V
E V

D D
E D
D V  V
D 
V
 D
   1

 


D
E V
D 
E
 D
 WACC :
rE  ( LR ) rP  (1  LR ) rD  rD   rP  rD  LR
Where: rE = Levered Equity Return,
rP = Property Return,
rD = Debt Return,
LR=Leverage Ratio (V/E).
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Using the WACC formula in real estate:
The "Weighted Average Cost of Capital" (WACC) Formula . . .
rP = (L/V)rD + [1-(L/V)]rE
(L/V) = Loan/value ratio
rD = Lender's return (return to the debt)
rE = Equity investor's return.
Apply to r, y, or g. . .
E.g., in previous numerical example:
E[r] = (.60)(.08) + (.40)(.13) = 10%
E[y] = (.60)(.08) + (.40)(.08) = 8%
E[g] = (.60)(0) + (.40)(.05) = 2%
(Can also apply to RP.)
In real estate,
Difficult to directly and reliably observe levered return,
But can observe return on loans,
and can observe return on property (underlying asset).
So, "invert" WACC Formula:
Solve for unobservable parameter as a function of the observable parameters:
rE = {rP - (L/V)rD} / [1 - (L/V)]
(Or in y or in g.)
(In y it’s “cash-on-cash” or “equity cash yield”)
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Note:
WACC based on accounting identities:
Assets = Liabilities + Owners Equity,
Property Cash Flow = Debt Cash Flow +
Equity Cash Flow
WACC is approximation,
Less accurate over longer time interval
return horizons.
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Using WACC to avoid a common mistake. . .
Suppose REIT A can borrow @ 6%, and REIT B @ no less than 8%.
Then doesn’t REIT A have a lower cost of capital than REIT B?
Answer: Not necessarily. Suppose (for example):
REIT A:
D/E = 3/7.
 D/V = L/V = 30%.
REIT B:
D/E = 1.
 D/V = L/V = 50%.
& suppose both A & B have cost of equity = E[rE] = 15%.
Then:
WACC(A)
=(0.3)6% + (0.7)15% = 1.8% + 10.5% = 12.3%
WACC(B)
=(0.5)8% + (0.5)15% = 4% + 7.5% = 11.5%
So in this example REIT A has a higher cost of capital than B,
even though A can borrow at a lower rate. (Note, this same
argument applies whether or not either or both investors are
REITs.) You have to consider the cost of your equity as well as the
cost of your debt to determine your cost of capital.
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13.4
“POSITIVE” & “NEGATIVE” LEVERAGE
“Positive leverage” = When more debt will
increase the equity investor’s (borrower’s)
return.
“Negative leverage” = When more debt will
decrease the equity investor’s (borrower’s)
return.
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“POSITIVE” & “NEGATIVE” LEVERAGE
Whenever the Return Component is higher in the
underlying property than it is in the mortgage
loan, there will be "Positive Leverage" in that
Return Component...
See this via The “leverage ratio” version of the
WACC. . .
rE = rD + LR*(rP-rD)
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Derivation of the Leverage Ratio Version of the WACC:
E = V-D

E

E

E
E

V V
E V
V

E

D

E
(V  E )  D
E
D
V V
E V

V V
E V

D D

E D
V V
E V

D D
E D
D V  V
D 
V
 D
   1

 


D
E V
D 
E
 D
 WACC :
rE  ( LR ) rP  (1  LR ) rD  rD   rP  rD  LR
Where: rE = Levered Equity Return,
rP = Property Return,
rD = Debt Return,
LR=Leverage Ratio (V/E).
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E x h ib it 1 3 -6 : T y p ic a l re la tiv e e ffe c t o f le v e ra g e o n
in c o m e a n d g ro w th c o m p o n e n ts o f in v e s tm e n t
re tu rn (n u m e ric a l e x a m p le )...
P ro p e rty to ta l re tu rn (r P ): 1 0 .0 0 %
C a p ra te (y P ): 8 .0 0 %
P o s itiv e c a s h -o n -c a s h le v e ra g e ...
L o a n In te re s t ra te (r D ): 6 .0 0 %
M o rtg a g e C o n s ta n t (y D ): 7 .0 0 %
E q u ity re tu rn c o m p o n e n t:
LR
LTV
yE
gE
rE
1
0%
8 .0 0 %
2 .0 0 %
1 0 .0 0 %
2
50%
9 .0 0 %
5 .0 0 %
1 4 .0 0 %
3
67%
1 0 .0 0 %
8 .0 0 %
1 8 .0 0 %
4
75%
1 1 .0 0 %
1 1 .0 0 %
2 2 .0 0 %
5
80%
1 2 .0 0 %
1 4 .0 0 %
2 6 .0 0 %
N e g a tiv e c a s h -o n -c a s h le v e ra g e ...
L o a n In te re s t R a te (r D ): 8 .0 0 %
M o rtg a g e C o n s ta n t (y D ): 9 .0 0 %
E q u ity re tu rn c o m p o n e n t:
LR
LTV
yE
gE
1
0%
8 .0 0 %
2 .0 0 %
2
50%
7 .0 0 %
5 .0 0 %
3
67%
6 .0 0 %
8 .0 0 %
4
75%
5 .0 0 %
1 1 .0 0 %
5
80%
4 .0 0 %
1 4 .0 0 %
rE
1 0 .0 0 %
1 2 .0 0 %
1 4 .0 0 %
1 6 .0 0 %
1 8 .0 0 %
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Section 13.5
e.g.:
Total: 10% =
(67%)*6%+(33%)*18%
Yield: 8% =
(67%)*7% + (33%)*10%
Growth: 2% =
(67%)*(-1%)+(33%)*8%
Leverage skews total
return relatively toward
growth component,
away from current
income yield.
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SUMMARY OF LEVERAGE EFFECTS...
(1) Under the typical assumption that the loan is less risky
than the underlying property, leverage will increase the ex
ante total return on the equity investment, by increasing
the risk premium in that return.
(2) Under the same relative risk assumption, leverage will
increase the risk of the equity investment, normally
proportionately with the increase in the risk premium
noted in (1).
(3) Under the typical situation of non-negative price
appreciation in the property and non-negative
amortization in the loan, leverage will usually shift the
expected return for the equity investor relatively away
from the current income component and towards the
growth or capital appreciation component.
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For example, across the four properties in the HBS “Angus Cartwright”
Case, the rule works in 11 out of 12 cases. (All except Fowler YC…)
Alison IRR Attribution: Property, Debt, Equity…
Stony IRR Attribution: Property, Debt, Equity…
Prop
Debt
Equity
LevgSign LevtEffect
Prop
Debt
Equity
IRR
IY
CFC
7.58%
3.90%
12.42%
Pos
Pos
6.27%
5.16%
8.12%
Pos
Pos
2.24%
0.64%
3.75%
Pos
YC
-0.91%
-1.89%
0.52%
Pos
Interaction
-0.03%
-0.02%
0.03%
7.31%
4.23%
11.77%
Pos
Pos
5.84%
6.80%
3.66%
Neg
Neg
Pos
IRR
IY
CFC
2.38%
1.08%
6.75%
Pos
Pos
Pos
YC
-0.88%
-3.59%
1.28%
Pos
Pos
Interaction
-0.03%
-0.06%
0.09%
Ivy IRR Attribution: Property, Debt, Equity…
LevgSign LevtEffect
Fowler IRR Attribution: Property, Debt, Equity…
Prop
Debt
Equity
LevgSign LevtEffect
Prop
Debt
Equity
IRR
IY
CFC
6.10%
4.09%
10.27%
Pos
Pos
5.72%
4.67%
8.07%
Pos
Pos
1.29%
0.81%
1.90%
Pos
YC
-0.90%
-1.38%
0.29%
Pos
Interaction
-0.01%
-0.01%
0.01%
LevgSign LevtEffect
6.54%
5.10%
10.41%
Pos
Pos
5.08%
5.72%
2.78%
Neg
Neg
Pos
IRR
IY
CFC
2.67%
1.03%
10.74%
Pos
Pos
Pos
YC
-1.18%
-1.63%
-2.81%
Pos
Neg
Interaction
-0.04%
-0.02%
-0.31%
Note that in the simple return, r = y + g, while in the IRR the correspondent to “g” breaks out into
two components, CFC + YC. That is:
IRR ≈ IY + (CFC + YC) ≈ y + g.
The pos/neg levg rule will always hold for IY, and therefore will always hold for the combination
(CFC+YC), but it may not always hold separately for both CFC and YC (as we see in Cartwright).
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Note: The preceding simple rule about what determines “pos” vs
“neg” leverage (relationship betw prop vs dbt return) does not
always hold with IRR attributes, because the WACC does not apply
exactly for multi-period returns or components…