rpv: present value reconsidered 0917 shimko.pdf · • no taxes, transaction costs, informational...
TRANSCRIPT
WHAT IF … ?
• Because of imperfect competition for profitable projects, you could earn
a return on idiosyncratic risk?
• Due to regulation, you had to allocate risk capacity optimally over time?
• You wanted to value an asset in the context of your nondiversified,
nonmarketable portfolio?
• How would you price assets in this case?
CAN YOU ANSWER THESE QUESTIONS?
• Assume (for now) that an asset’s cash flows are uncorrelated with every
priced market risk factor.
• If the expected cash flow is positive, should the value ever be negative?
• Should valuation ever be non-additive?
• Should the value ever depend on correlation across time?
• Should the order of cash flows (i.e. resolution of uncertainty) ever affect the
valuation?
• Should hedging ever add value?
• If you answered “yes” to any of these questions, a case can be made for RPV
JOHN LINTNER PRESENTED WAS ON THE RIGHT TRACK IN 1965
• Under the assumptions of the CAPM, and
• Given a set of one period projects with different risk characteristics,
• Select the projects to maximize value subject to a budget constraint
• This is an integer programming problem (ACCEPT/REJECT)
• Lagrangian multiplier on the constraint is the shadow price of internal capital,
i.e. the corporate hurdle rate
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BUT WHAT WE TEACH MBAS TODAY HAS GOTTEN OFF TRACK
• Assuming we can recapitalize the firm, then the capital constraint is nonbinding,
• and the CAPM acceptance criteria hold:
• Discount cash flows of each project at its CAPM-based hurdle rate
• Whenever NPV>0, accept the project
• What have we done?
• Effectively we have imposed all the CAPM assumptions
• Perfect markets
• No taxes, transaction costs, informational differences
• Unrestricted short sales allowed, divisible assets
• AND homogeneous investors with CRRA
• PLUS we have assumed that firms have infinite access to fairly priced capital
• PLUS we have assumed that risk is a constant percentage of value over time
• AND we’ve barely changed the model since the 1960s
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THE NEED FOR NEW VALUATION METHODSHAS NEVER BEEN GREATER
• Firms with limited capital need to maximize value subject to available capital, implying a cost of risk (i.e. the Lagrange multiplier on the risk constraint), especially true for financial institutions
• Lintner (1965)
• Incomplete markets support ranges of equilibrium prices which may be bounded by “good-deal” restrictions on the maximum Sharpe ratio, or other variables related to standard deviation
• Carr, Geman & Madan (2001), Campbell & Saá-Requejo (2000)
• In principal-agent formulations, risk averse managers maximize expected discounted utility, a concave function of future value, implying a “cost of risk”
• In practice, corporations use hurdle rates that far exceed predicted rates
• Jagannathan et al (2016, 2011) consistent with pricing of operational risk and investment timing options
• Maier & Tarhan (2004) – Hurdle rates in practice exceed arbitrage-based estimates by 5%
• The premium correlates with idiosyncratic risk
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SO WHY HAVEN’T WE DEVELOPEDCASH FLOW VALUATION METHODS THAT…
1. Allow different private valuations for heterogeneous market participants
2. Lead to negative valuations as risk becomes too large
3. Consider multiperiod cash flows, which are not spanned by market risk factors
• Be sensitive to correlations between cash flows across time, however they arise
• Be sensitive to risk and different patterns of risk resolution
4. Address firms with capital constraints, agency problems, undiversified ownership
structure and other market imperfections
5. Apply equally well to financial institutions and corporations generally: i.e. help
finance and insurance “converge”
6. Integrate benchmark pricing with the pricing of unspanned risks
7. Explain the difference between hurdle rates in theory and practice
8. Allocate risk charges appropriately by period
9. Be used to price both cash and derivative instruments
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Valuation Issues
Private
Negative
Multiperiod
Imperfections
Convergence
Integration
Hurdle rate puzzle
Allocation
Cash/Derivative
CAN WE AGREE ON A ONE PERIOD MODEL?
PROBLEM: C1 is normally distributed (,) and the risk-free rate is r. What is the value of C1 today?
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Should risk adjustments be based on covariance or variance?
Financial Economist
Under the single variable CAPM
or any single benchmark model,
𝑉0 =𝜇 − 𝜆 𝑐𝑜𝑣(𝐶1, 𝑟𝑚)
1 + 𝑟
P&C Insurance Actuary
According to the
equivalence principle,
𝑉0 =𝜇 − 𝜆
1 + 𝑟
AN INCLUSIVEGENERAL VALUATION EQUATION
Expected return of an asset
Expected capital gain
+ Expected cash flow
E[Vi+1|i] – Vi +
E[Ci+1|i]
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Required return of an
asset
Time value of money (risk-free)
+ Cost per unit of risk Risk
measure
rVi + k R[Vi+1|i]
=
=
=
MULTIPERIOD SEEMS SIMPLE AT FIRST…
• Value a set of cash flows at discrete times i{1..N} that are
multivariate normal with mean vector and covariance matrix
• Issues:
• Intertemporal correlations – may be strongly negative
• Incomplete markets
• Portfolio effects
• Nondiversifiable risks – even if idiosyncratic
• Constrained capital and balance sheets – implying a cost of risk
• CAPM/APT useful, but only for systemic risk pricing
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IS THERE A FINANCIAL ENGINEER IN THE ROOM?
• How would you value this?
• Reminder:
• Value a set of cash flows at discrete times i{1..N}
• Multivariate normal
• Mean vector and covariance matrix
• Assume for now
• Riskfree rate is zero
• Idiosyncratic risk, nondiversifiable
• Risk measure is the conditional dollar stdev of asset value
• Asset value (V) defined by sum of known and unknown
cash flows (C)
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USE BACKWARD INDUCTION TO VALUE CASH FLOWS
• Set required return equal to
expected return at time N-1 and
work backward
• i = information known at time i
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• VN = C1
• VN-1 = E(VN|N-1) – k Var(VN|N-1)½
• = Regression prediction of mean – k
(Regression residual variance) ½
• k = cost of risk as measured by standard
deviation
• VN-2 = E(VN-1|N-2) – k Var(VN-1|N-2)½
• …
• V0 = E(V1|0) – k Var(V1|0)½
• V0 = 1 – k (11)½
• Total risk = square root of the grand sum of the
covariance matrix
BUT WHAT IS K?
• Solving the GVE for k;
• This is the Sharpe Ratio, expressed in dollar terms. or the $Sharpe
Ratio
• In this measure of performance, risk need not be proportional to value.
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𝑘 =𝐸 𝑉𝑖+1 Φ𝑖 − 𝑉𝑖 − 𝑟𝑉𝑖
𝑅[𝑉𝑖+1|Φ𝑖]
HOW TO ALLOCATE RISK CHARGES OVER TIME14
𝐶 ≡𝐶𝑗
𝐶𝑁
𝜇 ≡𝜇𝑗
𝜇𝑁
Σ ≡Σ𝑗𝑗 Σ𝑗𝑁
Σ𝑁𝑗 Σ𝑁𝑁
Partition moments into known cash
flows (1..j) and unknown cash flows
(j+1..N)
𝐸(𝐶𝑁|𝐶𝑗) = 𝜇𝑁 + Σ𝑁𝑗Σ𝑗𝑗−1(𝐶𝑗 − 𝜇𝑗)
“Regression equation”
Σ𝑗 = Σ𝑁𝑁 − Σ𝑁𝑗Σ𝑗𝑗−1Σ𝑗𝑁 𝑤𝑖𝑡ℎ Σ0 = Σ
The (Issai) Schur complement of jj
Use conditional distributions of a
multivariate normal distribution to predict
unknown cash flows
The variance allocation to period j is the total variance reduction from j-1 to j: Variance Allocation = 1j-11 − 1j1
THE RPV PRESENT VALUE FORMULAAND ITS PROPERTIES
V0 = 1 – k (11)½
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• Selected Properties
• Idiosyncratic risk matters
• Correlation between cash flows matters
• Value is not generally additive
• Value is homogeneous of degree 1 in the
cash flows
• Incremental project valuation depends on
the order of the cash flows
• Negative values are possible under some
conditions
INCORPORATING BENCHMARKS
• The RPV model easily adapts to account for covariance risk and idiosyncratic risk
• Example: Energy loan
• Define for M time-neutralized market risk factors
• = (1, 2, … M) A vector of the cost of risk for each factor m=1..M
• CB = {cov(Ci,rim)} An (NxM) matrix of covariances of cash flows with returns on market
benchmarks
• R = Covariance matrix of residual cash flow risk by period, after controlling for priced
factors
• Then deduct the market risk charges to obtain the certainty equivalent valuation
• V0 = 1 – 𝚺𝐂𝐁′ 1 – k (1R1)½
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Sources of Return
Risk-free return on cash
Factor returns on traded
factors
Sharpe ratio for unspanned risks
EXAMPLE 1:CASH FLOW FOLLOWS A RANDOM WALK
• Suppose C0=100 and Ci+1=Ci+zi with =10 and r=0.
• Owner receives 5 payments C1…C5; Initial mean = $500
• Total variance equals sum of Schur compression
elements (right) in each period
• Allocation of risk by period
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Risk by period, Risk charges fall as
uncertainty isresolved
Schur Compression of Covariance
1
2
3
4
0
At time
EXAMPLE 1 CONTINUED:RPV CALCULATIONS
• Assume valuation requires a Sharpe Ratio of 2 for unhedgeable risks (k=2)
• The chart compares the traditional finance valuation (k=0) with the
RPV valuation
• Excess return per unit risk is verified to be constant and equal to 2
• The conventional IRR=13.01%
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EXAMPLE 2:CAPITAL STRUCTURE IN INCOMPLETE MARKETS
• Company has random walk cash flows from Example 1
• Debtholders and equity holders both value their claims with k=2
• Interest payments are tax deductible
• The corporate tax rate is 40%
• Default triggered if cash flow is insufficient to pay interest
• Penalty $25 per default
• What is the optimal debt level?
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• Steps
• Simulate cash flows and allocate
• Debt = Contractual
• Government = Tax proceeds
• Equity = Residual
• Use RPV to value equity
• Choose debt level (contractual
debt payment) to maximize equity
value to current equityholders
EXAMPLE 2 CONTINUED:RPV CALCULATIONS FOR CAPITAL STRUCTURE
• This is analogous to both…
• Merton (1974)
• DeAngelo and Masulis (1980)
• Optimal interest level (90-100)
maximizes equity value
• Equals firm value since equityholders get debt proceeds
• Debt credit spread is endogenous
to the model
• NOTE:
• Driven by cash flow modeling
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Debt Coupon
METHODOLOGY SUMMARY: SIMULATION IMPLEMENTATION
RPV: V0 = 1 – 𝚺𝐂𝐁′ 1 – k (1R1)½
1. Simulate cash flows, find covariances with priced benchmarks and over time
2. PV cash flow means and covariance matrices at the risk-free rate
3. Set k = your threshold Sharpe ratio, or calibrate to model
4. Add up the means
5. Subtract market risk charges
6. Subtract k times the square root of the grand sum of the covariance matrix
7. Accept the project if RPV>0
8. Reduce operational risks if reduction in risk charge exceeds cost of risk
reduction
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CONCLUSIONS
• RPV is a new valuation framework that provides a risk-free return on
cash, a covariance-based risk premium on market risks, and a
premium for residual risks based on a minimum required Sharpe ratio.
• The usual NPV is a nested special case, which obtains when the cost
of idiosyncratic risk, k=0.
• The RPV valuation framework is consistent with incomplete markets,
optimization of firm value with constraints on capital and agency-
theoretic firm valuation models.
• RPV is friendly to nonlinear relationships.
• RPV is tractable, flexible and straightforward to calibrate and
implement
• RPV addresses all the listed NPV issues mentioned on page 7.
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Valuation Issues
Private
Negative
Multiperiod
Imperfections
Convergence
Integration
Hurdle rate puzzle
Allocation
Cash/Derivative
NOTATION, ABBREVIATIONS & TERMINOLOGY
• Vi = Value of an asset at discrete time i, after cash flow is taken if applicable
• Time 0 being the present and time N being the maturity date
if applicable
• Ci = Stochastic cash flow at time i, mean i
• R = a defined risk measure
• k = the compensation to an investor per unit of risk taken
• i = Standard deviation of cash flow at time i
• ij = Correlation between cash flow at time i and time j
• i = information available at time i
• r = risk-free interest rate per period
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• = Expected return of an asset
• a, b: Regression coefficients for intercept and
slope
• M = a benchmark asset
• rM = Expected return on benchmark asset
• S = Stock price
• C = Call option price (sorry for repeat use of
letter!)
• T = Maturity date as applicable
REFERENCES
• Ammann, Manuel and Süss, Stephan and Verhofen, Michael, Sep 2010, “Do Implied Volatilities Predict Stock Returns?”
• Ang, Andrew, Robert J. Hodrick, Yuhang Xing and Xiaoyan Zhang. Feb 2006, “The Cross-Section of Volatility and Expected Returns" Journal of Finance 61, 259-299
• Bollerslev, Tim and Tauchen, George and Zhou, Hao, “Expected Stock Returns and Variance Risk Premia”, Feb 2009, Review of Financial Studies
• Burns, Fennell, David Carlson, Emilie Haynsworth and Thomas Markham, 1974, “Generalized Inverse formulas Using the Schur Complement”, SIAM J. Appl. Math., 26(2), pp 254–259.
• Carr, Peter, Helyette Geman and Dilip B. Madan, 2001, “Pricing and hedging in incomplete markets”, Journal of Financial Economics 62:1, pages 131-167
• Cochrane, John and Jesús Saá-Requejo, Feb 2000, “Beyond Arbitrage: Good-Deal Asset Price Bounds in Incomplete Markets”, Journal of Political Economy 108:1, pp 79-119
• DeAngelo, Harry and Masulis, Ronald W., “Optimal Capital Structure Under Corporate and Personal Taxation”, March 1980, Journal of Financial Economics 8:1, pp. 3-27
• Jagannathan, Ravi, David Matsa, Iwan Meier and Vefa Tarhan, June 2016, “Why do firms use high discount rates?”, Journal of Financial Economics 120:3, pp 445-463
• Jagannathan, Ravi, Iwan Meier and Vefa Tarhan, 2011, “The Cross-Section of Hurdle Rates for Capital Budgeting: An Empirical Analysis of Survey Data”, NBER Working Paper 16770.
• Lintner, John, “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets”, Feb 1965, Review of Economics and Statistics 47:1, p 13-37.
• McDonald, Robert and Daniel Siegel, Mar 1984, “Option Pricing When the Underlying Asset Earns a Below-Equilibrium Rate of Return: A Note”, Journal of Finance 39:1, pp 261-65
• McDonald, Robert, and Daniel Siegel, 1986, “The Value of Waiting to Invest”, Quarterly Journal of Economics 101(4), 707-728.
• Meier, Iwan and Vefa Tarhan, 2004?, “Corporate Investment Decision Practices and the Hurdle Rate Premium Puzzle”
• Merton, Robert C., May 1974, “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates”, Journal of Finance 29:2, pp 449-470.
• Ouellette, Diane Valérie, 1961, “Schur Complements and Statistics”, Linear Algebra and its Applications 36, pp. 187-295, Elsevier North Holland.
• Shimko, David, June 1997, “See Sharpe or Be Flat”, RISK Magazine.
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BONUS ROUND!25
VALUING A CASH FLOW IN INCOMPLETE MARKETS
• Use RPV to value a nonspannable cash flow ST at time T
• Let dS = S dz
• Let V(S,) = value of the cash flow with periods to maturity
• Consider the valuation V = S exp[-(r+k)]
• V earns a constant continuous Sharpe Ratio
• Proof
•𝐸 𝑑𝑉 −𝑟𝑉
𝑆𝑡𝑑𝐷𝑒𝑣(𝑑𝑉)=
𝑟+𝑘𝜎 𝑉 −𝑟𝑉
𝜎𝑉= 𝑘
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The cash flow is discounted usingThe riskless rate plus a risk premium of k
OPTION ON AN NONTRADED CASH FLOW27
• V = S exp[-(r+k)]
• Value of nontraded call option is C(V,)
• C(V,0)=Max(V-X,0)
• E(dC) = (r+k)VCV + ½2V2CVV − V
• StdDev(dC) = VCV
• Set the same $Sharpe ratio for C as V
• Valid since dC and dV are perfectly
correlated
• k =𝐸 𝑑𝐶 −𝑟𝐶
𝑆𝑡𝑑𝐷𝑒𝑣(𝑑𝐶)
• RESULTS
• k disappears from the differential
equation!
• The equation looks familiar
• rC = rVCV + ½2V2CVV − V
• Relative to V, this is Black-Scholes-
Merton
• Relative to W…
• The risk premium on the stock does
affect the option value, but only
through its effect on V
BSM WITHOUT RISK NEUTRALITY
• dS = S dt + S dW (now using S as the stock price!)
• Derivative value is C(S,)
• Required return (C is an unknown function) = Expected return (Ito’s Lemma)
• CC = SCS + ½2S2CSS − C (*)
• Beta of C wrt S = CS = CSS/C (elasticity)
• Then assume a single factor benchmark model, projecting the
instantaneous return of the call onto the stock:
• C = r + CS( - r)
• Substitute C and CS into (*) and the BSM differential equation
obtains immediately!
CONCLUSIONS TO THE BONUS ROUND
• BSM can still obtain when the underlying asset is untradeable
• This is because the value of an option does not depend on the expected
return of the underlying, even in this situation
• You don’t need a hedge portfolio, continuous trading or even tradability
to obtain the BSM differential equation for an option
• You only need assume one of:
• the underlying and the option have the same dollarized Sharpe ratio
• The single benchmark model applies in continuous time
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