A framework for modelling cash flow lags
Fredrik Armerin and Han-Suck Song
Working Paper 2020:17
Division of Real Estate Economics and Finance
Division of Real Estate Business and Financial Systems
Department of Real Estate and Construction Management
School of Architecture and the Built Environment
KTH Royal Institute of Technology
A framework for modelling cash flow lags
Fredrik Armerin
Division of Real Estate Economics and Finance
Department of Real Estate and Construction Management
Royal Institute of Technology, Stockholm, Sweden
Email: fredrik.armerin@abe.kth.se
Han-Suck Song
Division of Real Estate Economics and Finance
Department of Real Estate and Construction Management
Royal Institute of Technology, Stockholm, Sweden
Email: han-suck.song@abe.kth.se
Abstract:
Many irreversible investment problems studied in finance has the
property that the cash flow representing the cost and the revenue of the
investment occur at one time (either at the same time, or at two different
times). In this note we present a framework in which the cash flows are
allowed to be spread out in time, thus yielding a more realistic model. We
show the effect of this extension in an investment case study example.
Keywords: Optimal Stopping, Irreversible Investments, Cash Flow Lags,
Time-to-build
1
Introduction
In the standard optimal timing investment problem, as in the seminal paper by McDonald & Siegel [13], it is assumed that the investment cost is paid at the time the investment is done, and that all the revenues are also received at this time. Mathematically, the problem for the investor to solve is
sup
τ
EQe−rτ(X
τ− Iτ) .
Here τ is the time of the investment, Iτ is the cost of the investment, Xτ is the
revenue, r is the risk-free interest rate and Q is the risk-neutral measure; see below for details. This optimal investment problem is an example of an optimal stopping problem on the form
sup
τ
EQe−rτG(X
τ) . (1)
In many cases one or more cash flows occur later than at time τ . Examples where these types of lags occur include real estate investments and start-up companies.
It is well known that in problems such as the optimal investment problem described above, it is optimal to wait longer than to the first time until Xt− It
is positive; the difference between the revenue and the cost needs to be large enough before it is optimal to initiate the investment. When there is an extra delay imposed exogenously, the optimal time of investment can be both earlier (as in Bar-Ilan & Strange [4]) and later (as in the example in Section 3 below) than in the case without the delay in the cash flows.
Previous results generalizing the setup from McDonald & Siegel [13] to cases where the payoffs are delayed in time with respect to the time where the invest-ment is initiated include Øksendal [14], Alvarez & Keppo [2], Lempa [8], [9] and Sarkar & Zheng [15]. There are also cases where more complex models use time lags. Examples include Aguerrevere [1], Grenadier [5], [6], Majd & Pindyck [11] and Margsiri et. al. [12] (the same model for the time-to-build is used in Sarkar & Zheng [16]). For a non-technical discussion on real options and investment lags, see MacDougall & Pike [10].
The time span from the investment is initiated, or decided on, and until the investment is generating cash flows is, depending on the context, referred to as the time-to-build, an investment delay or an investment lag (Lempa [9]). We use the term ’cash flow lag’ to emphasize that we consider models where the cash flows are spread out in time.
The rest of this paper is organized as follows. In Section 2 the modelling setup of previous approaches to investment lags as well as our extension of existing models are presented, and Section 3 contains an application of our modelling framework.
2
Modelling cash flow lags
Let (Xt) be a strong Markov process defined on a suitable probability space.
We also assume the existence of a risk-neutral probability measure Q, locally equivalent to the original probability measure, such that the value of a fu-ture uncertain cash flow is given by the expected value using this measure and
discounting the cash flow using the constant bank account rate r > 0. See e.g. Jeanblanc et. al. [7] for the underlying theory.
Now consider the following class of investment problems. At time τ , at which the investment is initiated, there can be a lump sum cash flow. Then there is a time span τB under which there are no cash flows. At time τ + τB there can
also be a lump-sum payment. The investor’s goal is to maximize the total value of these cash flows, i.e. to solve the problem
sup
τ
EQx he−rτG(Xτ) + e−r(τ +τB)GB(Xτ +τB)
i
. (2)
Here G(Xτ) is the lump sum cash flow at time τ and GB(Xτ +τB) is the lump
sum cash flow at time τ + τB. The notation EQx means that we consider the
expected value under Q given that X0= x.
Example 2.1 Several of the models mentioned in the Introduction fits into the modelling setup descibed in Equation (2).
(i) McDonald & Siegel [13]: G(x) = x − I, τB= 0 and GB(x) = 0.
(ii) Øksendal [14]: G(x) = 0, τB= δ ∈ R+ and GB a general function.
(iii) Alvarez & Keppo [2]: G(x) = I, τB = ∆(Xτ) ≥ 0 with ∆(0) = 0 and
GB(x) = x.
(iv) Lempa [8]: G(x) = 0, τB is independent of X and exponentially
dis-tributed and GB is a general function.
(v) Margsiri et. al. [12]: G(x) = θ(x − I), τBa hitting time of X and GB(x) =
(1 − θ) · (x − I), where θ ∈ (0, 1).
(vi) Lempa [9]: G(x) = 0, τB is independent of X and has a phase-type
distribution and GB is a general function.
2 It should be noted that using the (strong if necessary) Markov property, all cases in the previous example can be reduced to a problem on the form given in Equation (1); see the referred literature for details in the respective case.
In practise, the cash flows occuring after the time τ + τBare not restricted to
one lump sum payment, but are in general spread out over time. The same may be true of the cash flows occuring at time τ , but here we foucs on the cash flows after time τ + τB (generalising to also include the other case is straightforward).
In order to allow for time lags in the cash flows after time τ + τB, we consider
optimal stopping problems on the form
sup τ EQx e−rτG(Xτ) + e−r(τ +τB) Z ∞ 0 e−rsGB(X(τ + τB+ s))dF (s) . (P)
When using this approach to model investments, τ is the time at which it is decided that the investment should be initiated, G(Xτ) the lump sum cash flow
at this time, τ + τB is the time at which further cash flows from the investment
is starting to appear (the time τB could e.g. be the time-to-build in a building
project) and F is a probability measure on [0, ∞) describing how the cash flows are distributed after the time τ + τB.
Example 2.2
(i) Lump sum payments. The cash flow(s) occur at given time spans t1, . . . , tn
after the time τ + τB:
F (s) =
n
X
i=1
wiδti(s).
Here the wi’s are strictly positive and sum to 1 and δxdenotes the Dirac
measure at x (here we only consider x ∈ R+). In this case the optimal
stopping problem (P) can be written
sup τ ExQ " e−rτG(Xτ) + e−r(τ +τB) n X i=1 wie−rtiGB(X(τ + τB+ ti)) # .
The choice w1 = 1 and F (s) = δ0(s) represents the optimal stopping
problem in Equation (2).
(ii) Continuous payouts. In this case dF (s) = f (s)ds for some density function f on [0, ∞). The optimal stopping problem can now be written
sup τ ExQ e−rτG(Xτ) + e−r(τ +τB) Z ∞ 0 e−rsGB(X(τ + τB+ s))f (s)ds . 2 To get a tractable, yet general, model we consider time delays τB that are
the sum of a random time U independent of the process X and exponentially distributed with mean 1/λ, and a constant time delay δ ≥ 0:
τB= U + δ.
Hence, we combine the models by Øksendal [14] and Lempa [8], and extend it to revenue cash flows spread out in time.
One feature of this model is that it is in fact on the form given in Equation (1). In the following proposition we use the standard operators Ptfor t ≥ 0 and
Rλfor λ > 0 defined by Ptf (x) = ExQ[f (Xt)] and Rλf (x) = ExQ Z ∞ 0 e−λsf (Xs)ds
respectively, and where we assume that each f used is such that the resepective operator is well defined.
Proposition 2.3 With notation and assumptions as above, Problem (P) can be written sup τ ExQe−rτH(Xτ) , where H(x) = G(x) + λe−rδPδRr+λGFB(x) and GFB(x) = Z ∞ 0 e−rsExQ[GB(Xs)] dF (s).
The proof is a straigthforward application of the strong Markov property and can be found in Appendix A.
3
An investment case study
We now present a concrete model of the cash flows generated by the invest-ment. Under the pricing measure Q, the revenue process X is assumed to be a geometric Brownian motion:
dXt= (r − q)Xtdt + σXtdWtQ.
Here q > 0 is the constant yield or implied yield (see Armerin & Song [3] for a discussion), σ > 0 is the volatility and WQ is a standard Wiener process under Q. At the inception of the investment, the cost I is paid, i.e.
G(x) = −I,
and all future cash flows are revenues form the investment. This is represented by the function
GB(x) = x,
and the revenue cash flows are distributed according to the deterministic distri-bution function F efter the random time τ + τB. Hence, the optimal investment
problem is sup τ ExQ −e−rτI + e−r(τ +τB) Z ∞ 0 e−rsX(τ + τB+ s)dF (s) . (3) When GB(x) = x we have GFB(x) = ExQ Z ∞ 0 e−rsXsdF (s) = x Z ∞ 0 e−qsdF (s)
and from this
λRr+λGFB(x) = x · λ λ + q Z ∞ 0 e−qsdF (s). Finally, λe−rδPδRr+λGFB(x) = λe −rδxe(r−q)δ· λ λ + q Z ∞ 0 e−qsdF (s) = xe−qδ· λ λ + q Z ∞ 0 e−qsdF (s).
It follows from Proposition 2.3 that the optimal stopping problem stated in Equation (3) can be written
sup τ ExQ e−rτ Xτe−qδ· λ λ + q Z ∞ 0 e−qsdF (s) − I . By introducing k = e qδ(1 + q/λ) R∞ 0 e−qsdF (s)
we can write the optimal stopping problem as
sup τ ExQ e−rτ 1 kXτ− I = 1 ksupτ ExQe−rτ(X τ− kI) .
We now define Vk(x; I) = sup τ ExQ e−rτ 1 kXτ− I . (4)
It then holds that
Vk(x; I) =
1
kV (x; kI) ,
where V is the value function of the standard optimal investment problem:
V (x; I) = ( (Vc− I) x Vc a when x < Vc x − I when x ≥ Vc, where Vc= a a − 1I and a = 1 2− r − q σ2 + s 1 2 − r − q σ2 2 + 2r σ2 > 1
(see e.g. McDonald & Siegel [13]). Hence,
Vk(x; I) = ( (Vc− I) x kVc a when x < kVc x k− I when x ≥ kVc.
An optimal stopping time for the optimal stopping problem given in Equation (3) is
τ = inf{t ≥ 0 | Xt≥ kVc}.
The factor k is always greater than or equal to 1, so the result of the delay of revenues is that the optimal level at which the investment should be done is increased compared to the non-delayed case. This in turn will lead to a delay in the time of investment. Even in the case where there is no time to build (this is represented by the case δ = 0 and λ = ∞), there will in general be a delay in the time at which the investment is done due to the fact that the revenue cash flows occur later in time.
Example 3.1 For a numerical example of the above model, we assume that the distribution of the revenue cash flows are according to an exponential dis-tribution:
f (s) = 1 γe
−s/γ, s ≥ 0,
for some γ > 0. We also consider the limiting case γ = 0, which represents the case when all revenue cash flows occur at the time τ + τB (i.e. F (s) = δ0(s)).
In this case Z ∞ 0 e−qsdF (s) = Z ∞ 0 e−qs1 γe −1 γsds = 1 γq + 1.
One way of interpreting the parameter γ is that it reflects the competitiveness of the market for the product the investment generates. In a market with many competitors, we expect competition to make the revenue cash flows occuring later than in a less competitive market. As γ increases, the revenue cash flows
are occuring later, making the interpretation that the higher the value on γ, the more competitive the market is, possible.
In Tables 1 to 3 below numerical value of the factor k is presented. The value λ = ∞ represents the case when there is no exponentially distributed waiting
time after τ + δ. 2 γ\λ ∞ 10 5 1 0 1.020 1.021 1.022 1.030 0.5 1.025 1.026 1.027 1.036 1 1.030 1.031 1.032 1.041 5 1.071 1.072 1.073 1.082
Table 1: The factor k when the yield is q = 0.01.
γ\λ ∞ 10 5 1
0 1.041 1.043 1.045 1.062 0.5 1.051 1.053 1.055 1.072 1 1.062 1.064 1.066 1.083 5 1.145 1.147 1.149 1.168
Table 2: The factor k when the yield is q = 0.02.
γ\λ ∞ 10 5 1
0 1.105 1.111 1.116 1.160 0.5 1.133 1.138 1.144 1.189 1 1.160 1.166 1.172 1.218 5 1.381 1.388 1.395 1.451
Table 3: The factor k when the yield is q = 0.05.
A
Proof of Proposition 2.3
Proof. We start by observing that
EQ x[e−rτBGB(X(s+τB+τ ))|Fτ] = λ R∞ 0 e −r(y+δ)EQ x[GB(X(s+δ+y+τ ))|Fτ]e−λydy = λe−rδR∞ 0 e −(r+λ)yEQ Xτ[GB(X(s+δ+y))]dy. It follows that EQ x[e−rτB R∞ 0 e −rsG B(X(τ +τB+s))dF (s)] = ExQ[ R∞ 0 e −rsEQ x[e−rτBGB(X(τ +τB+s))|Fτ]dF (s)] = ExQ[λe−rδR∞ 0 e −(r+λ)yR∞ 0 e −rsEQ Xτ[GB(X(s+δ+y))]dF (s)dy] = EQ x[λe −rδP δRr+λ R∞ 0 e −rs(P sGB)(Xτ)dF (s)] = EQ x[λe −rδP δRr+λR0∞e −rsEQ Xτ[G(Xs)]dF (s)],
where we have used ExQ[GB(s + δ + y))] = Ps+δ+yGB(x) = PδPyPsGB(x) for x = Xτ. With GFB(x) = Z ∞ 0 e−rsExQ[GB(Xs)] dF (s) we have ExQ e−r(τ +τB) Z ∞ 0 e−rsGB(X(τ + τB+ s))dF (s) = EQx e−rτλe−rδP δRr+λGFB(Xτ) . Finally we arrive at ExQ e−rτG(Xτ) + e−r(τ +τB) Z ∞ 0 e−rsGB(X(τ + τB+ s))dF (s) = ExQe−rτ G(X τ) + λe−rδPδRr+λGFB(Xτ) . 2
References
[1] Aguerrevere, F. L. (2003), ’Equilibrium Investment Strategies and Output Price Behavior: A Real-Options Approach’, The Review of Financial Stud-ies, Vol. 16, No. 4, pp. 1239-1272, DOI: 10.1093/rfs/hhg041.
[2] Alvarez, L. H. R. & Keppo, J. (2002), ’The impact of delivery lags on irreversible investment under uncertainty’, European Journal of Operational Research 136, pp. 173-180.
[3] Armerin, F. & Song, H.-S. (2018), ’Valuation of real options in incomplete models – an implied yield approach’, Fuzzy Economic Review, Vol. 23, No. 1, pp. 19-32, DOI: 10.25102/fer.2018.01.02.
[4] Bar-Ilan, A. & Strange, W. C. (1996), ’Investment Lags’, American Eco-nomic Review, Vol. 86, No. 3, pp. 610-622.
[5] Grenadier, S. R. (1995), ’The Persistence of Real Esate Cycles’, Journal of Real Estate Finance and Economics, p. 95-119.
[6] Grenadier, S. R. (2000), ’Equilibrium with Time-to-Build: A Real Options Approach’, Project Flexibility, Agency, and Competition: New Develop-ments in the Theory and Applications of Real Options, Oxford University Press.
[7] Jeanblanc, M. Yor, M. & Chesny, M. (2009), ’Mathematical Methods for Financial Markets’, Springer-Verlag London.
[8] Lempa, J. (2012), ’Optimal stopping with random exercise lag’, Math Meth Oper Res 75, pp. 273-286, DOI 10.1007/s00186-012-0384-7.
[9] Lempa, J. (2020), ’Some results on optimal stopping under phase-type dis-tributed implementation delay’, Mathematical Methods of Operations Re-search, https://doi.org/10.1007/s00186-019-00694-6.
[10] MacDougall, S. L. & Pike, R. H. (2003), ’Consider your options: changes to strategic value during implementation of advanced manufacturing tech-nology’, Omega, pp. 1-15.
[11] Majd, S. & Pindyck R. S. (1987),’Time to build, option value, and invest-ment decisions’, Journal of Financial Economics 18, pp. 2-27.
[12] Margsiri, W., Mello, A. S. & Ruckes, M. E. (2003), ’A Dynamic Anal-ysis of Growth via Acquisition’, Review of Finance, pp. 635-671, DOI: 10.1093/rof/rfn015.
[13] McDonald, R. & Siegel, D. (1986), ’The Value of Waiting to Invest’, The Quarterly Journal of Economics, 101(4), pp. 707-727.
[14] Øksendal, B. (2005), ’Optimal stopping with delayed information’, Stochas-tic and Dynamics, Vol. 5, No. 2, pp. 271-280.
[15] Sarkar, S. & Zhang, C. (2013), ’Implementation lag and the investment decision’, Economics Letters 119, pp. 136-140.
[16] Sarkar, S. & Zhang, C. (2015), ’Investment policy with time-to-build’, Jour-nal of Banking & Finance 55, pp. 142-156.
