The Incomplete Gamma Function Part IV - A Mean-Reverting, Return Model

Transcription

1 The Incoplete Gaa Function Part IV - A Mean-Reverting, Return Model Gary Schuran, MBE, CFA Noveber, 217 The base equation for a ean-reverting process fro Part II where the variable t is tie in years is... 1] ft = } d + c t a b t δt where... a >, b >, c <, n > 1 The solution to the base equation where Γx, y is the incoplete gaa function is... 1] ft = d a c b b 1 Γ cb, a b n Γ cb ], a b } In Part IV of the series on the incoplete gaa function we will build a return odel that incorporates ean reversion. To that end we will use the following hypothetical proble... Our Hypothetical Proble We are tasked with valuing a copany given the following odel paraeters... Table 1: Valuation Model Assuptions Description Value Current annualized revenue in dollars 1,, Current revenue growth rate % 18. Current return on assets % 15. Assets to annualized revenue % 8. Cost of capital % 1. The projected annual growth rate of noinal GDP over the next ten years is expected to be four percent. Assue that the ean-reversion half life is five years. We will use our odel to answer the following questions: Question: What is the value of this copany? Mean Reversion We are currently standing at tie zero where the current rate is unsustainable as arket forces will either increase or decrease that rate to the long-ter sustainable ean over tie. We will ake the following definitions... Ter Description current rate Market rate at tie zero unsustainable rate long-ter rate Market rate at tie infinity long-ter sustainable ean labda The rate of ean reversion where < λ < 1 2 Using the definitions in the table above we will define the arket rate at soe future tie t to be the following equation where the variable t is tie in years... arket rate at tie t = long-ter rate + current rate long-ter rate λ t 3 1

2 Using Equation 3 above note that at tie zero the arket rate is... arket rate = long-ter rate + current rate long-ter rate Using Equation 3 above note that at tie infinity the arket rate is... li arket rate = long-ter rate...because... li t t λ = current rate 4 =...if... λ > 5 In the equations above we defined the variable λ to be the rate of ean reversion. To calibrate λ we will choose soe future point in tie tie = T where the arket rate is halfway between the rate at tie zero and the rate at tie infinity i.e. the half life. The equation to calibrate λ is therefore... Revenue λ T =.5...such that... λ = ln.5 T We will define the variable ARGR to be the discrete tie annualized revenue growth rate, which is defined as the year over year growth rate of revenue. The generalized equation for the annualized revenue growth rate in discrete tie is... ARGR = Cuulative revenue for the current year Cuulative revenue fro the prior year 1 7 We will define the variable µ t to be the continuous tie revenue growth rate at tie t. The revenue growth rate at tie t is a function of the variable ω, which is the long-ter sustainable rate, the variable, which is the difference between the current unsustainable rate and the long-ter sustainable rate, the variable λ, which is the rate of ean reversion, and the variable t, which is tie in years. The equation for the revenue growth rate in continuous tie is... µ t = ω +...where... ω = µ...and... = µ µ 8 We will define the variable Γ t to be the cuulative revenue growth rate realized over the tie interval, t]. Using Equation 8 above the equation for the cuulative revenue growth rate at tie t is... Γ t = µ s δs = ω δs + 6 λ s δs 9 Using Appendix Equations 29 and 3 below the solution to Equation 9 above is... Γ t = ω t + 1 λ 1 We will define the variable R t to be annualized revenue at tie t. Using Equation 1 above the equation for annualized revenue is... } R t = R Γ t = R λ + ω t } λ 11 Using Appendix Equation 31 below the derivative of annualized revenue Equation 11 above with respect to tie is... δr t = ω + R δt λ + ω t } λ = ω R λ + ω t } λ + R λ + ω λ t } λ 12 2

3 Investent We will define the variable A t to be total assets i.e. investent at tie t, and the variable φ to be the ratio of total assets to annualized revenue. Using Equation 11 above the equation for total assets is... A t = φ R t 13 Using Equations 12 and 13 above the equation for the derivative of total assets with respect to tie is... δa t = φ δr t = φ ω R δt δt λ + ω t } λ + φ R λ + ω λ t } λ We will define the variable θ to be the after-tax return on assets, which is defined as the ratio of annualized net incoe to total assets. The generalized equation for the return on assets is... θ = Pre-tax revenue argin Annualized revenue 1 Tax rate Assets 15 We will define the variable θ t to be the continuous tie return on assets at tie t. The return on assets is a function of the variable η, which is the long-ter sustainable rate, the variable ψ, which is the difference between the current unsustainable rate and the long-ter sustainable rate, the variable λ, which is the rate of ean reversion, and the variable t, which is tie in years. The equation for the return on assets is... θ t = η + ψ...where... η = θ...and... ψ = θ θ 16 Cash Flow We will define the variable C t to be the present value at tie zero of annualized cash flow expected to be received at soe future tie t. The generalized equation for the present value of annualized cash flow is... C t = Net Incoe Investent i.e. the change in assets Discount Factor 17 We will define the variable κ to be the risk-adjusted discount rate. Using Equations 13, 14 and 16 above we can rewrite Equation 17 above as... C t = θ t A t κ t δt δa t κ t δt We will define the equation E 1 to be... E 1 = λ + ω t λ = λ + ω κ t λ We will define the equation E 2 to be... E 2 = = λ + ω t λ λ + ω λ κ t λ } κ t } κ t Using Equations 19 and 2 above we can rewrite Equation 18 above as... ] ] C t = φ R η E 1 + ψ E 2 ω E 1 E 2 = φ R η ω E 1 + ψ E Using Equation 21 above the equation for the present value of expected cash flow is... ] V = C t δt = φ R η ω E 1 δt + ψ E 2 δt 22 See Appendix Equations 36 and 41 below for the solution to each of the two integrals in Equation 22 above. 3

4 The Answer To Our Hypothetical Proble We will define the long-ter sustainable revenue growth rate to be the expected growth rate of noinal GDP over the next ten years, which is four percent. Using the odel paraeters in Table 1 above the continuous tie values for the long-ter sustainable revenue growth rate ω and the difference between the current unsustainable rate and the long-ter sustainable rate are... ω = ln1 +.4 = and... = ln ln1 +.4 = We will define the long-ter sustainable return on assets to be the cost of capital. Using the odel paraeters in Table 1 above the continuous tie values for the long-ter sustainable return on assets η and the difference between the current unsustainable rate and the long-ter sustainable rate ψ are... η = ln1 +.1 = and... ψ = ln ln1 +.1 = Using the odel paraeters in Table 1 above the value of the odel paraeter that represents the ratio of total assets to annualized revenue φ is... φ =.8 25 Using the odel paraeters in Table 1 above the value of the odel paraeter that represents the cost of capital κ is... κ = ln1 +.1 = Using Equation 6 above and the paraeters to our hypothetical proble the value of paraeter λ is... λ = ln.5 5. = Using Appendix Equations 36 and 41 below the solution to Equation 22 above, which is the answer to our hypothetical proble, is... ] V =.8 1,, = 1, 75, 28 References 1] Gary Schuran, The Incoplete Gaa Function - Part II, Deceber, 217 Appendix A. The solution to the following integral is... B. The solution to the following integral is... λ s δs = ω δs = ω s=t δs = ω s = ω t 29 s= λ s δs = λ s=t λ s = 1 s= λ C. The derivative of the following equation is... δ ω t + }] 1 = ω λ δt λ λ = ω + D. The solution to the first integral in Equation 22 above is... Using Equations 23 to 27 above the values for paraeters a, b, c and d in equation E 1 above Equation 19 are... a = =.911, b =.1386, c = =.561, d = =

5 Using Equation 32 above the solution to the first two paraeter values in base Equation 2 above are... d =.911 = and... a c b b 1 = = Using Equation 32 above the solution to the second incoplete gaa function in base Equation 2 above where = is... Γ.561, } = Γ.448,.911 = Using Equation 32 above the solution to the first incoplete gaa function in base Equation 2 above where n = is... Γ.561, } = Γ.448, = Using Equations 33, 34 and 35 above the value of the integral of Equation 22 above is... E 1 δt = = E. The solution to the second integral in Equation 22 above is... above Equa- Using Equations 23 to 27 above the values for paraeters a, b, c and d in equation E 1 tion 19 are... a = =.911, b =.1386, c = =.1947, d = = Using Equation 37 above the solution to the first two paraeter values in base Equation 2 above are... d =.911 = and... a c b b 1 = = Using Equation 37 above the solution to the second incoplete gaa function in base Equation 2 above where = is... Γ.1947, } = Γ 1.448,.911 = Using Equation 37 above the solution to the first incoplete gaa function in base Equation 2 above where n = is... Γ.1947, } = Γ 1.448, = Using Equations 38, 39 and 4 above the value of the integral of Equation 22 above is... E 1 δt = =