Homework Solution: Investment Science Solution Problem 5.2: The Road Project # or 4 are carried out Ê B# B% œ " Project ' or ( are carried out Ê B' B( œ " Add constraint: ÐB# B% ÑÖ" ÐB' B( Ñ× œ ! Observe: B# B % œ " Ê B ' B ( œ " Hence, if projects # or % are carried out, either project ' or ( must also be carried out. EMSE 6992 Chapter 5 : Investment Science - Leunberger : The Term Structure of Interest Rates Page 1 Notes by J. Rene van Dorp www.seas.gwu.edu/~dorpjr Homework Solution: Investment Science Cost ($1,000) NPV ($1,000) Optimal x-values Cost NPV Goals Road between Augen and Burger 1 Concrete, 2 lanes 2,000 4,000 0 0 0 2 Concrete, 4 lanes 3,000 5,000 0 0 0 3 Asphalt, 2 lanes 1,500 3,000 0 0 0 4 Asphalt, 4 lanes 2,200 4,300 1 2,200 4,300 500 1,000 0 0 0 6 Add Lane 1,500 1,500 1 1,500 1,500 7 New Structure 2,500 2,000 0 0 0 8 Traffic lights 100 300 0 0 0 9 Turn lanes 600 1,000 0 0 0 10 Underpass 1,000 At most one project can be selected for each major objective 2,000 1 1,000 4,700 2,000 7,800 1 Bridge at Cay Road 5 Repair Exisiting 1 Traffic Control in Downsburg EMSE 6992 Chapter 5 : Investment Science - Leunberger : The Term Structure of Interest Rates Totals 1 Page 2 Notes by J. Rene van Dorp www.seas.gwu.edu/~dorpjr Homework Solution: Investment Science Solution Problem 5.11: Growing Annuity W œ ∞ 5œ" œ œ œ " "< Ð"1Ñ5" Ð"<Ñ5 " "< "1 "< "1 "< Ð"1Ñ ∞ Ð"<Ñ5 Ð"1Ñ5" ∞ 5œ# 5# Ð"<Ñ Ð"1Ñ ∞ 5" "1 "< W Í WÐ" " "< 5" 5œ# " "< <1 Í W "< œ EMSE 6992 Chapter 5 œ " "< 5œ" Ð"<Ñ5 "1 "< Ñ ÍW œ : Investment Science - Leunberger : The Term Structure of Interest Rates œ " "< " <1 Page 3 Notes by J. Rene van Dorp www.seas.gwu.edu/~dorpjr Homework Solution: Investment Science Solution Problem 5.12: Two-Stage Growth It is common practice in security analysis to modify the basic dividend growth model by allowing more than one stage of growth, with the growth factor being different in different stages. As an example, consider company ^ , which currently distributed dividend of $"!Q annually. The dividends are expected to grow at a rate of "!% for the next & years and at a rate of &% thereafter. +Þ Using a dividend discount approach with an interest rate of "&% what is the approximate value of the company? Approximate Value: $"#(Þ*"Q (See spreadsheet) ,Þ Find a general formula for the value of a company satisfying a two-stage growth model. Assume a growth rate of K for 5 years, followed by a growth rate of 1 thereafter, and an initial dividend of H! . EMSE 6992 Chapter 5 : Investment Science - Leunberger : The Term Structure of Interest Rates Page 4 Notes by J. Rene van Dorp www.seas.gwu.edu/~dorpjr Homework Solution: Investment Science Ð" KÑ3 H! Ð" KÑ5 ∞ Ð" 1Ñ3 R T Z œ H! 3 5 3 Ð" <Ñ Ð" <Ñ Ð" <Ñ 3œ" 3œ" 5 Ð"Ñ For the first term in Ð"Ñ we have À 5 5 Ð" KÑ3 "K 3 "K 3 œ œ B ß where B œ Ð" <Ñ3 "< "< 3œ" 3œ" 3œ" 5 Utilizing B3 œ 8 3œ" BB8" "B Ð" KÑ3 œ 3 Ð" <Ñ 3œ" 5 EMSE 6992 Chapter 5 one obtains "K "< 5 +1 "K "< " "K "< " œ Ð" KÑ ‚ : Investment Science - Leunberger : The Term Structure of Interest Rates "K "< <K 5 Ð#Ñ Page 5 Notes by J. Rene van Dorp www.seas.gwu.edu/~dorpjr Homework Solution: Investment Science For the second term in Ð#Ñ we have ∞ Ð" 1Ñ3 "1 3 œ 3 Ð" <Ñ " < 3œ" 3œ" ∞ Utilizing B3 œ ∞ 3œ" B "B one obtains Ð" 1Ñ3 "1 " " "< œ œ ‚ œ Ð" 1Ñ ‚ Ð$Ñ "1 "1 3 Ð" <Ñ " < " "< <1 " "< 3œ" ∞ "1 Finally, substitution of Ð#Ñ and Ð$Ñ in Ð"Ñ yields: EMSE 6992 Chapter 5 : Investment Science - Leunberger : The Term Structure of Interest Rates Page 6 Notes by J. Rene van Dorp www.seas.gwu.edu/~dorpjr Homework Solution: Investment Science " R T Z œ H! Ð" KÑ ‚ 5 "K "< <K H! Ð" KÑ5 " Ð" 1Ñ ‚ Ð"Ñ Ð" <Ñ5 <1 Substitution of < œ !Þ"&ß K œ !Þ"ß 1 œ !Þ!&, H! œ "! and 5 œ & yields: R T Z œ $"#(Þ*#Q EMSE 6992 Chapter 5 : Investment Science - Leunberger : The Term Structure of Interest Rates Page 7 Notes by J. Rene van Dorp www.seas.gwu.edu/~dorpjr