Rate of return to Investment
Rate of Return to Investment
The Concept of investment For a one period investment, the ror can be evaluated each year R t+1 For investments that generate returns in many periods, there are two ways of describing the rate of return ROR in year t-yield/investment Often ask what is rate of return to an investment project?
The Concept of Investment • For a one period investment, the ROR can be evaluated each year. • For investments that generate returns in many periods, there are two ways of describing the rate of return. – ROR in year t = yield/investment. – Often ask: What is rate of return to an investment project? t t 1 t t p p R p + − =
The Example of Consol A consol (or perpetuity) that pays $100 annually can be bought and sold at the market for $1000 Want to compare the investment in the bond ($1000)with alternative investment opportunities(e.g, bank deposit) 0: What is the interest rater(or effective rield) consistent with this price?
The Example of Consol • A consol (or perpetuity) that pays $100 annually can be bought and sold at the market for $1000. • Want to compare the investment in the bond ($1000) with alternative investment opportunities (e.g., bank deposit). • Q: What is the interest rate R (or effective yield) consistent with this price?
Effective Yield of a bond A: Ther that equates the bond price with the present value of the bond Present value of the bond is 100 100 100 100 100 PV= (1+R)(1+R)(1+R)(1+ R Letting bond price equal to the Pv, 100 1000 R →R=100/1000=10
Effective Yield of a Bond • A: The R that equates the bond price with the present value of the bond. • Present value of the bond is • Letting bond price equal to the PV, • R=100/1000=10% 100 1000 R = 234 100 100 100 100 100 ... (1 ) (1 ) (1 ) (1 ) PV R R R R R = + + + + = + + + +
Generalize to all investment Effective yield(or rate of return)is the r that makes the present value of investment returns equal to the present value of the cost Our example: cost=1000 in period 1 return=100 each year starting from period 2 Education example is a little more complex
Generalize to All Investment • Effective yield (or rate of return) is the R that makes the present value of investment returns equal to the present value of the cost. • Our example: cost=1000 in period 1; return=100 each year starting from period 2. • Education example is a little more complex
Consider Investment in College education You graduate from high school at l8 You earns Wus if quitting school after high school · If you go to college pays direct cost H delays labor market entry by 4 years earns cot after college
Consider Investment in College Education • You graduate from high school at 18 • You earns if quitting school after high school • If you go to college, – pays direct cost H – delays labor market entry by 4 years – earns after college WHS WCOL
Potential Earnings Stream Faced by a high School Graduate Dollars COL Goes to College Quits after highs HS school Age 18 22 65
18 22 65 Goes to College Quits after highs school Age Dollars WHS WCOL -H Potential Earnings Stream Faced by a High School Graduate
The College Investment OL--H-H W P COL COL 十.+ COL (1+r)(1+r)2(1+r)3(1+r)4(1+r) +r) PVe=wue t W HS HS HS 十.+ (1+r)(1+r) (1+r) Effective yield of college education is the r when POor= py This is also called internal rate of return
The College Investment 2 3 4 5 4 6 (1 ) ... (1 ) (1 ) (1 ) (1 ) (1 ) r W r W r W r H r H r H PV H COL COL COL COL + + + + + + + + − + − + = − − 2 46 (1 ) ... (1 ) (1 ) r W r W r W PV W HS HS HS HS HS + + + + + + = + Effective yield of college education is the r when This is also called “internal rate of return”. PVCOL = PVHS
Benefits and Costs of College Alternative lllustration Benefits of College PVB COL (1+1)5++"c-Ws COL (1+r) 46 Costs of college PVCcoL =(h+ Wus)+ H+WHs H+WHs H+wus (1+r)(1+r)2(1+r)
Benefits and Costs of College • Alternative Illustration: • Benefits of College: • Costs of College 4 5 4 6 (1 ) ... (1 ) (1 ) r W W r W W r W W PVB COL HS COL HS COL HS COL + − + + + − + + − = 2 3 (1 ) (1 ) (1 ) ( ) r H W r H W r H W PVC H W HS HS HS COL HS + + + + + + + + = + +
The College decision a person chooses to go to college if PVCoL PVHs Or, equivalently, the rate of return is larger than returns from alternative investment
The College Decision A person chooses to go to college if Or, equivalently, the rate of return is larger than returns from alternative investment. PVCOL PVHS