Investment Science Part l:Deterministic Cash Flow Streams Dr.Xi CHEN Department of Management Science and Engineering International Business School Beijing Foreign Studies University 100089,Beijing.People's Republic of China 4口40+4之4至，至)只0 Xi CHEN (chenxi01090bfsu.edu.cn) Investment Science 1/174
Investment Science Part I: Deterministic Cash Flow Streams Dr. Xi CHEN Department of Management Science and Engineering International Business School Beijing Foreign Studies University 100089, Beijing, People’s Republic of China Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 1 / 174
Outline Principal and Interest ② Present Value Present and Future Value of Streams 4 Internal Rate of Return Evaluation Criteria Applications and Extensions 4口4+4三4至，至)只0 Xi CHEN (chenxi01090bfsu.edu.cn) Investment Science 2/174
Outline 1 Principal and Interest 2 Present Value 3 Present and Future Value of Streams 4 Internal Rate of Return 5 Evaluation Criteria 6 Applications and Extensions Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 2 / 174
Principal and Interest Outline Principal and Interest Present Value 3 Present and Future Value of Streams Internal Rate of Return Evaluation Criteria Applications and Extensions 4口40+4三4至，至)只0 Xi CHEN (chenxi01090bfsu.edu.cn) Investment Science 3/174
Principal and Interest Outline 1 Principal and Interest 2 Present Value 3 Present and Future Value of Streams 4 Internal Rate of Return 5 Evaluation Criteria 6 Applications and Extensions Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 3 / 174
Principal and Interest Example If you invest $1.00 in a bank account that pays 8%interest per year,then at the end of 1 year you will have in your account the principal(your original amount)at $1.00 plus interest at $0.08 for a total of $1.08. o What if you invest a larger amount,say A dollars,in the bank? o What if the interest rate is r? Xi CHEN (chenxi01090bfsu.edu.cn) Investment Science 4/174
Principal and Interest Example If you invest $1.00 in a bank account that pays 8% interest per year, then at the end of 1 year you will have in your account the principal (your original amount) at $1.00 plus interest at $0.08 for a total of $1.08. What if you invest a larger amount, say A dollars, in the bank? What if the interest rate is r? Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 4 / 174
Principal and Interest Simple Interest Under a simple interest rule,money invested for a period different from 1 year accumulates interest proportional to the total time of the investment. If an amount A is left in an account at simple interest r,the total value after n years is V=(1+n)A. o If the proportional rule holds for fractional years,then after any time t(measured in years),the account value is V=(1+t)A. The account grows linearly with time.The account value at any time is just the sum of the principal and the accumulated interest,which is proportional to time. 4口，4得+4之卡4至卡 Xi CHEN (chenxi01090bfsu.edu.cn) Investment Science 5/174
Principal and Interest Simple Interest Under a simple interest rule, money invested for a period different from 1 year accumulates interest proportional to the total time of the investment. If an amount A is left in an account at simple interest r, the total value after n years is V = (1 + rn)A. If the proportional rule holds for fractional years, then after any time t (measured in years), the account value is V = (1 + rt)A. The account grows linearly with time. The account value at any time is just the sum of the principal and the accumulated interest, which is proportional to time. Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 5 / 174
Principal and Interest Compound Interest Example Consider an account that pays interest at a rate of r per year.If interest is compounded yearly,then after 1 year,the first year's interest is added to the original principal to define a larger principal base for the second year. What is the account value after n years? The account earns interest on interest! Under yearly compounding,after n years,such an account will grow to V=(1+r)A. This is the analytic expression for the account growth under compound interest.This expression is said to exhibit geometric growth because of its nth-power form. 0Q0 Xi CHEN (chenxi01090bfsu.edu.cn) Investment Science 6/174
Principal and Interest Compound Interest Example Consider an account that pays interest at a rate of r per year. If interest is compounded yearly, then after 1 year, the first year’s interest is added to the original principal to define a larger principal base for the second year. What is the account value after n years? The account earns interest on interest! Under yearly compounding, after n years, such an account will grow to V = (1 + r) nA. This is the analytic expression for the account growth under compound interest. This expression is said to exhibit geometric growth because of its nth-power form. Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 6 / 174
Principal and Interest Rule (The seven-ten rule) Money invested at 7%per year doubles in approximately 10 years.Also, money invested at 10%per year doubles in approximately 7 years. 1200- 1000 Simple Compound 800 anjeA 600 400 200 0 02 A 681012141618202224 Years Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 7/174
Principal and Interest Rule (The seven-ten rule) Money invested at 7% per year doubles in approximately 10 years. Also, money invested at 10% per year doubles in approximately 7 years. Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 7 / 174
Principal and Interest Exercise (The 72 rule) The number of years n required for an investment at interest rate r to double in value must satisfy (1+r)P=2. By Taylor series,we have h+=x-写-+(-+(1<x≤ 。Using In2=0.69 and the approximation In(1+r）≈r valid for small r,show that n69/i,where i is the interest rate percentage,i.e., i=100r. oUsing the better approximation In(1+r)r-r2/2,show that for r≈0.08，there holds n≈72/i. )Q0 Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 8/174
Principal and Interest Exercise (The 72 rule) The number of years n required for an investment at interest rate r to double in value must satisfy (1 + r) n = 2. By Taylor series, we have ln (1 + x) = x − x 2 2 + x 3 3 − . . . + (−1)n−1 x n n + . . . (−1 < x ≤ 1). Using ln 2 = 0.69 and the approximation ln(1 + r) ≈ r valid for small r, show that n ≈ 69/i, where i is the interest rate percentage, i.e., i = 100r. Using the better approximation ln(1 + r) ≈ r − r 2/2, show that for r ≈ 0.08, there holds n ≈ 72/i. Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 8 / 174
Principal and Interest Compounding at Various Intervals Example (Quarterly compounding) Quarterly compounding at an interest rate of r per year means that an interest rate of r/4 is applied every quarter.Hence,money left in the bank for 1 quarter will grow by a factor of 1+(r/4)during that quarter.If the money is left in for another quarter,then that new amount will grow by another factor of 1+(r/4). What is the account value after 1 year? (+)>1+5 r>0. Right or not,why? ●What is the meaning? Xi CHEN (chenxi01090bfsu.edu.cn) Investment Science 9/174
Principal and Interest Compounding at Various Intervals Example (Quarterly compounding) Quarterly compounding at an interest rate of r per year means that an interest rate of r/4 is applied every quarter. Hence, money left in the bank for 1 quarter will grow by a factor of 1 + (r/4) during that quarter. If the money is left in for another quarter, then that new amount will grow by another factor of 1 + (r/4). What is the account value after 1 year? 1 + r 4 4 > 1 + r, ∀r > 0. Right or not, why? What is the meaning? Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 9 / 174
Principal and Interest Definition Effective interest rate is the equivalent yearly interest rate that would produce the same result after 1 year without compounding. Example An annual rate of 8%compounded quarterly will produce an increase of (1.02)41.0824;hence the effective interest rate is 8.24%.The basic yearly rate(8%in this example)is termed the nominal rate. Compounding can be carried out with any frequency. The general method is that a year is divided into a fixed number of equally spaced periods,say,m periods.The effective interest rate is the numberr' that satisfies 1+r=(+) Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 10/174
Principal and Interest Definition Effective interest rate is the equivalent yearly interest rate that would produce the same result after 1 year without compounding. Example An annual rate of 8% compounded quarterly will produce an increase of (1.02)4 ≈ 1.0824; hence the effective interest rate is 8.24%. The basic yearly rate (8% in this example) is termed the nominal rate. Compounding can be carried out with any frequency. The general method is that a year is divided into a fixed number of equally spaced periods, say, m periods. The effective interest rate is the number r 0 that satisfies 1 + r 0 = 1 + r m m . Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 10 / 174