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