Principal and Interest Continuous Compounding Imagine dividing the year into infinitely small periods and use m lim where e 2.7818...is the base of the natural logarithm.The effective interest rate is the number r'that satisfies 1+r/=e. Continuous Compounding Interest rate (% Nominal 1005001000200030005000750010000 Effective 10151310522214349964871117017183 Xi CHEN (chenxi01090bfsu.edu.cn) Investment Science 11/174
Principal and Interest Continuous Compounding Imagine dividing the year into infinitely small periods and use lim m→∞ 1 + r m m = e r , where e = 2.7818... is the base of the natural logarithm. The effective interest rate is the number r 0 that satisfies 1 + r 0 = e r . Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 11 / 174
Principal and Interest To calculate how much an account will have grown after any arbitrary length of time tk/m(measured in years).we use 1+)=+)"→m[+)们=r Continuous compounding leads to the familiar exponential growth curve 14 12 10 anje 6 2 0246B101214161B202224 Years Xi CHEN (chenxi01090bfsu.edu.cn) Investment Science 12/174
Principal and Interest To calculate how much an account will have grown after any arbitrary length of time t ≈ k/m (measured in years), we use 1 + r m k = 1 + r m mt ⇒ lim m→∞ h1 + r m mit = e rt . Continuous compounding leads to the familiar exponential growth curve. Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 12 / 174
Principal and Interest Debt and Money Markets Exactly the same thing happens to debt.If I borrow money from the bank at an interest rate r and make no payments to the bank,then my debt increases according to the same formulas.Specifically,if my debt is compounded monthly,then after k months my debt will have grown by a factor of [1+(r/12)]k Market Interest Rates interest rates (August 9,1995) US Ireasury bills and uotes 3-month bill 339 6-month bill 539 1-year bil 536 3-year tote (yteld) 605 10-year note (yield) 649 30-year bond (9h yteld) 692 Fed funds rate 56873 Discoutit tate 526 Prime iate 875 Cominercial paper 584 Certficates of deposit I month 517 2 miouths 524 I yenr 528 Banker's ucceptunees (30 days) 568 London late Eurodollars (I tnoith) 575 Londott Interbank offered tate (I month) 588 Federnl Home Loan Mortgage Corp (Freddie Mne)(30 yeis) 794 Xi CHEN (chenxi01090bfsu.edu.cn) Investment Science 13/174
Principal and Interest Debt and Money Markets Exactly the same thing happens to debt. If I borrow money from the bank at an interest rate r and make no payments to the bank, then my debt increases according to the same formulas. Specifically, if my debt is compounded monthly, then after k months my debt will have grown by a factor of [1 + (r/12)]k . Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 13 / 174
Present Value Outline Principal and Interest ② Present Value 3 Present and Future Value of Streams Internal Rate of Return Evaluation Criteria Applications and Extensions 4口，40+4立4至，三)及0 Xi CHEN (chenxi01090bfsu.edu.cn) Investment Science 14/174
Present Value 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 14 / 174
Present Value Example Consider two situations. O You will receive $110 in 1 year. You receive $100 now and deposit it in a bank account for 1 year at 10%interest. Clearly,these situations are identical after 1 year! We say that the $110 to be received in 1 year has a present value of $100.In general,$1 to be received a year in the future has a present value of $1/(1+r),where r is the interest rate. The process of evaluating future obligations as an equivalent present value is alternatively referred to as discounting.The factor by which the future value must be discounted is called the discount factor.The 1-year discount factor is 1 d= 1+r Xi CHEN (chenxi01090bfsu.edu.cn) Investment Science 15/174
Present Value Example Consider two situations. 1 You will receive $110 in 1 year. 2 You receive $100 now and deposit it in a bank account for 1 year at 10% interest. Clearly, these situations are identical after 1 year! We say that the $110 to be received in 1 year has a present value of $100. In general, $1 to be received a year in the future has a present value of $1/(1 + r), where r is the interest rate. The process of evaluating future obligations as an equivalent present value is alternatively referred to as discounting. The factor by which the future value must be discounted is called the discount factor. The 1-year discount factor is d1 = 1 1 + r . Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 15 / 174