Chapter 2 INTEREST BASIC APPLICATIONS Equation of Value

Chapter 2 INTEREST: BASIC APPLICATIONS • Equation of Value • Unknown Rate of Interest • Time-Weighted Rate of Return

2. 1 Equation of Value • Four numbers: • principal A(0) • accumulated value A(t) = A(0) ∙ a(t) • period of investment t (determine effective period in order to compute t) • rate of interest i • Time diagram • Bring all entries of the diagram to the same point in time and set equation of value

Examples 1. Find the accumulated value of 500 after 173 months at a rate of compound interest of 14% convertible quarterly (p. 30) 2. Alice borrows 500 from FF Company at a rate of interest 18% per year convertible semi-annually. Two years later she pays the company 3000. Three years after that she pays the company 2000. How much does she owe seven years after the loan is taken out? (p. 31) 3. Eric deposits 8000 on Jan 1, 1995 and 6000 on Jan 1, 1997 and withdraws 12000 on Jan 1, 2001. Find the amount in Eric’s account on Jan 1, 2004 if the rate of compound interest is 15% per year (p. 31)

More Examples… 4. (Unknown time) John borrows 3000 from FFC. Two years later he borrows another 4000. Two years after that he borrows an additional 5000. At what point in time would a single loan of 1200 be equivalent if i = 0. 18 ? (p. 32) 5. (Unknown rate of interest) Find the rate of interest such that an amount of money will triple itself over 15 years (p. 32)

2. 2 UNKNOWN RATE OF INTEREST • We need to find the rate of interest i • Set up equation of value and solve it for i • Very often the resulting equation is a polynomial equation in i of degree higher than 2 • In general, there is no formula for solutions of equation of degree ≥ 5 (and the formulas for degrees 3 or 4 are very complicated) • Use approximations (numerical methods)

Examples 1. Joan deposits 2000 in her bank account on January 1, 1995, and then deposits 3000 on January 1, 1998. If there are no other deposits or withdrawals and the amount of money in the account on January 1, 2000 is 7100, find the effective rate of interest. 2. Obtain a more exact answer to the previous question

2. 3 TIME-WEIGHTED RATE OF RETURN • Let B 0, B 1, … , Bn-1, Bn denote balances in a fund such that precisely one deposit or withdrawal (denoted by Wt) is made immediately after Bt starting from t=1 • Let W 1, … , Wn-1 denote the amounts of deposits (Wt > 0) or withdrawals (Wt > 0) and let W 0 = 0 • Determine rate of interest earned in the time period between balances: The time-weighted rate of return is defined by i = (1+i 1) (1+i 2) … (1+in) - 1

Example (p. 35) On January 1, 1999, Graham’s stock portfolio is worth 500, 000. On April 30, 1999, the value has increased to 525, 000. At that point, Graham adds 50, 000 worth of stock to this portfolio. Six month later, the value has dropped to 560, 000, and Graham sells 100, 000 worth of stock. On December 31, 1999, the portfolio is again worth 500, 000. Find the time-weighted rate of return for Graham’s portfolio during 1999.