Compound Interest: Further Topics and Applications. Chapter 9

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2 9-2 Compound Interest: Further Topcs and Applcatons Chapter 9

3 9-3 Learnng Objectves After letng ths chapter, you wll be able to:? Calculate the nterest rate and term n ound nterest applcatons? Gven a nomnal nterest rate, calculate ts effectve nterest rate? Gven a nomnal nterest rate, calculate ts equvalent nterest rate at another oundng frequency

4 9-4 Usng Fnancal Calculators n PV PMT represents the number of oundng perods represents the perodc nterest rate, j/m represents the prncpal or present value represents the maturty value or future value represents the perodc annuty payment (not used untl chapter 10) used to tell calculator to ute (CPT)

5 9-5 What amount must you nvest now at 6% ounded daly to accumulate to $5000 after 1 year? j= m = 6% n S = = $5000 =.06/365 n = 1 * 365 = 365 6/365= PMT PV Ans: $

6 Calculatng the perodc nterest rate, 9-6 = ( / PV) 1/n - 1

7 9-7 Suppose you had $4000 and you wanted t to grow to $9000 n 7 years so you could buy a car. what rate of nterest do you need to aclsh your goal? (Assume nterest s ounded annually.) Solvng for ths becomes, = (/PV) 1/n - 1 PV = = n = $4000 $ /x = sto 9000 / 4000 = y x rcl = - 1 = Ans: = 12.28% ounded annually = PV(1+) n 7 n /- PV 0 PMT

8 Suppose you had $4000 and you wanted t to grow to $9000 n 7 years so you could buy a car. What rate of nterest do you need to aclsh your goal? (Assume nterest s ounded monthly.) = (/PV) 1/n - 1 PV = = n = Same queston as 84 last tme, n but 84 now 1/x wth = sto nterest ounded monthly $4000 nstead of annually / 4000 = /- PV $9000 y x rcl = - 1 = 0 PMT 7*12 = 84 Ans: * 12 = 11.64% pa ounded monthly Fnancal Calculator sol n: Don t forget to multply by 12 to get nomnal rate 9-8

9 9-9 Suppose you had $8500 and you wanted t to grow to $13000 n order to buy a new car. You knew that you could get 8% ounded daly. How long wll t take to reach your goal? Reorganzng to solve for n: n = ln(/pv) / ln(1+ ) = PV(1+) n Fnancal Calculator sol n: PV = = = $8500 $ /365= /365+1 = lnx sto 13000/8500= lnx 8/365= /- PV / rcl = 0 PMT days = 5yr 3mo 22 days n

10 9-10 How long wll t take at 11.2% pa ounded monthly for $700 to grow to $1200? n = ln(/pv) / ln(1+ ) Fnancal Calculator sol n: PV = $ /12+1 = lnx 11.2/12= = = $ /12=.0093 sto 1200/700= lnx /- PV / rcl = 0 PMT n months = 4 yr 8.5 months

11 9-11 How long wll t take for money to trple when nvested at 7.2% pa ounded sem-annually? n = ln(s/p) / ln(1+ ) No PV or gven use easest numbers you can thnk of Fnancal Calculator sol n: PV = $1.072/2+1 = lnx 7.2/2= = = $3.072/2=.036 sto 3/1= lnx 3 1 +/- PV / rcl = 0 PMT n sem-annual perods = years

12 9-12 Nomnal and Effectve Rates of Interest Truth n Savngs Law Annual Percentage Yeld Nomnal Rate (Stated Rate) - The rate on whch the bank calculates nterest. Effectve Rate = Interest for 1 year Prncpal > the annually ounded rate that produces the same future value as the gven nomnal rate > use the symbol f

13 9-13 Nomnal and Effectve Rates of Interest Compared Begnnng Nomnal rate Compoundng End Effectve rate balance of nterest perod balance (APY) of nterest $1, % Invest for 1 year Annual Semannual Quarterly Daly $1, % $1, % $1, % $1, %

14 9-14 What s the effectve rate of nterest correspondng to 9% ounded daly? f = (1 + ) m - 1 j = m = = /365 Fnancal Calculator sol n: 100 +/- PV 12 n f = (1 +.09/365) =.0942 = 9.42% $ mples an effectve rate of 9.42% 9/365= 0 PMT

15 9-15 Whch s the most attractve of the followng nterest rates offered on 3 year GICs? A)5.8% cmpd sa B)5.68% cmpd mo C)5.4% cmpd daly It s faster to use the Fnancal Calculator sol n: 100 +/- PV 2 n 12 n 365 n 5.8/2 = 0 PMT 5.68/12= 5.4/365= $ $ $ The best rate s 5.8% cmpd sem-annually

16 9-16 Equvalent Interest Rates Interest rates that produce the same future value after one year

17 9-17 Clarfyng Termnology ounded quarterly =? per quarter refers to oundng frequency refers to oundng perod

18 9-18 For a gven nterest rate of 8% ounded quarterly, what s the equvalent nomnal rate of nterest wth monthly oundng? 2 = (1+ 1 ) m 1 /m and m 1 represent the gven values of and m 2 and m 2 represent the equvalent values of and m 2 = (1+.08/4) 4/12-1 = j = * 12 = 7.95% Calculator sequence 4 / 12 = sto.08 / = y x rcl = - 1 =

19 9-19 For a gven nterest rate of 8% ounded quarterly, what s the equvalent nomnal rate of nterest wth monthly oundng? Fnancal Calculator soluton We requre the equvalent perodc rate to produce the same future value of $100 after one year as the gven perodc rate /- 8/4 = PV Step 1: Calculate the of $100 nvested for 1 year as gven. 4 0 n PMT

20 9-20 For a gven nterest rate of 8% ounded quarterly, what s the equvalent nomnal rate of nterest wth monthly oundng? Fnancal Calculator soluton 7.95% pa cmpd monthly = 8% pa cmpd quarterly 12 Step 2: Compute the value of 2 that also makes $100 grow to the same n m 2 oundngs. Step 1: Calculate the of $100 nvested for 1 year as gven. n Don t forget to multply by 4 to get nomnal rate 100 8/4= 4 0 +/- PV n PMT

21 9-21 THE END