CHAPTER 4 Nominal and Effective Interest Rates

Transcription

1 CHAPTER 4 Nominal and Effective Interest Rates 4-1

2 4.1 Nominal and Effective Interest Rate Statements q q The time standard for interest computations One Year Interest can be computed more frequently than one time a year Ø Annually One time a year (at the end) Ø Every 6 months 2 times a year (semi-annual) Ø Every quarter 4 times a year (quarterly) Ø Every Month 12 times a year (monthly) Ø Every Week 52 times a year (weekly) Ø Every Day 365 times a year (daily) Ø Continuous infinite number of compounding periods in a year. 4-2

3 Three Time Based Units q Interest Period The period over which the interest is expressed (always stated). Ø Ex: 1% per month q Compounding Period (CP) The shortest time unit over which interest is charged or earned. Ø Ex: 8% per year, compounded monthly q Compounding Frequency The number of times m that compounding occurs within time period t. Ø Ex: 1% per month, compounded monthly has m = 1 Ø Ex: 10% per year, compounded monthly has m = 12 One Year is segmented into: 365 days, 52 weeks, 12 months One quarter is: 3 months with 4 quarters/year 4-3

4 Nominal Rate of Interest (r) q Definition: An interest rate that does not include any consideration of compounding r = interest rate per period X number of periods q For example, if interest rate is 1.5% per month Ø Nominal rate per year = 18% Ø Semi-annual nominal rate = 9% Ø 4.5% per quarter 4-4

5 Effective Interest Rate Definition: qthe effective interest rate is the actual rate that applies for a stated period of time. qthe compounding of interest during the time period of the corresponding nominal rate is accounted for by the effective interest rate. qthe effective rate is commonly expressed on an annual basis denoted as i a All interest formulas, factors, tabulated values, and spreadsheet relations must have the effective interest rate to properly account for the time value of money. 4-5

6 Interest statement q r % per time period, compounded m-ly Ø 12% per year, compounded monthly -> the effective rate is 1% per month Ø 12% per year, compounded quarterly -> the effective rate is 3% per quarter Ø 3% per quarter, compounded monthly -> the effective rate is 1% per month q If the compounding frequency is not stated, it is assumed to be the same as the time period of r, in which case the nominal and effective rates have the same value Ø 12% per year or 2% per month 4-6

7 The Effective Rate per CP The Effective rate per compounding period (CP) is: r% per time period t r = m compounding periods per t m Ex: 9% per year, compounded quarterly: r = 9%, m = 4(4 quarters in a year), CP=quarter i per month = 0.09/4 = 2.25% per quarter 4-7

8 4.2 Effective Annual Interest Rate q Notation r = the nominal interest rate per year. m = the number of compounding periods within the year. i = the effective interest rate per compounding period (r /m) i a or i e = the true, effective annual rate given the value of m. 4-8

9 Derivation of the relationship q Interest could be compounded more than one time within the year! m Assume the one year is now divided into m compounding periods. 4-9

10 Derivation of the relationship q Compounding frequencies per year = m, Effective rate per CP = r/m, q Solving for i a yields; 1 + i a = (1+r/m) m i a = (1 + r/m ) m 1 Thus, i a = ( 1+ r/m) m

11 Example q Given: interest is 18% per year, compounded monthly P=$1,000. What is the future value after 1 year? 4-11

12 Example: 12% Nominal No. of EAIR EAIR Comp. Per. (Decimal) (per cent) Annual % semi-annual % Quartertly % Bi-monthly % Monthly % Weekly % Daily % Hourly % Minutes % seconds % See Table 4.3 for more figures 4-12

13 Example 4.2: Credit card case q APR(annual percentage rate): nominal rate q APR = 14.24% with no CP mentioned -> CP= 1year q However credit card payments are required monthly (a) Effective interest rate if CP=year and CP=month 4-13

14 Example 4.2: Credit card case (b) If he accepts the card and completes the $1,000 transfer, what is the total balance he owes after one month later? 4-14

15 Example 4.4: Credit card case q 1년간연체시상환하여야할총액과실질이자율? 연체시이자율 : 29.99%/12 = 2.499% per month 연체수수료 : $

16 Example 4.4: Credit card case To find an effective monthly rate =1000(F/P, i, 12) = 1000(1+i) 12 i= 5.278% -> compare with 1.187%! Effective annual rate? 4-16

17 Section Equivalence Relations: Comparing Payment Period and Compounding Period Lengths (PP vs. CP) 4-17

18 Payment Period (PP) q Recall: Ø CP is the compounding period q PP : frequency of payments or receipts Ø PP is the payment period q Why CP and PP? Ø Often the frequency of depositing funds or making payments does not coincide with the frequency of compounding. 4-18

19 Equivalence: Comparing PP to CP q Reality: ØPP and CP s do not always match up; q Savings Accounts for example; ØMonthly deposits with, ØQuarterly interest earned or paid; ØThey don t match! q Make them match! (by adjusting the interest period to match the payment period.) 4-19

20 Ex. 4.7 Single Amounts: PP >= CP q r = 12% per year, compounded semiannually q Future worth at 10-year? F 10 =? r = 12%/yr, c.s.a $1,000 $1,500 $3,

21 Ex. 4.7 Single Amounts: PP >= CP q Method 1. (n relates to months) Ø Determine effective rate per CP and set n equal to the number of CPs between P and F 4-21

22 Ex. 4.7 Single Amounts: PP >= CP q Method 2. (n relates to years) Ø Determine the effective rate for the time period t of the nominal rate, and set n equal to the total number of periods 4-22

23 Ex. 4.8 Series : PP >= CP F 7 =?? q Consider: A = $500 every 6 months Find F 7 if r = 8%/yr, c.q. (PP > CP) q Fine the effective rate per PP! q Determine n as the total number of PP 4-23

24 Ex. 4.8 Series : PP >= CP q We need i per 6-months effective q Now, the interest matches the payments. 4-24

25 Example 4.9 Credit card case q Want to payoff $1,030 by monthly automatic checking account transfer for 2 years q What amount? And APY? q APY(Annual percentage Yield): effective rate 4-25

26 Section 4.8 Effective Interest Rate for Continuous Compounding 4-26

27 Continuous Compounding q Recall: ØEAIR = i = (1 + r/m) m 1 ØWhat happens if we let m approach infinity? ØThat means an infinite number of compounding periods within a year or, Ø The time between compounding approaches 0. ØWe will see that a limiting value will be approached for a given value of r 4-27

28 Derivation of Continuous Compounding q We can state, in general terms for the EAIR: i r = (1 + ) m -1 m m é ù r r m æ r ö (1 + ) - 1 = ê 1+ ú -1 m ê ç è m ø ú ë û r 4-28

29 Derivation of Continuous Compounding q Recall that h æ 1 ö lim ç 1+ = e = h è h ø m r æ r ö lim ç 1 + = e, m è m ø So that: m é ù æ r ö r r i = lim ê 1+ ú - 1 = e -1. m ê ç è m ø ú ë û 4-29 r

30 Derivation of Continuous Compounding q The EAIR when interest is compounded continuously is then: EAIR = e r 1 Where r is the nominal rate of interest compounded continuously. This is the max. interest rate for any value of r compounded continuously. 4-30

31 Ex (a) q r = 18% per year, compounded continuously n Effective annual rate? e = = 19.72%/year The 19.72% represents the MAXIMUM EAIR for 18% compounded anyway you choose! n Effective monthly rate? r/month = 0.18/12 = 1.5%/month effective monthly rate is e = 1.511%/month 4-31

32 Finding r from the cont. comp. EAIR q To find the equivalent nominal rate given the EAIR when interest is compounded continuously, apply: i = e r -1 e r = i + 1 \ r = ln( i + 1) 4-32

33 Ex (b) q An investor requires an effective return of at least 15% per year. q What is the minimum annual nominal rate that is acceptable? q Nominal rate with continuously compounding! r = ln(1.15) = = 13.98% A rate of 13.98% per year, with c.c. generates the same as 15% true effective annual rate. 4-33

34 Ex q P=$5,000, n= 10 years, r=10% per year q Marci receives annual compounding, while Suzanne continuous compounding 4-34