(2.4)

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2.4 NOMINAL, PERIODIC AN D EFFECTIVES RATES The term \"effective" is used for rates of interest (and discount) in which interest is paid once a year. either at the end of the year or at the beginning of the year, as the case may be. When interest is paid more frequently than once year. it is called nominal rate of interest (or discount}. That is. Nominal rate,j. Nominal means \"in name only". This is sometimes called the quoted rate. Periodic rate, % . The amount of interest you are charged each period. Eectlve annual rate, l. The rate that you actually get charged on an annual basis. Illustration 2: Lender A might charge an annual effective rate of 18% on loans while Lender B might charge 18% compounded quarterly. Lender A is charging an effective rate i while Lender B is charging at a nominal rate of interest of 18% compounded quarterly. Eighteen percent compounded quarterly is equivalent to saying 4.5% per quarter which is called the periodic rate. Example 11. Assume that a credit card has an interest rate of 396 per month. What is the annual rate that you are actually charged? Solution: The nominal rate is 36% compounded monthly. The periodic rate is 3% per month. (You are charged 3% interest on your balance every month). The effective annual rate is 42.57% When interest is compounded annually. the effective rate i is equal to the nominal rate j but when it is compounded more than once a year, the effective rate is higher than the nominal rate. These rates are equivalent in case P under each method gives the same compounded amount at the same period of time. Hence. to nd the m equivalent effective rate of a nominal rate, use the fonnula(1+l)=[1+%] . From this formula different formulas may be derived. A. Deriving the formula for i (1 +l)= [1 +m J m |=[1+] 1 m B. Deriving the formula forj: 46 m (1 +1)1/m - 1 = j 1 = m (1+1)1/m -1] To summarize, NOMINAL RATE, EFFECTIVE RATE (1 + 1) = (1+1)" i= 1+ 1 - 1 j =m [(1+ 1)/m -1 ] From example 11, 1= 1+4 -1 1= 1+ 0.36 )12 12 -1 1 = (1.03) 2 -1 i = 0.425760x100 i = 42.58% effective Example 12. If 100 is invested for a year at 8% compounded quarterly, find the effective rate of interest. Solution: Given: P = 100, j = 8%, m = 4, t = 1 year, n = 4 1= ( 1 + j )m "m) -1 1= 14 0.08 )4 4 - 1 i = (1.02)4 -1 i = 0.082432x100 i = 8.24% effective Example 13. What effective rate is equivalent to each of the following nominal rates compounded semiannually.: A) 5.5%? B) 8% C) 9%? D) 10% Solutions: A) Given: j = 5.5%, m =2 1= 1 +4 -1 1= 1+ 0.055 )2 2 -1 i= (1.0275)2 -1 i = 0.0558x100 i= 5.58% effective 47B) Given: j = 8%, m = 2 1= 1+4 -1 1= 1+ 0.08 2 i = (1.04)2-1 i = 0.0816x100 i = 8.16% effective C) Given: j = 9%, m = 2 1= 1 +4 -1 1= 1+ 0.09 )2 2 i= (1.045) - 1 i = 0.092025x100 i =9.20% effective D) Given: j = 10%, m = 2 1= 1 +4 -1 1= 1+ 0.10 2 -1 i = (1.05)2 -1 i = 0. 10250000x100 i=10.25% effective Example 14. Find the equivalent nominal rate corresponding to each conversion period of the effective rate 5%. A) annually; B) semiannually; C) quarterly; and D) monthly. Solutions: A) Given: i = 5%, m = 1 1 = m (1+1)1/m -1] j =1 (1.05) -1 j = 0.05 x 100 j = 5% compounded annually B) Given: i = 5%, m =2 j = m (1+1)1/m -1] j = 2 (1.05)1/2-1 j = 0.049390 x 100 j = 4.94% compounded semiannually 48i 2(1.05\"(1+2)1)x100=4.94 C) Given: i = 5%, m = 4 J=m[(1+1)\""'1] j=4[(1.c+5)1M 4] j = 0.049038 x 100 j = 4.91% compounded quarterly D) Given: i =5%, In: 12 =m[(1+l]\"m1] J=12[(1.cs}\"'21] j = 0.048889 x 100 j = 4.89% compounded monthly EXERCISES 2.4 01. 33593353 3 .8 10. Find the effective rate corresponding to the rate 7%. converted A) annually; B) semiannually; C) quarterly and D) monthly. Find the effective rate if the nominal rate is 5%, compounded A) annually: B) semiannually: C) quarterly and D) monthly. Which yields more interest. {3%. m = 4} or {3%. effective}? Which yields more interest. {5%. m = 2} or [5%. effective}? If interest is converted semiannually. nd the nominal rate. which will yield an effective rate of 5 1-596. If interest is compounded quarterly. find the nominal rate. which will yield an effective rate of 4.65%. What effective rate will yield a nominal rate of A) 6% compounded quarterly; B) 7% compounded monthly and C) 8.5% compounded annually. What effective rate will yield a nominal rate of A) 5% compounded semiannually; B) 9% compounded monthly and C) 1.5% compounded mutually. Which yields more interest. {6.25%. m = 2} or {6%, effective}? Which yields more interest. {5.75%. m = 2} or {6%, effective}? 49