NOMINAL AND EFFECTIVE INTEREST RATE

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NOMINAL AND EFFECTIVE INTEREST RATE
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Nominal and Effective Interest Rate
This topic is very important. Sometimes one
interest rate is quoted, sometimes another is
quoted. If you confuse the two you can make a
bad decision.  A bank pays 5 compounded
semi-annually. If you deposit 1000, how much
will it grow to by the end of the year? Solution
The bank pays 2.5 each six months. You get
2.5 interest per period for two periods. The
result is 1000 ? 1000(1.025) 1,025 ?
1025(1.025) 1,050.60 With i 0.05/2, r
0.05, P ? (1 i) P ? (1r/2)2 P (1 0.05/2)2 P
(1.050625) P
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Nominal and Effective Interest Rate
Terms the example illustrates r 5 is called
the nominal interest rate per interest period
(usually one year) i 2.5 is called the
effective interest rate per interest period ia
5.0625 is called the effective interest rate per
year (annum) m 2 is the number of
compounding subperiods per time period. In the
example, ia (1.050625) 1 (1.025)2 1 (1
r/m)m 1.
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Nominal and Effective Interest Rate

• Conclusion Effective interest rate per year
•  
• with
• r nominal interest rate per year 
• m number of compounding sub-periods per year
• i r/m effective interest rate per compounding
• sub-period.
• Remark. The term i we have used up to now is
  more precisely defined as the effective interest
  rate per interest period. If the interest period
  is one year (m 1) then i r.

ia (1 r/m)m 1 (1 i)m 1
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Nominal and Effective Interest Rate

• Example 4-14. A bank pays 1.5 interest every
  three months. What are the nominal and effective
  interest rates per year?
• Solution 
• Nominal interest rate per year r 4 ? 1.5
  6 a year
• Effective interest rate per year
• ia (1 r/m)m 1 (1.015)4 1 0.06136 ?
  6.14 a year.
• See Table 4-1, Nominal Effective Interest for
  various compounding period lengths. (Excel
  sheet).

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Nominal and Effective Interest Rate
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Nominal and Effective Interest Rate

• Example 4-15. Joe Loan Shark lends money on the
  following terms. Duh, If I gib you 50 dolla on
  Monday, den youse guys owes me 60 dolla da
  following Monday.
• 1.What is the nominal rate, r?
• We first note Joe charges i 20 a week, since
  60 (1i)50 ? i 0.2. Note we have solved F
  50(F/P,i,1) for i.  
• We know m 52, so r 52 ? i 10.4, or 1,040 a
  year.  
• 2. What is the effective rate, ia ? 
• From ia (1 r/m)m 1 we have ia
  (110.4/52)52 1 ? 13,104. This means about
  1,310,400 a year.

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Nominal and Effective Interest Rate

• 3. Suppose Joe can keep the 50, as well as all
  the money he receives in payments, out in loans
  at all times? How much would Joe L. Shark have
  at the end of the year?
• We use F P(1i)n to get F 50(1.2)52 ?
  655,232 (not bad, but probably illegal). 
• Words of Warning. When the various compounding
  periods in a problem all match, it makes
  calculations much simpler. When they do not
  match, life is more complicated.
• Recall Example 4-3. We put 5000 in an account
  paying 8 interest, compounded annually. We want
  to find the five equal EOY withdrawals. We used
  A P(A/P,8,5) 5000 ? (0.2505) ? 1252.
  Suppose the various periods are not the same in
  this problem.

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Nominal and Effective Interest Rate

• Example 4-16. Sally deposits 5000 in a CU
  paying 8 nominal interest, compounded quarterly.
  She wants to withdraw the money in five equal
  yearly sums, beginning Dec. 31 of the first year.
  How much should she withdraw each year? 
• Note effective interest is i 2 r/4 8/4
  quarterly, and there are 20 periods.
• Solution

W
W
W
W
W
i 2, n 20
5000
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Nominal and Effective Interest Rate

• Note the withdrawal periods and the compounding
  periods are not the same. If we want to use the
  formula
• A P (A/P,i,n)
• then we must find a way to put the problem into
  an equivalent form where all the periods are the
  same.
• Solution 1. Suppose we withdraw an amount A
  quarterly. (We dont, but suppose we do.). We
  compute 
• A P (A/P,i,n) 5000 (A/P,2,20) 5000
  (0.0612) 306.

i 2, n 20
5000
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Nominal and Effective Interest Rate

• These withdrawals are equivalent to P 5000.
  Now consider the following
• Consider a one-year period.
• This is now in a standard form that repeats every
  year.
• W A(F/A,i,n) 306 (F/A,2,4) 306 (4.122)
  1260.
•  Sally should withdraw 1260 at the end of each
  year.

A
W
W
W
W
i 2, n 4
W
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Nominal and Effective Interest Rate

• Solution 2. Probably the easiest way.
• ia (1 r/m)m 1 (1 i)m 1 (1.02)4 1
  0.0824 ? 8.24
• Now use
• W P(A/P,8.24,5) P i (1i)n/(1i)n 1
  5000(0.252)
• 1260 per year.

W
5000
ia 8.24, n 5
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Continuous Compounding

• Continuous compounding can sometimes be used to
  simplify computations, and for theoretical
  purposes. The table above illustrates that er -
  1 is a good approximation of (1 r/m)m for large
  m. This means there are continuous compounding
  versions of the formulas we have seen earlier.
• For example,
• F P er n is analogous to F P (F/P,r,n) 
• P F e-r n is analogous to P F (P/F,r,n)
• We will pay little attention to continuous
  compounding in this course. You are supposed to
  read the material on continuous compounding in
  the book, but it will not be included in the
  homework or tests.

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Summary. Notation

•  i effective interest rate per interest period
  (stated as a decimal) 
• n number of interest periods 
• P present sum of money  
• F future sum of money an amount, n interest
  periods from the present, that is equivalent to P
  with interest rate i 
• A end-of-period cash receipt or disbursement
  amount in a uniform series, continuing for n
  periods, the entire series equivalent to P or F
  at interest rate i. 
• G arithmetic gradient uniform period-by-period
  increase or decrease in cash receipts or
  disbursements 
• g geometric gradient uniform rate of cash flow
  increase or decrease from period to period
• r nominal interest rate per interest period
  (usually one year)
• ia effective interest rate per year (annum) 
• m number of compounding sub-periods per period

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Summary Formulas

• Single Payment formulas
• Compound amount F P (1i)n P (F/P,i,n)
• Present worth P F (1i)-n F (P/F,i,n)
• Uniform Series Formulas
• Compound Amount F A(1i)n 1/i A
  (F/A,i,n)
• Sinking fund A F i/(1i)n 1 F
  (A/F,i,n)
• Capital recovery A P i(1i)n/(1i)n 1
  P (A/P,i,n)
• Present worth P A(1i)n 1/i(1i)n A
  (P/A,i,n)

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Summary Formulas

• Arithmetic Gradient Formulas
• P G (1i)n i n 1/i2 (1i)n (present
  worth)
• G 1/i n/(1i)n 1 G (P/G,i,n)
• A G (1i)n i n 1/i (1i)n i
  (uniform series)
• G 1/i n/(1i)n 1 G (A/G,i,n)
• Geometric Gradient Formulas 
• If i ? g,
• P A1 1 (1g)n(1i)-n/(i-g) A1
  (P/A,g,i,n)
• If i g,
• P A1 n (1i)-1 A1 (P/A,g,i,n)
•  

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Summary Formulas

• Nominal interest rate per year, r the annual
  interest rate without considering the effect of
  any compounding  
• Effective interest rate per year, ia
• ia (1 r/m)m 1 (1i)m 1 with i ? r/m