NOMINAL - PowerPoint PPT Presentation
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Title: NOMINAL
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NOMINAL EFFECTIVE INTEREST RATECONTINUOUS
COMPOUNDING
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NOMINAL EFFECTIVE RATES
- Payment Period, Tp - Length of time during which
cash flows are not recognized except as end of
period cash flows. - Compounding Period, Tc - Length of time between
compounding operations. - Interest Rate Period, T - Interest rates are
stated as per time period. T is the time
period.
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NOMINAL EFFECTIVE RATES
- Nominal Interest Rate - rate at which money
grows, /T, without considering compounding. - For example if the nominal interest rate, r,
were - r 1.5 per month,
- the nominal rate would also be
- r 0.05 per day if 30 days/mo.
- r 4.5 per
quarter - r 9 semiannually
- r 18 per year
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NOMINAL EFFECTIVE RATES
- Effective Interest Rate - rate at which money
grows, /T, considering compounding. - Effective Interest Rate, i, equals the nominal
interest rate, r, if the interest rate period and
the compounding period are equal. (TTc)
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NOMINAL EFFECTIVE RATES
- Example
- A Bank pays a nominal 12 per year compounded
semiannually. What is the effective interest rate
for 6 mo.? 12 mo? 24 mo.? - The nominal interest is 12 per year so the
nominal interest rate is also 6 per 6 mo. - The effective interest rate is also 6 per 6 mo.
since 6 mo. is the compounding period.
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NOMINAL EFFECTIVE RATES
- What is the effective interest rate for 12 mo.?
- F P(1 i)2 where i .06
- F P(1 .06)2 P (1.124)
- i is 12.4 per year
- What is the effective interest for 24 mo.?
- FP(1.06)4P(1.262)
- i is 26.2 per 24 months
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NOMINAL EFFECTIVE RATES
- In general for m compounding periods per time
period where r is the nominal rate per time
period - F P(1 r/m)m
- r is in /T and m T/Tc
- If i is the effective interest rate per time
period - F P(1 i)
- Hence (1 i) (1 r/m)m
- or i (1 r/m)m - 1
- i and r are in /T
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NOMINAL EFFECTIVE RATES
- Example
- The nominal rate is 1 per month and
compounding occurs monthly - What is the effective rate per 12 months
- r must be in /T for T12 months. r.12
- m T/Tc m 1 year / 1 month 12
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NOMINAL EFFECTIVE RATES
- Example -
- The nominal rate is 6 per quarter
- What is the effective semi-annual interest rate
if compounded monthly? - 6 per quarter is 12 per semiannual period
- effective and nominal rate per month is r such
that
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Continuous Compounding
- Using the relationship between r and i, it is
possible to compute the effective interest rate
if compounding occurred in an infinitely small
period. This is called continuous compounding. - Taking the limit as m approaches infinity
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Continuous Compounding
- Example -
- For a nominal interest rate of 12 per year what
is the effective Monthly and yearly rate for
continuous compounding. - First find the nominal rate of the time period in
question, month, r 12 / year 1 / month - i per month e.01-1 .01005
or 1.005 - The nominal annual rate is 12
- i per year
e.12 -1 - .1275 or
12.75
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NOMINAL EFFECTIVE RATES
- Adjusting i, r, and n for Single Payment Factors
- Two requirements
- i must be effective rate of interest
- payment period measured in the same units as the
interest rate, which in the simplest form means - n equals the number of compounding periods
- compounding period and payment period are equal
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NOMINAL EFFECTIVE RATES
- EXAMPLE -
- A deposit of 3000 is made after 3 years and
5000 after 5 years, what is the value of the
account after 10 years if the interest rate is
12 per year compounded semiannually?
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NOMINAL EFFETIVE RATES
- Select payment period of one year
- Find the effective interest rate per year
- r 12 per year
- i (1 .12/2 )2 - 1 .1236 or
12.36 - Contribution of 3K, 3000 (FP, .1236, 7)
- i (FP, i,
7) - 12
2.2017 - 13
2.3526 - 3000 (2.2017.36(2.3526-2.2
017)) - 3000(2.256)
6768.07 -
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NOMINAL EFFECTIVE RATES
- Contribution of 5K, 5000(FP, .1236, 5)
- i
(FP, i, 5) - 12
1.7623 - 13 1.8424
- 5000 (1.7623 .36(1.8424 - 1.7623))
- 5000 (1.791) 8955.68
- F 6768.07 8955.68 15,723.75
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NOMINAL EFFECTIVE RATES
- Adjusting i, r, and n for Uniform Series Factors
- If the Payment Period is gt Compounding Period
- n is defined as the number of payments
- i is effective interest over one payment period
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NOMINAL EFFECTIVE RATES
- EXAMPLE 100 is deposited each quarter for 5
years. What is the present value of the account
if interest is 12 per year compounded monthly. - Payment Period is one quarter
- Nominal rate per month is .12/12 .01 which is
also the effective interest rate - The effective interest per quarter is then
- i (1
.03/3)3 - 1 .0303
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NOMINAL EFFECTIVE RATES
- With the effective rate per quarter F may be
calculated - F 100(FA, .0303, 20)
- 100
(26.8704 .03 (29.778 - 26.8704)) - 100
(26.9576) 2695.76
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NOMINAL EFFECTIVE RATES
- Adjusting i, r, and n for Uniform Series Factor
- If Payment Period lt Compounding Period
- In this case the method for handling payments
made between compounding periods must be defined
and the cash flow diagram modified to reflect
equivalent end of period payments
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NOMINAL EFFECTIVE RATES
- Example -
- Deposits are made as follows 5000 at the end
of February, 1000 at the end of April, and 7000
at the end of Sept. If interest is 12 per year
compounded semiannually and no interest is paid
on inter-period deposits, find the value of the
payment at years end.
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NOMINAL EFFECTIVE RATES
EQUIVALENT PAYMENTS
PAYMENTS
J
F
M
A
M
J
J
A
S
O
N
D
5000
1000
7000
6000
7000
F 6000(FP, .06, 1) 7000 6000 (1.06)
7000 13,360
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NOMINAL EFFECTIVE RATES
- Same Problem except simple interest is paid on
inter-period payments.
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NOMINAL EFFECTIVE RATES
J
F
M
A
M
J
J
A
S
O
N
D
5000
1000
7000
F6
F12
F6 5000 ( 1 (4/6) .06) 1000 (1 (2/6) .06)
6220
F12 7000 (1 (3/6) .06) 7210
F F6 (FP, .06, 1) F12 6220 (1.06)
7210 13,803.20
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NOMINAL EFFECTIVE RATES
- SUMMARY
- Payment Period - Time period during which
payments are recognized only as end of period
payments. - Compounding Period - time between compounding
operations - Set T Tp
- then if TTc , i r
- T gt Tc then T set equal to Tp with i (1 r/m)m
-1 - T lt Tc then T set equal to TcTp and methods of
dealing with inter-period payments are defined
