Industrial Operations 233

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Industrial Operations 233

• Investment Analysis

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Time value of Money

• Time Line
• P present value
• Fn Future value
• A Annuity (payments of equal amount)

Fn
P A A A A A

0 1 2 ... n-1 n
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Basic Formulas

• Fn P(1i)n
• P Fn/(1I)n
• n Log(Fn/P)/Log(1i)
• i (Fn/P)1/n - 1
• Above formulas are only valid for compound
  interest and with Fn and P (no annuities)

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Some examples

• (1) Find P of 10,000, 18 years hence, if the
  interest rate is 10 per annum compounded (i)
  half-yearly, (ii) quarterly, and (iii) monthly.
• Solution
• (i) n 36 i 0.05 p Fn/(1i)n
• P 10000/(1.05)36 1726.57
• (ii) n 72 i 0.025 Therefore P 1689.98
• (iii) n 216 i 0.0083 P 1677.30

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Examples Contd.

• (2) At 10 per annum converted quarterly, how
  long will it take for 750 to increase to 1000.
• Solution
• i 0.10 P 750 Fn 1000 n ?
• n Log(Fn/P)/Log(1i)
• n Log(1.333)/Log(1.025) n 11.64 quarters
• ie. 2.9 years, or 3 years.

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Examples Contd.

• (3) 500 has to be turned into 1200 in 7 years.
  What must be the annual interest rate compounded
  quarterly?
• Solution
• P 500 F28 1200 n 28 i ??
• i (Fn/P)1/n - 1 i (1200/500)1/28 - 1
  0.0317
• ie. 3.17 per quarter ie. 3.17 x 4 12.68 per
  year compounded quarterly.

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Time equivalence of money

• An important concept
• eg. 100 today is equivalent to 110 one year
  from now at i 10
• All the amounts above are equivalent
• The basis for project comparisons
• Comparisons have to be at the same point in time

100 110 121 133.1 146.4
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Interest rates

• Simple - ie. not compounding
• eg. 10 per annum
• Compound - any rate which is compounded
• Nominal
• the advertised interest rate by banks financial
  institutions
• eg. 10 per annum compounded quarterly
• Effective - the real int rate per period

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Effective Interest rate

• Two types
• effective per period effective per annum
• eg. Nominal, i 10 per annum comp quarterly
• effective per quarter (period) 10/4 2.5
• What is the effective rate per annum?

0 1 2 3 4
quarters
year
0 1
1 F?
I 10 p. a. comp quart
10
Effective Rate

• For quarters
• F 1(1.1/4)4 1.0254 1.1038 .. (1)
• For years
• F 1 (1i)1 1I .(2)
• Therefore, 1.1038 1i i 0.1038
• ie. 10.38 per annum effective (or, ie 10.38)
• From (1) (2)
• (1r/k)k (1ie) where, r nominal rate, k
  no of compoundings per annum
• Therefore, ie (1r/k)k - 1 ..(3)

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Examples

• (1) r 8.5 p.a. comp monthly. Find the
  effective rates.
• Effective rate per month 8.5/12 0.70833 per
  month
• Effective rate per annum, ie (1r/k)k - 1
• ie (1 .085/12)12 -1 0.08839 ie. 8.839
  p.a. effective
• (2) Bank A offers 9.5 interest credited half
  yearly. Bank B offers 9.25 interest comp daily.
  Where should we put our money?

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Examples Contd.

• Approach 1 Find effective interest rate
• Bank A ie (1.095/2)2 -1 0.09725, ie. 9.725
  
• Bank B ie (1.0925/365)365 -1 0.0969, ie.
  9.69
• Bank A gives higher effective interest
• Approach 2 Invest 100 for one year
• Bank A F 100(1 0.095/2)2 109.725
• Bank B F 100(1 0.0925/365)365 109.69
• Money grows faster in Bank A

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A Complete Example

• I wish to save 5000 for a boat, which I intend
  to buy 2 years from now
• (a) How much should I invest today if int rate is
  8 p.a. comp quarterly?
• Solution F 5000 n8 quarters i0.02 P?
• P5000/(1.02)8 4267.45
• (b) 6 months after my investment rate drops to 6
  p.a. quarterly. How much will I be short of 5000
  at the end of 2 years?

0 2 4 6 8
i.02 i.015
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Complete Example Contd.

• Solution P4267.45 i.02 for the first 2
  quarters and .015 for the remaining 6 quarters
  F?
• F4267.45(1.02)2(1.015)6 4854.72
• Shortfall 5000 - 4854.72 145.28
• (c) How much longer would it take to reach 5000
• Solution P4854.72 F5000 i0.015 per quar
  n?
• n Log(Fn/P)/Log(1i) Log(5000/4854.72)/Log(1.0
  15)
• n 1.98 quarters

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Tutorial

• (1) Describe the concept of time equivalence of
  money. Why it is so important.
• (2) How long will it take for a sum of money to
  double itself if the interest rate is 12 p.a.
  comp monthly?
• (3) A loan of 10000 is obtained at 10 p.a. comp
  quarterly. After 2 years a partial repayment of
  6000 is to be made, with the balance a further 2
  years later. How much is the balance?
• (4) Bill borrowed 3750 at 6 p.a. comp
  quarterly, and settled his debt by a payment of
  4500. When was the payment made?
• (5) Max borrows 10000 at 12 p.a. converted
  quarterly. He is to repay 2000 at the end of
  each of the first 4 years, with a final payment
  at the end of 5 years. What is the size of the
  final payment?