Chapter 2: The Theory of Interest

1
Chapter 2 The Theory of Interest

• Basic Interest Idea

1.00 Principal
0.08 interest
1.08
First of Year
End of Year
Interest rate expressed as a decimal r 0.08
(1 r) 1.00 1.08
2
Simple Interest Rate
3
Simple Interest Rate (Contd)

• If the proportional rule holds for fractional
  years, after any time t (measured in years) then
• V (1 r t) A
• Certificates of Deposit sometimes pay simple
  interest.

4
Compound Interest Rate
5
Compound Interest Rate (Contd)

• Question. Would you rather have simple interest,
  or compound interest?
• Guesses Suppose you can get 10 compound
  interest yearly on a 2,000 IRA. How much would
  it be worth after 10, 20, 30 or 40 years?

6
Seven-Ten Rule

• (1.1)7 1.949. Money invested at 10 per year
  doubles in about 7 years.
• (1.07)10 1.967. Money invested at 7 per year
  doubles in about 10 years.
• Twenty-Eighty Rule

7
Compound Interest (Compound.xls)
8
Compound Versus Simple Interest

• Compound interest is said to exhibit geometric
  growth.
• Given the same annual rate, compound interest
  always gives a greater value than simple
  interest, since simple interest grows linearly.

9
Effective Interest Rate

• Compounding at other intervals than yearly e.g.,
  daily, monthly, quarterly.
• Quarterly compounding of an interest rate r for a
  year applied to an amount A results in (1r/4)4 A
  in a year. In this case, r satisfying 1 r
  (1r/4)4 is called the effective interest rate
  and r is nominal rate
• Example r 0.08, r/4 0.02, 1 r (1.02)4
  1.0824, r 0.0824 is the effective interest
  rate. Note r gt r.

10
Continuous Compounding

• useful for modeling simplifications
• If an interest rate r is compounded m times per
  year, after m periods, the result is
• (1 r/m)m
• Further,
• lim m-gt ? (1 r/m)m er , where e ? 2.7818
• Example e0.08 1.0833, compared with (1.02)4
  1.0824

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Continuous Compounding (Contd)

• Continuous Compounding for t years gives
• (1 r/m)m t -gt er t

12
Debt

• You can also be on the paying end of compound
  interest. It works both ways.
• If you borrow money and do not make payments
  after one year your debt is
• (1r/12)12
  (interest is charged monthly)
• Interest rates change regularly and depend upon
  the borrowing period
• Question. Is paying credit card interest a good
  idea?
• (explain credit card tricks with grace period
  and compound interest)

13
Credit Card Example (CrdCard.xls)
14
Present Value

• Present Value (reverse the Future Value Idea)
• r
• Future Value Idea A ? A (1r) A
• 
  d1
• Present Value Idea A 1/(1r) A d1 A ?
  A
• Example. If 100 is worth 108 in one year, then
  the present value of that 108 is 100.

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Example

• If 100 is worth 108 in one year, then the
  present value of that 108 is 100.
• Receiving 108 in one year is no different than
  putting 100 in a bank and letting it earn 8
  interest for one year.
• If you owe 108 in one year, its present value is
  just 100, since you could put 100 in the bank
  today at 8 interest and have the money to pay
  the debt in a year.
• If you owe an amount A in one year, you could
  put A/(1r) in the bank today at interest rate r
  and have the money to pay the loan in a year.

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Discounting

• (1-year) Discount Factor d1 1/(1r).
• Example r 0.08, d1 1/(10.08) ? 0.926
• Discounting the process of evaluating future
  obligations as an equivalent present value.
• Note, at 8, in two years
• 100 ? 108 ? 116.64

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k-Period Discounting

• At 8 a year, the present value of a debt of
  116.64 due in two years is 100.
• More generally, with compounding of m equal
  periods per year, and in k periods
• dk 1/1 (r/m)k (k-period discount
  factor)
• Present value of a payment A to be received k
  periods in the future is dk A.

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PV FV of Streams

• Ideal Bank A bank which does the following
• applies the same rate of interest to both
  deposits and loans
• has no service charges or transactions fees
• its interest rate does not depend on the size of
  the principal
• separate transactions in an account are
  completely additive in their effect on future
  balances.

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PV FV of Streams (Contd)

• Example of Ideal Bank
• 2-year CD offers same rate as a loan payable in 2
  years.
• Rates for 1-year and 2-year CDs need NOT be the
  same.

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Constant Ideal Bank

• Constant ideal bank is an ideal bank with
• an interest rate independent of the length of
  time for which it applies
• interest compounded according to normal rules
  (ones we have presented for compound interest)
• A constant ideal bank is the reference point
  used to describe the outside financial market
  the public market for money.

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FV of Cash Flow Stream

• Cash Flow Stream (x0,x1, , xn)
• We have a fixed time cycle for compounding
  (e.g., yearly), a period is the length of this
  cycle. Cash flows occur at the end of each
  period (some can be zero). We deposit each cash
  flow in a constant ideal bank, paying r per
  period.

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FV of Cash Flow Stream (Contd)

• x0 grows to x0 (1r)n
• x1 grows to x1 (1r)n-1
• x2 grows to x2 (1r)n-2
• xn grows to xn (1r)n-n
• Future value of cash flow stream with interest
  rate r per period is
• FV x0 (1r)n x1 (1r)n-1 x2 (1r)n-2
  xn

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FV Example

• 2K invest in IRA each year for n years

24
FV Example (Contd)
25
PV of Cash Flow Stream

• Cash Flow Stream (x0,x1, , xn)
• We have a fixed time cycle for compounding (e.g.,
  yearly), a period is the length of this cycle.
  Cash flows occur at the end of each period (some
  can be zero). We deposit each cash flow in a
  constant ideal bank, paying r per period.

26
PV of Cash Flow Stream (Contd)

• x0 is now worth x0
• x1 is now worth x1 /(1r)1
• x2 is now worth x2 /(1r)2
• xn is now worth xn /(1r)n
• Present Value of cash flow stream with interest
  rate r per period is
• PV x0 x1 /(1r)1 x2 /(1r)2 xn/(1r)n

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Example

• Stream (-2,1,1,1), r 0.10
• PV -2 1/(1.10) 1/(1.10)2 1/(1.10)3
• -2 0.9090 0.8264 0.7513 ? 0.4869.
• This present payment amount of 0.487 is
  equivalent to the entire stream.

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Relation of PV and FV

• Question
• FV x0 (1r)n x1 (1r)n-1 x2 (1r)n-2
  xn
• PV x0 x1 /(1r)1 x2 /(1r)2
  xn/(1r)n
• What do you get if you multiply PV by (1r)n ?
• Conclusion For a cash flow stream with rate r,

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Relation of PV and FV (Contd)

• The FV and PV functions are both in Excel.
• Example Stream (-2,1,1,1), r 0.10.
• PV 0.487, FV 0.648,
• PV FV/(1.1)3 0.487.
• Both PV and FV are equivalent ways to think about
  the stream.

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Frequent and Continuous Compounding

• Cash Flow Stream (x0,x1, , xn)
• r nominal annual interest rate, interest
  compounded at m equally spaced periods per year
• PV x0 x1 /1(r/m)1 x2 /1(r/m)2
  xn/1(r/m)n
• Continuous Compounding at times t0, t1, , tn and
  stream (x(t0),x(t1), , x(tn))
• PV x(t0)e-rt0 x(t1)e-rt1 x(tn) e-rtn

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Equivalent Cash Streams

• Definition X and Y are equivalent if both
  streams have the same PV, evaluated at the banks
  interest rate r
• Ideal bank can transform X to Y (Y to X)
• Example. With r 0.1, the following are
  equivalent
• (1,0,0), (0,0,1.21), (0,1.1,0), (a,b,c) for any
  a, b, c with
• 1 a b/(1.1) c/(1.1)2

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Internal Rate of Return

• Motivating Example. An ideal bank offers you the
  following
• Deal 1. Invest 2,000 today. At the end of
  years 1, 2, and 3 get 100, 100, and 500 in
  interest at the end of year 4, get 2,200 in
  principal and interest.
• Question. Is this a good deal?
• Approach find out the implicit interest rate you
  would be receiving that is, solve for
• 2000 100 /(1r)1 100 /(1r)2 500/(1r)3
  2200/(1r)4
• Solution r 10.7844 . This is the interest
  rate for the PV of your payments to be 2,000.

33
Internal Rate of Return (Contd)

• Deal 2 At the end of years 1, 2, and 3 get 100,
  100, and 100 in interest at the end of year 4,
  get 2,000 in principal only.
• We find r for which
• 2000 100 /(1r)1 100 /(1r)2 100/(1r)3
  2000/(1r)4
• Solution r 3.8194.
• Which deal would you prefer?
• Would either deal be attractive?

34
Internal Rate of Return (Contd)

• How do we even know that the following equation
  has a solution?
• 2000 100 /(1r)1 100 /(1r)2
• 500/(1r)3 2200/(1r)4
• Let c 1/(1r). Rewrite the equation as
• -2000 100 c 100 c2 500 c3 2200 c4 0.
• This is now a 4th degree polynomial in c. We
  know, from algebra, that an nth degree polynomial
  has exactly n roots. However, we need to be sure
  the roots are real numbers (and not imaginary).
  We would also like them to be positive.

35
Internal Rate of Return (Contd)

• If c 0, f(c) -2000 lt 0. For c sufficiently
  large, f(c) gt 0. Further, f is a continuous
  function, and strictly increasing. The c value,
  say c0, where f(c0) 0, (where the graph
  intersects the origin) is therefore the only
  real, positive root. Also f(1) 900 gt 0, so the
  root satisfies c0 lt 1.

36
Internal Rate of Return (Contd)

• Observation. The number r satisfying
• 2000 100 /(1r)1 100 /(1r)2 500/(1r)3
  2200/(1r)4
• i.e.,
• 0 -2000100 /(1r)1 100 /(1r)2 500/(1r)3
  2200/(1r)4
• is also the number r for which the cash flow
  stream
• (-2000,100,100,500,2200) has PV 0, since
• PV x0 x1 /(1r)1 x2 /(1r)2 xn/(1r)n

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Internal Rate of Return (Contd)

• Let (x0,x1, , xn) be a cash flow stream. The
  internal rate of return (IRR) of this stream is a
  number r satisfying
• 0 x0 x1 /(1r)1 x2 /(1r)2 xn/(1r)n
• Remark. We use the word internal because the
  rate of return depends only on the stream. It is
  not defined with reference to the financial
  world. It is the rate an ideal bank would have
  to apply to generate the given stream from an
  initial balance of zero.

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IRR in EXCEL
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Main Theorem of IRR

• Suppose the cash flow stream (x0,x1, , xn) has
• x0 lt 0, and xk ? 0 for k 1, , n with at
  least one of x1, , xn positive. Then there is a
  unique positive root to the equation
• f(c) x0 x1 c x2 c2 xn cn
• Further, if x0 xn gt 0 (the total return
  then exceeds the initial investment, x1 xn
  gt -x0), then the corresponding IRR r (1/c) 1
  is positive.

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Proof of Main Theorem of IRR

• f(0) x0 x1 0 x2 02 xn 0n x0 lt0
• f(1) x0 x1 1 x2 12 xn 1n
• x0 x1 x2 xn gt 0
• since f is strictly increasing and continuous,
• root c0 satisfies the 0 lt c0 lt1 and
• IRR r0 (1/ c0) 1 is positive

41
Internal Rate of Return (Contd)

• Remark. One can make up cash flow streams where
  no positive real root exists, or even where roots
  are complex numbers. To do so you must violate
  the hypotheses of the IRR Theorem. These
  hypotheses are usually reasonable.
• Note. Entries in CFS can be 0.

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Net Present Value (NPV)

• Definition
• NPV PV of benefits - PV of costs

43
NPV in EXCEL
44
Evaluation Criterion NPV or IRR

• Example. Cutting Trees.
• Units in terms of 100,000.
• a) (-1,2) cut in one year
• b) (-1,0,3) cut after two years
• NPV Approach, assuming r 10
• a) NPV -1/1 2/(1.1) 0.82
• b) NPV -1/1 0/(1.1) 3/(1.1)2 1.48 gt 0.82
• Cut after two years choose option b)

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Evaluation Criterion NPV or IRR (Contd)

• IRR Approach, c 1/(1r) ? r 1/c - 1
• a) 0 -1 2 c ? c ½ ? r 1, i.e., 100
• b) 0 -1 0 c 3 c2 ? c2 1/3 ? c 1/?3 ?
• r ?3 1 ? 0.732 lt 1.
• Cut after one year choose option a).

46
Evaluation Criterion NPV or IRR (Contd)

• Different decisions depending on evaluation
  criterion
• NPV gives b), IRR gives a).
• Insight. Use a longer planning horizon, reinvest
  all earnings. Which looks better in 6 years?
• a) 1 gt 2 gt 4 gt 8 gt 16 gt 32
  gt 64
• cash flow stream (-1,0,0,0,0,0,64)
• b) 1 in 2 yearsgt 3 in 2 yearsgt 9
  in 2 years gt 27
• cash flow stream (-1,0,0,0,0,0,27)
• a) Business doubles every year.
• b) Business triples every two years ? increases
  by ?3 every year (in long run).
• Multiplication coefficient is called grows rate
  (GR)
• since 2 gt ?3, a) has larger grows rate.

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Evaluation Criterion NPV or IRR (Contd)

• a) IRR 2-1 for cut every year
• b) IRR ?3 1 for cut every 2 years
• This is no accident, but a general result. The
  project with largest IRR also has the highest
  growth rate.

48
Evaluation Criterion NPV or IRR (Contd)

• Conclusion 1. When the proceeds of the
  investment can be repeatedly reinvested in the
  same type of project but scaled in size, it makes
  sense to choose the project with largest IRR to
  get the greatest growth of capital.
• Conclusion 2. NPV makes more sense for
  one-shot projects.

49
Evaluation Criterion NPV or IRR (Contd)

• If we studied the cutting tree project for a
  longer horizon than 1 year, the NPV approach
  would also choose a). (Note with a longer
  horizon there is ALSO more uncertainty. Further,
  these rates of growth cannot continue
  indefinitely.)
• a) (-1,0,0,0,0,0,64) NPV 35.13
• b) (-1,0,0,0,0,0,27) NPV 14.24
• Theorists agree that, overall, the best criterion
  is based on NPV. It provides consistency and
  rationality.

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Other Evaluation Factors

• 1) Which interest rate should we choose for NPV?
• the rate the bank pays on CDs
• the rate paid by the 3-month US Treasury bill
• the rate paid by the highest grade commercial
  bond
• the cost of capital rate the company must offer
  to potential investors in company
• 2) PV is not the whole story about the rate of
  return.
• Alternative 1. Invest 1,000, get a NPV of 100.
• Alternative 2. Invest 1,000,000, get a NPV of
  100.
• If you had 1,000,000, which alternative would
  you choose?

51
Simplico Gold Mine

• You can lease a gold mine from its owners for up
  to ten years. The lease is an up-front payment.
• You can extract up to 10,000 oz. per year
• Each oz. costs 200.
• Market selling price for gold is 400/oz.
• Interest rate is 10
• All this information remains constant over the
  10-year period (heroic assumption).

52
Simplico Gold Mine (Contd)

• Question. What is the most you would be willing
  to pay for the lease?

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Simplico Gold Mine (Contd)

• Answer. Compute PV of income from the mine,
  assuming you operate it at full capacity for all
  10 years.
• Annual profit is (400 - 200) ? 10,000 oz/year
• 2 M.
• PV of annual profit for 10 years
• PV (2 M)/(1.1) (2M)/(1.1)2
  (2M)/(1.1)10 12.29 M
• If you paid more than 12.29M for the lease, you
  would lose money on this undertaking.

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Warnings

• Cash Flow Warning For each period t,
• cash flow income expense
• i.e., net income. Dont forget the net idea.
• Apples Oranges Warning for Cycle Problems. Be
  certain to compare alternatives over the same
  time horizon. The horizon can either be finite
  or, in some cases, of indefinite length.

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Automobile Purchase

• Alternative A.
• Car A costs 20,000
• Has annual maintenance cost of 1,000
• Has a useful lifetime of 4 years
• Has no salvage value
• Alternative B.
• Car B costs 30,000
• Has annual maintenance cost of 2,000
• Has a useful lifetime of 6 years
• Has no salvage value

56
Automobile Purchase (Contd)

• These alternatives are NOT over the same time
  horizon. We can try a planning horizon of 12
  years (3 Car As, 2 Car Bs).
• Car A Analysis (3 cycles of 4 years each)
• One Cycle PV
• PVA 20,000 1,0001/(1.1) 1/(1.1)2
  1/(1.1)3
• 22,487.
• Three Cycle PV (note the trick used here)
• PVA3 PVA 1 1/(1.1)4 1/(1.1)8 48,336.

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Automobile Purchase (Contd)

• Car B Analysis (2 cycles of 6 years each)
• One Cycle PV
• PVB 30,000 2,0001/(1.1) 1/(1.1)2
• 1/(1.1)5
• Two Cycle PV (similar trick)
• PVB2 PVB 1 1/(1.1)6 58,795.
• Conclusion. For the 12-year period, using 3 of
  Car A is cheaper than using 2 of Car B.

58
Machine Replacement

• How often should we replace a machine? Planning
  horizon length is not known in advance
• Purchase cost is 10,000 (would need to adjust
  for inflation in reality)
• First year operating cost is 2,000, no salvage
  value.
• Machine operating costs increases by 1,000 each
  year (these costs incurred at end of year).

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Machine Replacement (Contd)

• Replacing Machine Every Year
• Machine 1 CFS (-10,-2)
• Machine 2 CFS (0,-10,-2)
• Machine 3 CFS (0,0,-10,-2)
• etc
• Each stream looks identical to the one before,
  except it starts one year later (and so must be
  discounted by 10).
• Conclusion. For an indefinite number of 1-year
  replacements,
• PV 10 2/(1.1) PV/(1.1) ? PV 130
• (i.e., 130,000)

60
Machine Replacement (Contd)

• Replacing Machine Every 2 Years
• Machine 1 CFS (-10,-2,-3)
• Machine 2 CFS (0,0,-10,-2,-3)
• Machine 3 CFS (0,0,0,0,-10,-2,-3)
• etc
• Each stream looks identical to the one before,
  except it starts two years later (and so must be
  discounted by 10 for 2 years).
• PV 10 2/(1.1) 3/(1.1)2 PV/(1.1)2
• 14.2975 PV/1.21
• 1.21 PV PV 1.21 ? 14.2975 ? PV 82.381

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Replacing Machine Every k Years

• PV (cost of one cycle of k years) PV/(1.1)k ?
• (1.1)k 1 PV (1.1)k ? cost of 1 cycle of k
  years ?
• PV (1.1)k ? cost of 1 cycle of k years /
  (1.1)k-1
• Just compute the cost of one cycle of k years,
  and then compute the PV using the latter formula.

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Replacing Machine Every k Years

• See Table 2.3, p. 31. Replacing machine every 5
  years is cheapest.

63
Taxes

• Suppose you can get 100/year in dividends
  annually for two years. You are in the 28
  income tax bracket. The time value of money is
  10.
• Not considering taxes, the PV of dividends is
• PVno tax 100/1.1 100/(1.1)2 90.91
  82.64
• 173.55
• Due to taxes, 100 gt (1-0.28)100 72 ,
• the actual PV of the dividends is
• PVtaxes 72/1.1 72/(1.1)2 65.45 59.50
• 124.95.

64
Taxes (Contd)

• Consequently
• PVno tax 100 / 1.1 100 / (1.1)2
• PVtaxes (1-.28)100 / 1.1 (1-.28)100 / (1.1)2
  
• (1-.28) PVno tax
• To get PV with taxes
• You might now reject a project you would
  otherwise accept.

65
Taxes (Contd)

• Assuming this tax rate is applied to all revenues
  and expenses (it might not be), IRR would be
  identical with or without taxes, e.g.,
• k 100/(1r) 100/(1r2) ?
• (1-t) k (1-t) 100/(1r) (1-t) 100/(1r2)

66
Taxes (Contd)

• When a uniform tax rate is applied to all
  revenues and expenses as taxes and credits
  respectively, recommendations from before-tax and
  after-tax analyses are identical.
• Example. Pay 150, today. Int. rate is 10, tax
  rate 28.
• Alternative A. Get 100 in 1 year, 100 more in
  2 years.
• Alternative B. Get 150 in 1 year, 50 more in 2
  years.

67
Taxes (Contd)

• Alternative A.
• PVno tax 100/1.1 100/(1.1)2 90.91 82.64
  173.55
• PVtaxes 72/1.1 72/(1.1)2 65.45 59.50
  124.95.
• Alternative B.
• PVno tax 150/1.1 50/(1.1)2 136.36
  41.32 177.68
• PVtaxes 108/1.1 36/(1.1)2 98.18 29.75
  127.93.
• For comparison purposes, with, or without taxes,
  Alternative B looks better. However, if you want
  to know the actual PV with taxes, then you must
  use the tax rate.

68
Depreciation Example

• We buy a machine for 10,000.
• It has a useful life of 4 years.
• It generates a cash flow of 3,000 each year.
• It has a salvage value of 2,000.
• The government requires depreciating the cost of
  the machine over its useful life, e.g., if its
  net cost to us over 4 years is 10,000 2,000
  8,000, with straight-line depreciation we report
  its annual cost each year as 8,000/4 2,000 .

69
Depreciation Example (Contd)

• Thus the taxable income it produces each year is
  3,000-2,000 1,000.
• If the combined federal and state tax rate is
  43, we pay 430 tax each year. NPV is computed
  at 10.

70
Depreciation Example (Contd)

• Bad News. Without taxes, NPV 876, so this
  operation would be profitable. With taxes, NPV
  -487, so it would not be.

71
Points About Inflation

• A new Volkswagen bug cost about 2,200 in 1962.
  Today it costs about 16,000.
• A good starting masters salary in IE was about
  10,000 in 1962. What is it today?
• Inflation is now running 2 to 3 a year in the
  US. Some years it has been as much as 7 to 8
  in the US during the last twenty years.
• Some other countries have experienced
  hyper-inflation. In Germany just after WW I, a
  weeks wages could not be carried in both arms.

72
Inflation

• Inflation is an increase in general prices over
  time
• Inflation rate f prices are multiplied by (1f
  ) .
• Constant (also called real) dollar keeps the
  same purchasing power.
• Actual (also called nominal, inflated, market )
  dollars are used in transactions.
• Let r be nominal interest rate, than the real r0
  interest rate is given by
• 1r0 (1r )/(1f )
• and
• r0 (r - f )/(1f )

73
Inflation (Contd)

• Money in the bank increases (nominally) by 1r ,
• but purchasing power is deflated by 1/(1f)
• Example. Inflation rate f0.04 ,
• nominal rate r 0.1, real rate r0 equals
• r0 (r - f )/(1f )(0.1- 0.04 )/(1
  0.04)0.0577

74
Inflation (Contd)

• Question. What happens to the real rate of
  interest if the rate of inflation exceeds the
  nominal rate of interest?
• Warning. Dont mix nominal dollars and real
  dollars! Work consistently with one or the
  other.

75
Inflation (Contd)
76
Inflation (Contd)

• The real cash flow has all values in todays
  dollars. The nominal cash flows has values in
  future years adjusted for inflation, e.g., 5000
  5200/(1.04), so 5200 in a year is worth 5000
  today. 5000 5408/(1.04)2, so 5408 in two years
  is worth 5000 today, etc.
• We get the same result either way (apart from
  rounding errors).

77
Inflation (Contd)

• Note. It is common to estimate cash flow in
  constant/real dollars relative to the present,
  because then ordinary price increases (ones due
  to inflation) can be neglected in a simple
  estimation of cash flows.
• Effects of Inflation on a 2,000 annual IRA
  example is in the file infln-bp.xls