LECTURE 3'Time value of money

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LECTURE 3.Time value of money

• 1. Time value of money the concept
• 2. Future value and present value
• 3. Future and present value of stream of cash
  flow
• 4. Simple rules for financial decisions with time
  factor of money
• 5. How financial instruments are valued

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1. Time value of money the concept

• All financial decisions are based on approach
  that they must by justified by the expected
  benefits in the future.
• In such way the problem of changes of money value
  in time is arising.
• Concept of time value of money based on
• the money in hand today is worth more than in
  future
• the future value of money must be higher then
  present equivalent.
• There are 3 reasons why it is so

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1. Time value of money the concept

• First. Future value of present money must be
  higher because they could be invested to earn
  some interest. That is why alternative to hold
  cash now is to compare with another to invest
  and earning interest.
• Second. Purchasing power of money as rule changes
  in time. It is due to inflation. But even in
  times than inflation is absent individual price
  are fluctuated changing purchasing power of
  money.
• Third. Receipt of money back is uncertain.
  Investing person take a risk of getting money
  back.

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1. Time value of money the concept

• Concept of time value of money postulate
• All operations with money must be compared
  between alternatives to find the best result.
• Interest rate is a simple but prominent
  equivalent of any change of time value of money.

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2. Future value and present value

• Changing in time value of money gets future and
  present nomination
• Getting from present value to future value is
  called compounding.
• Getting from future value to present value is
  called discounting.

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2. Future value and present value

• Future value (FV)
• Calculation of FV is based on next elements
• (PV) Present value of amount of money 1000
• (i) Interest rate in percent per year 10
• (n) Number of years 3
• So FV 1000 1,1 for first year
• FV 1000 1,1 1,1 for second year
• FV 1000 1,1 1,1 1,1 for third year.
• So FV 1000 1,1³ 1331
• Formula for FV PV (1 i)n

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2. Future value and present value

• Future value.
• (1 i)n is a future value factor (fvf)
• To simplify calculations of FV use table of fvf.

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2. Future value and present value

• Future value
• For example if i 5, n 3 years, PV 2000 ,
  so FV 2000 fvf 2000 1,16 2320
• if i 6, n 4 years, so FV 2000 1,26
  2520.
• Future value is arising function. The more
  interest rate and time the more coefficient of
  future value factor so the more future value.

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2. Future value and present value

• Present value (PV)
• To calculate PV is necessary than we know value
  of money in the future but we must compare it
  with today amount. Formula of PV is opposite to
  FV, so
• PV FV / (1 i)n
• 1 / (1 i)n is a present value factor (pvf) that
  is opposite to fvf. To simplify calculations use
  table of pvf

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2. Future value and present value

• Table of present value factor

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2. Future value and present value

• Present value.
• For example. If we get 1000 by 5 years what
  present value of them will be. i 6.
• PV 1000 pvf 1000 0,75 750.
• Opposite to FV, PV is declining function. The
  more n and i the less present value.

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3. Future and present value of stream of cash flow

• Often cash flow we pay or receive is regular in
  time and has same amount. Such streamed cash
  flows called annuity. Such model of payment is
  very widely used in financial operations.
• The calculation of FV and PV of annuity is differ
  than in a case of ordinary FV and PV.

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3. Future and present value of stream of cash flow

• The model of annuities future value calculation
  is
• FVa P (1 i)¹ P (1 i)² P (1 i)³
  P (1 i)n-1
• there P is amount of payment, n number of
  periods, i interest rate.
• To simplified formula is
• FVa P (((1 i)n - 1) / i).
• (((1 i)n - 1) / i) is a future value of annuity
  factor (that also could be found in table) (fvfa)

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3. Future and present value of stream of cash flow

• The model of annuities present value calculation
  is
• PVa P / (1 i)¹ P / (1 i)² P / (1 i)³
  P / (1 i)n-1
• there P is amount of payment, n number of
  periods, i interest rate.
• To simplified formula is
• PVa P (1 / i - 1 / (i (1 i)n).
• (1 / i - 1 / (i (1 i)n) is a present value of
  annuity factor (that also could be found in
  table) (pvfa)

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3. Future and present value of stream of cash flow

• Table of future value factor of annuity

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3. Future and present value of stream of cash flow

• Table of present value annuity factor

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3. Future and present value of stream of cash flow

• For example
• If we save 1000 per year during 5 years on 5,
  what FVa will be
• 1000 fvfa 1000 5,53 5530 .
• If we save 1000 per year during 5 years on 5,
  what PVa will be
• 1000 pvfa 1000 4,33 4330 .
• In a case of stream of cash what is for forever
  such annuity called perpetual annuity. In such
  case we can not get to know future value of
  perpetual payment. We only can get to know
  present value. PV of perpetual annuity is amount
  of one payment divided on interest rate
• PVap P / i.
• For example, if P 100 per year, i 10, PVap
  100 / 0,1 1000

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4. Simple rules for financial decisions with time
factor of money

• The main idea of efficiency of financial decision
  is that present value of future cash flow will be
  more than present value of any costs or cash
  out-flows.
• Difference between present value of in-flows and
  out-flows is net present value (NPV). Operation
  will be worth only if NPV be positive. So NPV
  rule says project with positive NPV is acceptable
  and project with negative NPV should be rejected.
  
• How to choose i? Interest rate to discount cash
  flow is a cost of capital for project or next
  alternative for investment, first of all banking
  discount rate.

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4. Simple rules for financial decisions with time
factor of money

• For example
• What is better? To buy bond 750 for 3 years and
  receive 1000 or to invest in bank deposit with
  interest rate of 5.
• Bonds FV is 1000. What PV of bond if discount
  rate is 5 - your possible alternative. PV of
  bond 1000 pvf 1000 0,86 860.
• 860 of bonds PV is more than 750 of initial
  amount so NPV is positive 860-750 110 so to
  buy bond is to increase wealth.

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4. Simple rules for financial decisions with time
factor of money

• Alternative solution of this problem is future
  value rule.
• Future value rule says choose that project FV of
  which is higher then alternative.
• In our case (To buy bond 750 for 3 years and
  receive 1000 or to invest in bank deposit with
  interest rate of 5) we must calculate FV of 750
  for 3 years on 5 which deposit can bring.
• FV of deposit 750 fvf 750 1,16 870.
• FV of bond is higher than FV of deposit. 1000 -
  870 130, so bond is preferable.

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4. Simple rules for financial decisions with time
factor of money

• Next alternative solution is calculation of
  Internal Rate of Return (IRR).
• IRR is such rate of earning under which NPV is
  zero. It is mean that IRR is threshold. If IRR is
  higher than discount rate the project brings
  positive NPV. If IRR is lower than discount rate
  the project brings negative NPV. So IRR rule says
  that investment project with higher IRR then
  discount rate is preferable but borrowing project
  is preferable when IRR is less then discount
  rate.
• IRR is calculated on the transformation of FV
  formula. FV PV (1 i)n. i (FV / PV)¹/n -
  1.
• IRR for bond is (1000 / 750)¹/³ - 1 0,1 or 10.
• 10 of IRR for bond is higher then 5 for deposit.

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4. Simple rules for financial decisions with time
factor of money

• NPV of annuity and investment decision.
• Is it worthwhile 5000 investment in insurance
  certificate that bring 1100 per 5 years if
  alternative is to open deposit account at 7.
• NPV case
• In this case we have to calculate annuity PV
  discounting 5 years cash flow of 1100 by rate at
  7 and compare with 5000 of amount we have to
  invest.
• PVa 1100 pvfa 1100 4,10 4510.
• PVa of 4510 is less than 5000. NPV of annuity
  is negative 4510 5000 -490. So deposit on 5
  years at 5 is preferable.
• FV case
• In this case we have to calculate FV of deposit
  and FV of annuity and compare it.
• FV of deposit 5000 fvf 5000 1,40 7000
• FV of annuity 1100 fvfa 1100 5,75 6325
• FV of deposit is higher than FV of annuity on
  675 so deposit is preferable.
• But in the case of FV we used two operations and
  in the case of NPV only one. Method of NPV is
  better.

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4. Simple rules for financial decisions with time
factor of money

• NPV of annuity and financial decision.
• The main idea is that annuities one payment would
  be less than in alternative case. From formula of
  PVa we get P and compare. Project with less P is
  to accept.

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5. How financial instruments are valued

• All financial and real assets may be set at money
  equivalence value. All of assets has face value
  but it is not product of market forces. Only in
  market condition value of any assets become a
  category of finance.
• Three major factors determine assets value
• Cash flow that assets can generate
• Growth rate of the cash flow
• Risk about future cash flow or assets itself.

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5. How financial instruments are valued

• Real efficiency (profitability, liquidity, growth
  of sales, increase in earnings per share) produce
  financial efficiency viewed as value increase.
  Only a few cases produce reverse situation.
• In such approach 3 factors of value increasing
  doing together. Raise of profits positively
  affect cash flow and decline risk so value of
  assets go upward. Decline of profits reduce cash
  flow determining increase in risk so value of
  assets go downward.

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5. How financial instruments are valued

• Valuation of bonds
• Valuation of bond determined from cash flow
  structure.
• Bond value Coupon / (1 i)¹ Coupon / (1
  i)² Coupon / (1 i)³ Coupon / (1 i)n
  Nominal / (1 i)n or
• Bond value Coupon pvfa Nominal pvf.
• If bond value is more than nominal its mean that
  bond is traded with discount or discount rate is
  higher than coupon rate.
• If bond value is less than nominal its mean that
  bond is traded with premium or discount rate is
  lower than coupon rate
• Such situations may be consequence of
• risk about financial performance of bond issuer
• market tendencies about supply and demand for
  capital
• risk attitude of market participants.

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5. How financial instruments are valued

• Valuation of preference stock
• Preference stock brings fixed dividend in
  infinite time-line. So its value is based on
  formula
• Value of preference stock FD / (1 i)¹ FD /
  (1 i)² FD / (1 i)³ FD / (1 i)8 or
• Value of preference stock Fixed Dividend / i.
• Determinacy of difference between value of
  preference stock and its face value is the same
  as in bond case.

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5. How financial instruments are valued

• Valuation of common stock
• Value of common is based on formula
• Value of common stock D1 / (1 i)¹ D2 / (1
  i)² D3 / (1 i)³ D 8 / (1 i)8 or
• Value of common stock Dividend / i.
• Determinacy of difference between value of common
  stock and its face value is the same as in above
  cases.

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5. How financial instruments are valued

• The main problem of assets valuation is how to
  choose discount rate (i) correctly.
• Main recommendation
• discount rate may be define as rate of return of
  same assets class
• discount rate may be define as rate of return on
  the same financial instruments issued by similar
  firms
• discount rate may be define as rate of return on
  alternative financial instrument that could be
  chosen by investor.
• In a case of deep and liquid financial market
  with many instruments which are traded comparison
  must be based on same class of assets or similar
  firms instruments. In another case discount rate
  would be chosen incorrectly.
• In a case of shallow and illiquid financial
  market discount rate may be define as bank
  deposit rate that available for investor.