ACS310a: Finance

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ACS310a Finance

• Time Value of Money
• (Lecture 4)

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Overview of Lecture

• Future vs. Present Value
• Time Value of Money Tables
• Compounding
• Annuities, Perpetuities
• Growth and Interest Rates
• Loan Amortization
• Periodic Investments to accumulate future sum

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Role of Time Value in Finance

• The timing of cash flows has important economic
  consequences that are recognized as the Time
  Value of Money.
• Time value is based on the belief that a dollar
  today is worth more than a dollar that will be
  received at some future date.

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Future vs. Present Value

• Future Value is cash you will receive at a given
  future date.
• Present Value is the equivalent of cash on hand
  today.
• A time line can be used to depict the cash flows
  associated with a given investment.
• Since financial managers make decisions at time
  zero, they tend to rely on present value
  techniques.

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Figure 5.2 Compounding and Discounting
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Computational Aids

• Financial Tables are commonly used as quick
  reference tools for determining present and
  future values at various interest or discount
  rates of a range of time periods.
• Modern calculators are programmed to perform the
  complete computational analysis using the
  underlying formulas for present and future value.

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Figure 5.3 Financial Tables
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Future Value of Single Amount

• Principal is the amount of money on which
  interest is paid.
• Compound Interest is the interest earned on a
  given deposit that becomes part of the principal
  at the end of a specified period.
• Future Value of a present amount is found by
  applying compound interest over a specified
  period of time.

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Equation of Future Value

• FVn future value at the end of period n.
• PV initial principal, or present value.
• k annual rate of return.
• n number of periods the money is left on
  deposit.
• (5.4) FVn PV ? (1k)n

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Using Tables Calculators

• Table A-1 provide values for the Future Value
  Interest Factor (FVIF) which simplifies the
  process of calculating FV in equation (5.4).
• (5.5) FVIFk,n (1k)n
• (5.6) FVn PV ? FVIFk,n

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Graphic View of Future Value
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Question 5.6 Inflation and FV

• Given
• Want to buy new car in 5 years
• Purchase price 14,000 today
• Car expected to increase in price by 2 to 4
  over next 5 years
• A) Estimate the price of the car if inflation is
  2 and if it is 4
• B) How much more expensive is the car if
  inflation is 4?

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Question 5.6 Answer
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Compounding More Frequently Than Annually

• Semiannual Compounding involves two compounding
  periods within the year.
• Quarterly Compounding involves four compounding
  periods within the year.
• (5.7) FVn PV ? (1k/m)mn
• Where m is the number of compounding periods
  within the year.

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Continuous Compounding

• Continuous Compounding involves compounding over
  every microsecond.
• (5.8) FVn(continuous) PV ? (ekn)
• (5.9) FVIFk,n(continuous) ekn

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Nominal and Effective Annual Rates of Interest

• The Nominal, or State, Annual Rate is that
  charged by a lender or promised by a borrower.
• The Effective Annual Rate (EAR) is the interest
  actually paid or earned due to compounding.
• (5.10) EAR (1k/m)m - 1

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Question EAR

• Given
• Investing 2,000 in an investment account today
• Expect nominal annual ROR of 8
• ROR applies for all future years
• What is the EAR, for semiannually and continuous
  compounding at the end of 10 years?

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Answer to EAR Question
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Future Value of An Annuity

• An Annuity is a stream of equal annual cash
  flows, either inflows or outflows.
• There are two basic types of annuities
• Ordinary Annuity where the cash flow occurs at
  the end of each period, and
• Annual Due Annuity where the cash flow occurs at
  the beginning of each period.

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FV of Ordinary Annuity

• The Future Value Interest Factor for an Annuity
  (FVIFA) is
• (5.14)
• (5.15) FVAn PMT ? (FVIFAk,n)
• Where PMT is the amount of each cash flow
  payment.

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FV of Annuity Due

• Since an Annuity Due requires the cash flow at
  the beginning of the period only a simple
  adjustment to the FVIFA is needed.
• (5.16) FVIFAk,n(Annuity Due)FVIFAk,n? (1k)

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Q5.16 Ordinary Ann. vs. Ann. Due

• Given
• Two 10 year annuities (C D)
• Ordinary annuity - 2,500/yr for 10 yrs
• Annuity Due - 2,200/yr for 10 yrs
• Want to select the better of the two annuities
• Which annuity has the greater value over the 10
  years based on an annual interest rate of 10?
  What about 20?

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Q5.16 Answer (Parts (a) and (b))
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Present Value of Single Amount

• Present Value is the current dollar value of a
  future amount the amount of money that would
  have to be invested today at a given rate of
  return over a specified period to equal the
  future amount.
• The process of finding Present Value is often
  referred to as Discounting Cash Flows.
• The annual rate of return used is referred to as
  the discount rate, required return, cost of
  capital, or opportunity cost.

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Equation for Present Value

• (5.19) PV FVn FVn ? ( 1 )
• (1k)n (1k)n
• Tables may also be used to look up the Present
  Value Interest Factor (PVIF).
• (5.21) PVIFk,n ( 1 )
• (1k)n

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Graphic View of Present Value
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PV of Mixed Stream

• A Mixed Stream is cash flows of different amounts
  during the future periods.
• To determine the Present Value of a Mixed Stream
  we must calculate the present value of each
  future amount, then sum the total of the present
  value calculations.

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Q5.34 PV of Mixed Stream

• Given
• Payments received
• End of year 1 30000
• End of year 2 25000
• Annual year end payments yrs 3 9 15000
• End of year 10 10000
• Q) if the required ROR is 12, what is the PV of
  the series of payments?
• Q) Given the option b/w a one-time payment of
  100,000 and PV, which offer should the company
  take?

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Q5.34 Answer (b) and (c)
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PV of an Annuity

• An Annuity is a series of uniform future cash
  flows. We may determine the Present Value of an
  Annuity using a Present Value Interest Factor for
  an Annuity (PVIFA).
• (5.26)
• (5.27) PVAn PMT ? (PVIFAk,n)

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PV of Mixed Stream with Embedded Annuity

• Three steps to determine the Present Value of a
  Mixed Stream with an Embedded Annuity.
• Find the present value of the annuity at
  specified discount rate.
• Add the present value calculated to any other
  cash flow occurring in the period just before the
  start of the annuity to determine a revised cash
  flows.
• Discount the revised cash flows back to time zero
  in the normal fashion at specified discount rate.

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PV of a Perpetuity

• A Perpetuity is an annuity with an infinite life.
• Adjusting the PVIFA where n? we have
• (5.28) PVIFAk,? 1
• k

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Investments Required to Accumulate a Future Sum

• To determine the Payments necessary to accumulate
  a Future Sum, we simply rearrange the formula for
  the future value of an annuity (5.15)
• (5.30) PMT FVAn
• FVIFAk,n

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Loan Amortization

• Loan Amortization is the determination of the
  equal annual loan payments necessary to provide a
  lender with a specified interest return and to
  repay the loan principal over a specified period.
• Rearranging the formula for PVA (5.27)
• (5.32) PMT PVAn
• PVIFAk,n

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Q5.49 Loan Amortization

• Given
• Borrowed 15,000 at 14
• Repaid over 3 yrs
• Loan amortized into 3 equal annual end of year
  payment
• What is the annual end of year loan payment?
• Figure out loan amortization schedule showing
  interest and principal breakdown
• Why does the interest portion of each payment
  decline with the passage of time?

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Q5.48 Answer
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Growth or Interest Rates

• It is often necessary to calculate the compound
  annual growth rate of a series of cash flows.
• Either future value or present value interest
  factors can be used depending on the situation.
• Financial Calculators can determine the precise
  annual interest rate.
• This rate is called the Internal Rate of Return
  (IRR).