Lecture 2Time value of Money

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Lecture 2Time value of Money
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Net Present Value/Time Value of Money

• Basic (and important) idea Money received now is
  worth more than money in the future.
• Why?
• We can invest in risk-free investments and gain
  more money. For example, say a one year T-bill
  has a return of 2.8 currently. Thus, if 1000 is
  invested now in a 1 year T-bill, the return at
  the end of the year will be
• 1000(0.028 1000) 1000 1.028 1028
• So if we were offered 1000 now, we could turn
  that into 1028 at the end of the year.

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• Now, how much would we have to invest (at rate r)
  to obtain C1 in one year?
• From the formula for interest,
• C1 C0 (1r)
• Where C0 is the principal (original investment).
• Solving algebraically for C0
• C0 then is the Present Value of obtaining the
  payout C1 in one year.
• What if we go multiple years into the future?

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• Assume an interest rate of r, and initial
  investment C0. Then C1, the money in our account
  after one year is
• C1 then becomes our principal for the next
  years compounding, so
• In general then,
• Where Cn is the Future Value of C0 in n years.

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What about multi-period Present Value?

• Similarly to the single period case, solve for
  C0. Since
• the Present Value of Cn will be

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Aside Compounding. Assume 5 interest

• Also, 127.63 100 (1.05)5 100 1.2763. In
• 30 years the 100 would grow to 100(1.05)30
• 432.19

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• Note at 12 interest, the 100 would grow to
  2995.99 after 30 years. If we found the value
  after 3 years and rounded the answer, and then
  brought it forward another 27 years, we would be
  slightly off (six cents, in this case)
• This shows the power of both the interest rate
  and compounding. 5 interest gives 432.19 over
  the same time period.

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Another aside Doubling time

• Law of 72 Doubling time.
• , so
• This is approximately
• For 6 interest, the law of 72 gives a doubling
  time of 12 years. The exact answer is 11.90
  years.

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Present Values are additive

• This means we can add the present value of cash
  flows at different times together to get the Net
  Present Value for a single project, and we can
  add different projects together to get the NPV
  from several projects.
• Projects with positive net present value will
  make money and should be undertaken. If only a
  subset of projects can be undertaken, pick the
  highest NPV possible.

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Example 1, from the previous lecture

• If we assume a 10 interest rate,
• At a 60 interest rate, NPV is -4.98

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• Example 2 Project 1 100 a year for 4 years, or
  
• Project 2 500 at the end of 4 years.
• NPV of Project 1
• NPV of Project 2
• At an interest rate of r 10, NPV of project 1
  is 316.99. NPV of project 2 is 341.51.

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• At an interest rate of r 10, NPV of project 1
  is 316.99. NPV of project 2 is 341.51.
• We should undertake both projects
• If only one is possible, project 2 should be
  chosen.
• If there was a cost at the start of 330, only
  project 2 should be undertaken, even if both
  could be done.
• Which project has a higher value will depend on
  the interest rate. The switching rate is found by
  setting the NPV of the projects equal to each
  other, and solving for r.

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A few words about compounding periods

• A 10 stated annual interest rate, compounded
  semi-annually means that 5 interest is computed
  every six months.
• What is the effective annual interest rate to
  which this corresponds?
• Answer (1.05)2-1 0.1025, so the effective
  annual interest rate is 10.25.

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• General result with m compounding periods in one
  year, stated rate r
• Continuous compounding
• From Calculus,

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• The power of compounding periods Assume 100
  in the bank for one year with a SAIR of 10.

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• Why is continuous compounding useful in finance?
  The returns with continuous compounding can be
  simply added together, but discrete compounding
  rates cannot be added together.
• For example

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Annuities and Perpetuities

• A Perpetuity gives a constant cash flow C at the
  end of every year in the future (infinitely
  long). The cash flow looks like
• Two ways to calculate the NPV of this perpetuity.
  First, we can think of depositing a sum of money
  V in the bank at an interest rate r. The
  principal must remain untouched since the
  perpetuity lasts forever. Thus, C must be the
  annual interest on V. So, C V r, or the NPV is
  C/r.

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• The other (mathematical) way to calculate the NPV
  is to sum the infinite series
• (1)
• (2)
• Subtracting (1) from (2),
• And we obtain the previous result (thankfully!)

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• An Annuity is like a Perpetuity, except the
  payments cease after a certain number of years.
  Lets look at an annuity that pays C at the end
  of the next T years.

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Growing Perpetuity

• If the cash flows grow at a rate g, and the
  interest rate is r, the cash flows will be
• Using the previous algebraic method to sum this
  series, except we multiply by (1r)/(1g) to
  obtain equation (2). The result, after more
  algebra is