SASF CFA Quant. Review

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SASF CFA Quant. Review
Investment Tools Time Value of Money
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Time Value of Money

• Concepts Covered in This Section
• Future value
• Present value
• Perpetuities
• Annuities
• Uneven Cash Flows
• Rates of return

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• Time lines show timing of cash flows.

Interest Rate
Cash Flows

• Tick marks at ends of periods.
• Time 0 is today Time 1 is the end of Period 1
  or the beginning of Period 2.
• 90 of getting a Time Value problem correct is
  setting up the timeline correctly!!!

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Whats the FV of an initial 100 after 3 years if
i 10?
Future Values
FV ?

• Finding FVs (moving to the right on a time line)
  is called compounding.
• Compounding involves earning interest on
  interest for investments of more than one period.

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Single Sum - Future Present Value

• Assume can invest PV at interest rate i to
  receive future sum, FV
• Similar reasoning leads to Present Value of a
  Future sum today.

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Single Sum FV PV Formulas
PV Calculation for 100 received in 3 years if
interest rate is 10
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Question on PV of a given FV

• Ex 1. An investor wants to have 1 million when
  she retires in 20 years. If she can earn a 10
  percent annual return, compounded annually, on
  her investments, the lump-sum amount she would
  need to invest today to reach her goal is closest
  to
• A. 100,000.
• B. 117,459.
• C. 148,644.
• D. 161,506.

• This is a single payment to be turned into a set
  future value FV1,000,000 in N20 years time
  invested at r10 interest rate.
• PV 1/(1r) N FV
• PV 1/(1.10) 20 1,000,000
• PV10 0.14864(1,000,000)
• PV10 148,644

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Perpetuities
Perpetuity is a series of constant payments, A,
each period forever.
PVperpetuity ?A/(1i)t A ?1/(1i)t A/i
Intuition Present Value of a perpetuity
is the amount that must invested today at the
interest rate i to yield a payment of A each
year without affecting the value of the
initial investment.
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Annuities

• Regular or ordinary annuity is a finite set of
  sequential cash flows, all with the same value A,
  which has a first cash flow that occurs one
  period from now.
• An annuity due is a finite set of sequential cash
  flows, all with the same value A, which has a
  first cash flow that is paid immediately.

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Time line for an ordinary annuity of 100 for 3
years.
Ordinary Annuity Timeline
i
100
100
100
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Ordinary Annuity vs. Annuity Due
Difference between an ordinary annuity and an
annuity due?
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Annuity Formula and Perpetuities
Intuition Formula for a N-period annuity of A
is PV of a Perpetuity of A today minus PV
of a Perpetuity of A in period N
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Annuities Perpetuities Again

• Rather than memorize the annuity formula I find
  it easier to calculate it as the difference
  between two perpetuities with the same payment.
• PV of an N-period annuity of A per period is
• PVN
• (A/i) x 1 1/(1i)N

• Calculating the PV of an annuity has 3 steps
• Calculate (A/i)
• PV of a Perpetuity with payments of A per
  period.
• Calculate 1/(1i)N
• Discount factor associated with end of the
  annuity.
• Calculate PVN
• (A/i) x 1 - 1/(1 i)N
• I think this is easier under pressure than
  memorizing the formula.

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Question on FV of Annuity Due

• Ex 2.An individual deposits 10,000 at the
  beginning of each of the next 10 years, starting
  today, into an account paying 9 percent interest
  compounded annually. The amount of money in the
  account at the end of 10 years will be closest
  to
• A. 109,000.
• B. 143,200.
• C. 151,900.
• D. 165,600.

• This is an annuity due of A10,000 for N10
  years at i9 interest rate.
• Annuity due must be adjusted by (1i) to reflect
  payment is made at beginning rather than end of
  period.
• Also must adjust PV formula by (1i)N for FV of
  annuity.
• PVN (1i)N(1i)(A/i) 1 1/(1i)N
• PV10 (1.09)11 (10K/.09) 1 1/1.0910
• PV10 (2.58)(111,111)1 0.42
• PV10 165,601

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Time line for uneven CFs 100 at end of Year 1
(t 1), 200 at t2, and300 at the end of Year
3.
Uneven Cash Flows
300
100
200
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Question on Uneven Cash Flows

• Ex 3.An investment promises to pay 100 one year
  from today, 200 two years from today, and 300
  three years from today. If the required rate of
  return is 14 percent, compounded annually, the
  value of this investment today is closest to
• A. 404.
• B. 444.
• C. 462.
• D. 516.

• This is a set of unequal cash flows. You could do
  it as a sum of annuities but it is easier to
  calculate it directly in this case.
• Interest rate is i 14.
• PV ? 1/(1i) t FVt
• PV 100/(1.14) 200/(1.14)2 300/(1.14)3
• PV 87.72 153.89 202.49
• PV 444.10

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Uneven Cash Flows
Intuition PV of uneven cash flows is equal to
the sum of the PVs of regular cash flows that
sum to the uneven cash flows.
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Interest Rate Definitions

• Stated Annual interest rate or quoted interest
  rate is m x ip where ip is the periodic
  interest rate times the number of periods in a
  year, m.
• Stated in contracts.
• Does not account for effects of compounding
  within the year.
• Periodic interest rate ip is /m x ip where is
  is the stated annual interest rate divided by the
  number of periods in a year, m.
• Used in calculations, shown on time lines.
• Effective Annual interest rate or EFF the amount
  to which a 1 grows to in year with compounding
  taken into account.
• Use EAR or EFF only for comparisons when payment
  periods differ between investments.
• Given a stated annual interest rate, iS, the
  periodic rate is iP iS/m, where m is the
  number of periods a year.
• Effective annual interest rate is computed as
  (1 ip)m 1

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Comparison of Compounding Periods
Annually FV3 100(1.10)3 133.10.
Semiannually FV6 100(1.05)6 134.01.
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Questions on Time Value

• Develop an approach to problems on Time Value.
• Draw the Time line for the cash flows.
• Put in the cash flows from the problem.
• Identify if single payment, annuity, annuity due,
  or perpetuity.
• If uneven cash flows can you break it into sums
  of annuities?
• Identify what is to be calculated PV, FV, N or
  i ?
• Write out the appropriate formula, put in values
  for the variables, and calculate.
• Best Study Tip Do the problems, and then do some
  more and then do some more!! Practice using your
  calculator!!

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Possible Time Value Questions

• Present Value Formula
• Given FVN, i, N solve for PVN
• Given PVN , i, N solve for FVN
• Given PVN, FVN, N solve for i
• Given PVN, FVN, i solve for N
• Perpetuity Formula
• Given A, i solve for PVper
• Given PVper, i solve for A
• Given PVper, A solve for i
• Annuity Formula
• Given A, i, N solve for PV
• Given A, i, N solve for FV
• Given PV, i, N solve for A

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Bonds and Their Valuation

• Key features of bonds
• Bond valuation
• Measuring yield
• Assessing risk

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Key Features of a Bond

• Par value Face amount paid at maturity.
  Assume 1,000.
• Coupon interest rate Stated interest rate.
  Multiply by par value to get dollars of interest.
  Often fixed but can float with market rate.
• Maturity Years until bond must be repaid.
  Declines.
• Issue date Date when bond was issued.
• Default risk Risk that issuer will not make
  interest or principal payments.

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Valuing a 5-Period Bond
Time 0
1
2
3
4
5
6
7

• Discounted Cash Flow Approach
• Current Bond Price Present value of all future
  Cash Flows (Interest Principal) at required
  return, kB.

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The Right Discount Factor

• The discount rate (ki) is the opportunity cost of
  capital, i.e., the rate that could be earned on
  alternative investments of equal risk.
• ki k IP DRP MRP LP
• k Real rate of interest
• IP Inflation risk premium
• DRP Default risk premium
• MRP Maturity premium
• LP Liquidity risk premium

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Whats the value of a 10-year, 10 coupon bond if
kd 10?
Bond Valuation Example
VB ?
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Stocks and Their Valuation

• Features of common stock
• Determining common stock values

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Features of Common Stock

• Represents ownership.
• Ownership implies control.
• Stockholders elect directors.
• Directors hire management.
• Managements goal Maximize stock price.

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Valuing Common Stock
Uncertain Dividends, Dti
Time 0
1
2
3
4
5
6
7

• Dividend Discount Model
• Current Stock Price Present value of all future
  Expected Cash Flows (Dividends) at required
  return, kS.

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Stock Value PV of Dividends

• Constant Growth stock
• One whose dividends are expected to grow
  forever at a constant rate, g.
• Can link this to earnings by assuming that firm
  pays out a fixed percentage of earnings as
  dividends
• i.e. Dt k x Et where k equals payout ratio

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For a constant growth stock
If g is constant, then
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What is a stocks market value if D0 2.00, ks
13, g 6?
Constant growth model
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Supernormal Growth

• If we have firm expected to have
• Supernormal growth of 30 for 3 years,
• then a long-run constant growth of g 6
• What is P0? Assume that ks is still 13.
• Can no longer use constant growth model.
• However, growth becomes constant after 3 years.
• Treat firms value as the sum of two parts
• PV of the Cash flows for the 3 years of
  supernormal growth at 13.
• PV of the Cash Flows thereafter at long-run
  growth of 6.

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Timeline for Supernormal Growth
D0 2.00 2.60 3.38
4.394 4.6576
PV1
PV2
PV3
PV4