FINANCE 3

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FINANCE3 . Present Value

• Professor André Farber
• Solvay Business School
• Université Libre de Bruxelles
• Fall 2007

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Using prices of U.S. Treasury STRIPS

• Separate Trading of Registered Interest and
  Principal of Securities
• Prices of zero-coupons
• Example Suppose you observe the following prices
• Maturity Price for 100 face value
• 1 98.03
• 2 94.65
• 3 90.44
• 4 86.48
• 5 80.00
• The market price of 1 in 5 years is DF5 0.80
• NPV - 100 150 0.80 - 100 120 20

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Present Value general formula

• Cash flows C1, C2, C3, ,Ct, CT
• Discount factors DF1, DF2, ,DFt, , DFT
• Present value PV C1 DF1 C2 DF2
  CT DFT
• An example
• Year 0 1 2 3
• Cash flow -100 40 60 30
• Discount factor 1.000 0.9803 0.9465 0.9044
• Present value -100 39.21 56.79 27.13
• NPV - 100 123.13 23.13

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Several periods future value and compounding

• Invests for 1,000 two years (r 8) with annual
  compounding
• After one year FV1 C0 (1r) 1,080
• After two years FV2 FV1 (1r) C0 (1r)
  (1r)
• C0 (1r)² 1,166.40
• Decomposition of FV2
• C0 Principal amount 1,000
• C0 2 r Simple interest 160
• C0 r² Interest on interest 6.40
• Investing for t years FVt C0 (1r)t
• Example Invest 1,000 for 10 years with annual
  compounding
• FV10 1,000 (1.08)10 2,158.82

Principal amount 1,000Simple interest
800Interest on interest 358.82
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Present value and discounting

• How much would an investor pay today to receive
  Ct in t years given market interest rate rt?
• We know that 1 0 gt (1rt)t t
• Hence PV ? (1rt)t Ct gt PV Ct/(1rt)t
  Ct ? DFt
• The process of calculating the present value of
  future cash flows is called discounting.
• The present value of a future cash flow is
  obtained by multiplying this cash flow by a
  discount factor (or present value factor) DFt
• The general formula for the t-year discount
  factor is

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Discount factors
Interest rate per year Interest rate per year Interest rate per year
years 1 2 3 4 5 6 7 8 9 10
1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091
2 0.9803 0.9612 0.9426 0.9246 0.9070 0.8900 0.8734 0.8573 0.8417 0.8264
3 0.9706 0.9423 0.9151 0.8890 0.8638 0.8396 0.8163 0.7938 0.7722 0.7513
4 0.9610 0.9238 0.8885 0.8548 0.8227 0.7921 0.7629 0.7350 0.7084 0.6830
5 0.9515 0.9057 0.8626 0.8219 0.7835 0.7473 0.7130 0.6806 0.6499 0.6209
6 0.9420 0.8880 0.8375 0.7903 0.7462 0.7050 0.6663 0.6302 0.5963 0.5645
7 0.9327 0.8706 0.8131 0.7599 0.7107 0.6651 0.6227 0.5835 0.5470 0.5132
8 0.9235 0.8535 0.7894 0.7307 0.6768 0.6274 0.5820 0.5403 0.5019 0.4665
9 0.9143 0.8368 0.7664 0.7026 0.6446 0.5919 0.5439 0.5002 0.4604 0.4241
10 0.9053 0.8203 0.7441 0.6756 0.6139 0.5584 0.5083 0.4632 0.4224 0.3855
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Spot interest rates

• Back to STRIPS. Suppose that the price of a
  5-year zero-coupon with face value equal to 100
  is 75.
• What is the underlying interest rate?
• The yield-to-maturity on a zero-coupon is the
  discount rate such that the market value is equal
  to the present value of future cash flows.
• We know that 75 100 DF5 and DF5
  1/(1r5)5
• The YTM r5 is the solution of
• The solution is
• This is the 5-year spot interest rate

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Term structure of interest rate

• Relationship between spot interest rate and
  maturity.
• Example
• Maturity Price for 100 face value YTM (Spot
  rate)
• 1 98.03 r1 2.00
• 2 94.65 r2 2.79
• 3 90.44 r3 3.41
• 4 86.48 r4 3.70
• 5 80.00 r5 4.56
• Term structure is
• Upward sloping if rt gt rt-1 for all t
• Flat if rt rt-1 for all t
• Downward sloping (or inverted) if rt lt rt-1 for
  all t

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Using one single discount rate

• When analyzing risk-free cash flows, it is
  important to capture the current term structure
  of interest rates discount rates should vary
  with maturity.
• When dealing with risky cash flows, the term
  structure is often ignored.
• Present value are calculated using a single
  discount rate r, the same for all maturities.
• Remember this discount rate represents the
  expected return.
• Risk-free interest rate Risk premium
• This simplifying assumption leads to a few useful
  formulas for
• Perpetuities (constant or growing at a constant
  rate)
• Annuities (constant or growing at a constant
  rate)

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Constant perpetuity
Proof PV C d C d² C d3 PV(1r) C C
d C d² PV(1r) PV C PV C/r

• Ct C for t 1, 2, 3, .....
• Examples Preferred stock (Stock paying a fixed
  dividend)
• Suppose r 10 Yearly dividend 50
• Market value P0?
• Note expected price next year
• Expected return

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Growing perpetuity

• Ct C1 (1g)t-1 for t1, 2, 3, .....
  rgtg
• Example Stock valuation based on
• Next dividend div1, long term growth of dividend
  g
• If r 10, div1 50, g 5
• Note expected price next year
• Expected return

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Constant annuity

• A level stream of cash flows for a fixed numbers
  of periods
• C1 C2 CT C
• Examples
• Equal-payment house mortgage
• Installment credit agreements
• PV C DF1 C DF2 C DFT
• C DF1 DF2 DFT
• C Annuity Factor
• Annuity Factor present value of 1 paid at the
  end of each T periods.

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Constant Annuity

• Ct C for t 1, 2, ,T
• Difference between two annuities
• Starting at t 1 PVC/r
• Starting at t T1 PV C/r 1/(1r)T
• Example 20-year mortgage
• Annual payment 25,000
• Borrowing rate 10
• PV ( 25,000/0.10)1-1/(1.10)20 25,000 10
  (1 0.1486)
• 25,000 8.5136
• 212,839

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Annuity Factors
Interest rate per year Interest rate per year Interest rate per year
years 1 2 3 4 5 6 7 8 9 10
1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091
2 1.9704 1.9416 1.9135 1.8861 1.8594 1.8334 1.8080 1.7833 1.7591 1.7355
3 2.9410 2.8839 2.8286 2.7751 2.7232 2.6730 2.6243 2.5771 2.5313 2.4869
4 3.9020 3.8077 3.7171 3.6299 3.5460 3.4651 3.3872 3.3121 3.2397 3.1699
5 4.8534 4.7135 4.5797 4.4518 4.3295 4.2124 4.1002 3.9927 3.8897 3.7908
6 5.7955 5.6014 5.4172 5.2421 5.0757 4.9173 4.7665 4.6229 4.4859 4.3553
7 6.7282 6.4720 6.2303 6.0021 5.7864 5.5824 5.3893 5.2064 5.0330 4.8684
8 7.6517 7.3255 7.0197 6.7327 6.4632 6.2098 5.9713 5.7466 5.5348 5.3349
9 8.5660 8.1622 7.7861 7.4353 7.1078 6.8017 6.5152 6.2469 5.9952 5.7590
10 9.4713 8.9826 8.5302 8.1109 7.7217 7.3601 7.0236 6.7101 6.4177 6.1446
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Growing annuity

• Ct C1 (1g)t-1 for t 1, 2, , T r ? g
• This is again the difference between two growing
  annuities
• Starting at t 1, first cash flow C1
• Starting at t T1 with first cash flow C1
  (1g)T
• Example What is the NPV of the following project
  if r 10?
• Initial investment 100, C1 20, g 8, T 10
• NPV 100 20/(10 - 8)1 (1.08/1.10)10
• 100 167.64
• 67.64

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Review general formula

• Cash flows C1, C2, C3, ,Ct, CT
• Discount factors DF1, DF2, ,DFt, , DFT
• Present value PV C1 DF1 C2 DF2
  CT DFT

If r1 r2 ...r
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Review Shortcut formulas

• Constant perpetuity Ct C for all t
• Growing perpetuity Ct Ct-1(1g)
• rgtg t 1 to 8
• Constant annuity CtC t1 to T
• Growing annuity Ct Ct-1(1g)
• t 1 to T

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Compounding interval

• Up to now, interest paid annually
• If n payments per year, compounded value after 1
  year
• Example Monthly payment
• r 12, n 12
• Compounded value after 1 year (1 0.12/12)12
  1.1268
• Effective Annual Interest Rate 12.68
• Continuous compounding
• 1(r/n)n?er (e 2.7183)
• Example r 12 e12 1.1275
• Effective Annual Interest Rate 12.75

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Juggling with compounding intervals

• The effective annual interest rate is 10
• Consider a perpetuity with annual cash flow C
  12
• If this cash flow is paid once a year PV 12 /
  0.10 120
• Suppose know that the cash flow is paid once a
  month (the monthly cash flow is 12/12 1 each
  month). What is the present value?
• Solution 1
• Calculate the monthly interest rate (keeping EAR
  constant)
• (1rmonthly)12 1.10 ? rmonthly 0.7974
• Use perpetuity formula
• PV 1 / 0.007974 125.40
• Solution 2
• Calculate stated annual interest rate 0.7974
  12 9.568
• Use perpetuity formula PV 12 / 0.09568
  125.40

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Interest rates and inflation real interest rate

• Nominal interest rate 10 Date 0 Date 1
• Individual invests 1,000
• Individual receives 1,100
• Hamburger sells for 1 1.06
• Inflation rate 6
• Purchasing power ( hamburgers) H1,000 H1,038
• Real interest rate 3.8
• (1Nominal interest rate)(1Real interest
  rate)(1Inflation rate)
• Approximation
• Real interest rate Nominal interest rate -
  Inflation rate