FINANCE
3 . Present Value
Professor André Farber
Solvay Business School
Université Libre de Bruxelles
Fall 2007
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:
• Discount factors:
C1, C2, C3, … ,Ct, … CT
DF1, DF2, … ,DFt, … , DFT
• Present value:
PV = C1 × DF1 + C2 × DF2 + … + CT × DFT
• An example:
• Year
• Cash flow
• Discount factor
• Present value
0
-100
1.000
-100
1
2
3
40
60
30
0.9803 0.9465 0.9044
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 × (1+r) = 1,080
• After two years
FV2
= FV1 × (1+r)
= C0 × (1+r) × (1+r)
•
= C0 × (1+r)² = 1,166.40
•
•
•
•
Decomposition of FV2
C0
C0 × 2 × r
C0 × r²
Principal amount
Simple interest
Interest on interest
1,000
160
6.40
• Investing for t years FVt
= C0 (1+r)t
• Example: Invest €1,000 for 10 years with annual compounding
Principal amount
1,000
• FV10 = 1,000 (1.08)10 = 2,158.82
Simple interest
800
Interest on interest
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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 => (1+rt)t €t
• Hence
PV  (1+rt)t = Ct => PV = Ct/(1+rt)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:
DFt 
1
(1  rt ) t
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Discount factors
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/(1+r5)5
100
• The YTM r5 is the solution of:
75 
• The solution is:
 100 
r5  

 75 
1
5
(1  r5 ) 5
 1  5.92%
• 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 > rt-1 for all t
• Flat if rt = rt-1 for all t
• Downward sloping (or inverted) if rt < 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
• Ct =C for t =1, 2, 3, .....
Proof:
PV = C d + C d² + C d3 + …
PV(1+r) = C + C d + C d² + …
PV(1+r)– PV = C
PV = C/r
C
PV 
r
• Examples: Preferred stock (Stock paying a fixed dividend)
• Suppose r =10%
Yearly dividend =50
50
• Market value P0?
P 
 500
0
• Note: expected price next year =
• Expected return =
.10
P1 
50
 500
.10
div1  ( P1  P0 )
50  (500  500)

 10%
P0
500
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Growing perpetuity
• Ct =C1 (1+g)t-1 for t=1, 2, 3, .....
r>g
C1
PV 
rg
• Example: Stock valuation based on:
• Next dividend div1, long term growth of dividend g
• If r = 10%, div1 = 50, g = 5%
P0 
50
 1,000
.10  .05
• Note: expected price next year =
P1 
• Expected return =
52.5
 1,050
.10  .05
div1  ( P1  P0 ) 50  (1,050  1,000)

 10%
P0
1,000
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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
C
1
PV  [1 
]
T
r
(1  r )
• Difference between two annuities:
– Starting at t = 1
PV=C/r
– Starting at t = T+1
PV = C/r ×[1/(1+r)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
# 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 (1+g)t-1 for t = 1, 2, …, T
r≠g
T

C1
1 g  
PV 
 
1  
r  g   1  r  
• This is again the difference between two growing annuities:
– Starting at t = 1, first cash flow = C1
– Starting at t = T+1 with first cash flow = C1 (1+g)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
PV 
Ct
C1
C2
CT


...


...

(1  r1 ) (1  r2 ) 2
(1  rt ) t
(1  rT )T
PV 
Ct
C1
C2
CT


...


...

(1  r ) (1  r ) 2
(1  r ) t
(1  r )T
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Review: Shortcut formulas
•
Constant perpetuity: Ct = C for all t
C
PV 
r
•
Growing perpetuity: Ct = Ct-1(1+g)
r>g
t = 1 to ∞
C1
PV 
rg
•
Constant annuity: Ct=C
•
Growing annuity: Ct = Ct-1(1+g)
t = 1 to T
t=1 to T
PV 
C
1
(1 
)
T
r
(1  r )
C1
(1  g )T
PV 
(1 
)
T
rg
(1  r )
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Compounding interval
• Up to now, interest paid annually
• If n payments per year, compounded value after 1 year :
r n
(1  )
n
• 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:
1. Calculate the monthly interest rate (keeping EAR constant)
(1+rmonthly)12 = 1.10 → rmonthly = 0.7974%
2. Use perpetuity formula:
PV = 1 / 0.007974 = 125.40
Solution 2:
1. Calculate stated annual interest rate = 0.7974% * 12 = 9.568%
2. 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%
(1+Nominal interest rate)=(1+Real interest rate)×(1+Inflation rate)
Approximation:
Real interest rate ≈ Nominal interest rate - Inflation rate
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