MBA Finance Part-Time
Present Value
Professor Hugues Pirotte Spéder
Solvay Business School
Université Libre de Bruxelles
Fall 2002
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
1
1
Present Value
Objectives for this session :
1.
2.
3.
4.
18/11/2002
Introduce present value calculation in a simple 1-period setting
Extend present value calculation to several periods
Analyse the impact of the compounding periods
Introduce shortcut formulas for PV calculations
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
2
Present value is the workhorse for measuring value creation. It captures two
basic financial principles:
A euro today is worth more than a euro tomorrow
A safe euro is worth more than a risky one
The first principle is based on the existence of financial markets. Borrowing
and lending are used to move cash flows over time. Excess cash can be
invested and earns interest: one euro invested today will be worth more than
one euro in the future. Stated differently, the On this other hand, cash
deficit can be met with borrowing. When borrowing, the future repayment
will be greater than the amount borrowed as it will include both the
principal and the interest.
The second principle is based on the idea that individual are risk averse:
they prefer a safe cash flow to a uncertain one with the same expected value.
Stated differently, risk averse individual would pay more for the safe cash
flow.
2
Present Value: teaching strategy
• 1-period:
• FV, PV, 1-year discount factor DF1
• NPV, IRR
• Several periods: start from Strips
– Zero-coupons & disc. Factors
• General PV formula
– From prices to interest rates: spot rates
– Shortcut formulas: perpetuities and annuities
– Compounding interval
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
3
In a one-period framework, both teaching strategies are equivalent.
Starting from prices presents both advantages (+) and drawbacks (-):
+ prices are what we are looking for; starting from zc is natural
+ multiple discount rates are easy to introduce (term structure)
+ starting from price is more natural
-
3
Interest rates and present value: 1 period
• Suppose that the 1-year interest rate r1 = 5%
?
?
• €1 at time 0
€1.05 at time 1
• €1/1.05 = 0.9523 at time 0
€1 at time 1
• 1-year discount factor: DF1 = 1 / (1+r1)
• Suppose that the 1-year discount factor DF1 = 0.95
?
?
• €0.95 at time 0
€1 at time 1
• € 1 at time 0
€ 1/0.95 = 1.0526 at time 1
• The 1-year interest rate r1 = 5.26%
• Future value of C0 : FV1(C0) = C0 ×(1+r1) = C0 / DF1
• Present value of C1: PV(C1) = C1 / (1+r1) = C1 × DF1
• Data: r1 ? DF1 = 1/(1+r1) or
18/11/2002
A.Farber & H.Pirotte
Data: DF1 ? r1 = 1/DF1 - 1
MBA Finance – 03 Present Value
4
We now introduce the 1-year interest rate.
Starting from the 1-year discount factor (the market value of a 1-year zerocoupon with face value of 1), it is straightforward to calculate the 1-year
interest rate. You pay €0.95 for a 1-year bond. You receive €1 one year
later. The interest you earn over one year is €0.5. This correspond to an
interest rate of 0.5/0.95 = 5.26%
We are now in a position to calculate the present value of a cash flow one
year from now.
If we know the 1-year discount factor, we proceed as before by multiplying
the future cash flow by the 1-year discount factor.
If we know the 1-year interest rate, we get the PV as:
PV = C1 / (1+r1)
Note the relationship between present value and future value:
PV = FV1 * DF1
4
Let’s define the goal à Time value of money: using PV
• Consider simple investment project:
– Interest rate r = 5%, DF1 = 0.9523
125
0
1
-100
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
5
We begin with the simplest possible setting: analyzing a 1-period project
under certainty.
The key point to realize is that, because of the interest rate, adding cash
flows is not sufficient to reach a decision.
To see this, imagine that the current 1-year interest rate is 10% and that you
analyze a project with the following cash flows: C0 = -100 C1 = +105
The future cash inflow is greater than the initial cash outflow. But this is not
a sufficient condition for this project to be profitable. With an interest rate
of 10%, investing 100 in financial market would give a future cash inflow of
110.
Comparing the cash flows associated with these two alternative strategies
shows that you are better off lending at 10% rather than investing your
money in the project.
5
Thus, we can compute the « Net Future value »…
•
•
NFV = +125 - 100 × 1.05 = 20
= + C1 - I (1+r)
•
Decision rule: invest if NFV>0
•
Justification: takes into account the
cost of capital
– cost of financing
– opportunity cost
+125
+100
0
-100
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
1
-105
6
In this simple setting (1-period, certainty), several possible interpretations of
the interest rate are possible.
Cost of capital: if you don’t have the require amount for the investment, you
would have to borrow it. The repayment one year later would be 110 =
I(1+r). The net future value tells you how much you will have in cash one
year later if you go ahead with the project and borrow to fund the initial
investment.
Opportunity cost of capital: suppose that you have in cash the required
amount for the project. You don’t need to borrow but you still need to take
interest rate in account because, if the projected is accepted, your money
will not be available to earn interest. Committing the cash to the project has
an opportunity cost: the interest foregone because cash in invested in the
project rather than being lent.
6
… or inversely the « Net Present Value »
• NPV = - 100 + 125/1.05 = + 19
•
= - I + C1/(1+r)
•
= - I + C1 × DF1
•
= - 100+125 × 0.9524
•
= +19
+119
+125
• DF1 = 1-year discount factor
•
a market price
• C1 × DF1 =PV(C1)
• Decision rule: invest if NPV>0
-100
-125
• NPV>0 ⇔ NFV>0
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
7
Application: Valuing a 1-year bond
Suppose that the current 1-year rate r = 5%
How much would you be ready to pay for a bond with following
characteristics?
Maturity
Face value
Coupon
1 year
100
8%
Solution:
The price to pay is the present value of the future cash flow
P0 = 108/1.05 = 108 × 0.9524 = 102.86
Check: the expected return on this bond should be equal to the 1-year
interest rate (5%)
Expected return = [Coupon + (P1-P0)]/P0
= [8+(100-102.86)]/102.86
=5%
7
Economic foundations of net present value
Euros next year
I. Fisher 1907, J. Hirshleifer 1958
165
Perfect capital markets
Separate investment decisions
from consumption decisions
105
Slope = - (1 + r) = - (1 + 5%)
NPV
-50
50
18/11/2002
A.Farber & H.Pirotte
100
200
MBA Finance – 03 Present Value
207
Euros now
8
8
Entreprise Value Maximisation
Euros next year
Investment
opportunities
Investment
NPV
0
Euros today
Market value of company
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
9
9
Internal Rate of Return
• Alternative rule: compare the internal rate of return for the project to the
opportunity cost of capital
• Definition of the Internal Rate of Return IRR : (1-period)
IRR = Profit/Investment = (C1 - I)/I
• In our example:
IRR = (125 - 100)/100 = 25%
à The Rate of Return Rule: Invest if IRR > r
• In this simple setting, the NPV rule and the Rate of Return Rule lead to
the same decision:
• NPV = -I+C1/(1+r) >0 ⇔ C1>I(1+r) ⇔ (C1-I)/I>r ⇔ IRR>r
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
10
The definition of the IRR in this slide is only valid for 1-period projects. A
more general definition is suggested by noting that, if discounted at the IRR,
the NPV is always equal to 0.
NPV=-I+C1/(1+IRR)= -I + C1/[1+(C1-I)/I]=0
The leads to a general definition. The internal rate of return is defined as the
rate of discount which makes NPV=0
NPV
IRR
Discount rate
10
IRR: a general definition
•
•
The Internal Rate of Return is
the discount rate such that the
NPV is equal to zero.
30.00
25.00
20.00
-I + C1/(1+IRR) ≡ 0
•
•
•
Net Present Value
15.00
In our example:
-100 + 125/(1+IRR)=0
⇒ IRR=25%
10.00
5.00
IRR
0.00
0.0%
2.5%
5.0%
7.5%
10.0%
12.5%
15.0%
17.5%
20.0%
22.5%
25.0%
27.5%
30.0%
-5.00
-10.00
Discount Rate
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
11
Application: Yield to maturity on a 1-year bond
The yield to maturity on a bond is the discount rate such that the current
market value is equal to the present value of future cash flows.
For a 1-year bond, the bond’s yield y is obtained by solving the following
equation
P0 = [Coupon + Face Value]/(1+y)
The can be written as:
-P0 + [Coupon + Face Value]/(1+y) = 0
Comparing this later expression to the definition of the IRR show that the
yield to maturity is the IRR for an investment the bond at the current market
price.
Example: Suppose that a 1-year 8% coupon bond with face value 100 sells
for 101.50
The yield to maturity on this bond is:
y = (8+100)/101.50 – 1=6.40%
11
Present Value Calculation with Uncertainty
Consider the following project:
• C0 = -I = -100
Cash flow year 1:
+50 with probability ½
+200 with probability ½
•
•
•
•
•
18/11/2002
The expected future cash flow is C1 = 0.5 * 50 + 0.5 * 200 = 125
The discount rate to use is the expected return of a stock with similar risk
r = Risk-free rate + Risk premium
=
5%
+
6%
(this is an example)
NPV = -100 + 125 / (1.11) = 12.6
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
12
Calculating present value when future cash flows are uncertain is achieved
using the same formula as before with two main differences.
The first one is that, as the future cash flow is uncertain, the amount to
discount is the expected cash flow. The expected cash flow is obtained as
the average of the possible realizations of the future weighted by their
respective probabilities.
The second difference is that the discount rate to use is no longer equal to
the 1-year risk-free interest rate. If the future cash is uncertain, risk averse
investors will require a higher expected return than when the cash flow is
certain. This higher expected return of obtained by adding a risk premium to
the risk-free interest rate to obtain the discount rate.
The exact determination of this risk premium is a difficult topic that we will
analyze in a later session.
12
A simple investment problem
• Consider the following project:
• Cash flow t = 0
C0 = - I = - 100
• Cash flow t = 5
C5 = + 150 (risk-free)
• How to calculate the economic profit?
• Compare initial investment with the market value of the future
cash flow.
• Market value of C5
•
•
•
= Present value of C5
= C5 * Present value of $1 in year 5
= C5 * 5-year discount factor
= C5 * DF5
• Profit = Net Present Value = - I + C5 * DF5
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
13
When doing financial calculations, we have to juggle with market values
and interest rates. The traditional presentation of present value starts with
one single interest rate. It then proceeds to explain how to calculate prices
(or present values) based on this interest rate. This is what Ross, Westerfield
and Jaffee do in their Chapter 4. Brealey and Myers (Chap 3) recognize first
that interest rate vary depending on the maturity of cash flow. But they then
finesse the issue by assuming that “the interest rate is the same regardless of
the date of the cash flow”. The analysis of the term structure (the
relationship between interest rates and maturities) is deferred to an appendix
(Appendix 5A in RWJ) or to a later chapter (Chap 23 in BM).
In this presentation, we take a slightly different approach. We start from
market prices of risk-free zero coupons. We show how these market prices
can be used to calculate the net present value of a project involving certain
cash flows. We then examine how interest rates can be determined based on
these market prices.
Of course, at the end, we will be able to take whatever starting point is more
appropriate. But starting from prices will, hopefully, make clear that there is
no such thing at The interest rate. Present value calculation taking into
account the current shape of the term structure is not more difficult than
assuming a flat term structure. It has the huge advantage to fit with current
prices observed on the market.
13
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
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
14
A zero-coupon is a pure discount bond. You pay a price today and you
receive one single payment (the face value of the zero-coupon) at one future
date (the maturity of the zero-coupon).
In the early 1980s, several U.S. investment banks started “stripping” U.S.
Treasury issues. They would buy a bond (consisting of a number of coupons
and the principal) and sell each element separately as a zero-coupon using
feline names as CATS, TIGRS, COUGARS and LIONS1. The idea was
taken over by the U.S. Treasury in 1985 who started offering its own zerocoupon instruments under the denomination STRIPS (nothing to do with
strip tease, at least as far as I know..).
Prices of U.S. Treasury strips are regularly published in the WSJ. Note that
prices are quoted with colons that represent 32nds. For instance, 93:16
means 93 + 16/32 = 93.50.
Strips were later on introduced in various European markets. In the UK, Gilt
strips exist (a gilt is a long term British government bond) and quotations
can be found in the FT. France introduced stripped OATs (OAT means
Obligation Assimilée du Trésor). In Belgium, strips on Belgian government
bonds are known as “obligations démambrées”.
1
CATS: Certificates of Accrual on Treasury Securities (Salomon Brothers)
TIGRS: Treasury Investment Growth Receipts (Merrill Lynch)
COUGARS: Certificates on Government Receipts (A. G. Beckers Paribas)
LIONS: Lehman Investment Opportunity Notes (Lehman Brothers)
14
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
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
15
We now have a general formula to compute present values. The formula is
very simple. You simply multiply each cash flow by the corresponding
discount factor and you take the sum.
Keep in mind that the discount factors are market prices for zero-coupon.
An analogy is helpful to understand the logic.
Suppose that you are offered the following deal. You pay 100 EUR and you
receive 30 USD (US dollar), 40 GBP (British pound) and 60 CHF (Swiss
franc). Is this a good deal?
Before accepting, you will convert all cash flows into one unit: the euro
using current spot exchange rates. If the spot rates are:
1 USD = 1 EUR
EUR
1 GBP = 1.50 EUR
1 CHF = 0.70
the net present value is NPV = - 100 + 30 * 1 + 40 * 1.50 + 40 * 0.70
= - 100 + 132 = +32
This is a good deal.
Present value calculation is similar. Instead of cash flows being
denominated in different currencies, they take place at different points in
time. The NPV is obtained by valuing each cash flow at the appropriate
“exchange rate”.
15
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
• FV10 = 1,000 (1.08)10 = 2,158.82
Principal amount
1,000
Simple interest
Interest on interest
18/11/2002
A.Farber & H.Pirotte
800
358.82
MBA Finance – 03 Present Value
16
Two important points to note:
1. Note that reinvestment takes place at the same rate r
2. Reinvestment is done once a year (annual compounding)
The Power of Compounding
The power of compounding increases with the investment horizon. interest
on interest represents an increasing fraction of the future value as the
investment horizon become longer.
As an example, consider an investment of €1 at 8% for 50 years.
The future value is FV50 = (1.08)50 = 46.90
The decomposition of this future value is:
Principal amount
Simple interest
Interest on interest
1
4
41.90
Interest on interest is 89% of future value!
For those of you having a mathematical background, note that the future
value is a convex function of time to maturity:
f(t) = C0 (1+r)t
f’(t) = t C0 (1+r)t-1 > 0
f”(t) = t(t-1) C0 (1+r)t-2 > 0
16
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 =
18/11/2002
A.Farber & H.Pirotte
1
(1 + rt ) t
MBA Finance – 03 Present Value
17
This is the standard formula to calculate a discount factor.
Note that the discount rate used rt is, in general, a function of the maturity.
17
Discount factors
Interest rate per year
# years
18/11/2002
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
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
18
18
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
• The YTM r5 is the solution of:
100
75 =
(1 + r5 ) 5
• The solution is:
1
 100 
r5 = 

 75 
5
− 1 = 5.92%
• This is the 5-year spot interest rate
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
19
A spot transaction is a transaction where both delivery and payment take
place immediately. In a forward transaction, the terms of the transaction are
set now but the transaction itself is executed at some future date.
The spot interest rate is the rate on an loan starting immediately. For
example, you borrow 75 and you will repay 100 five year later. The spot
interest rate is simply the annual interest rate charged on this transaction.
The final payment (100) is the future value of the amount borrowed:
100 = 75 (1+r5)5 ? r5 = 5.92%
There are no intermediate interest payment.
The previous formula can also be written as:
75 = 100 / (1+r5)5
If 75 is the current price of a 5-year zero-coupon with face value equal to
100, then this second presentation shows that the spot interest rate is the
yield-to-maturity of the zero-coupon.
19
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
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
20
Starting from the prices of zero-coupons with different maturities, we
calculate the yield-to-maturity of each of them to obtain the various spot
interest rates.
In general, spot rates vary with maturity. The term structure of interest rates
is the relationship between spot interest rates (yields of zero-coupons) and
maturity.
The term structure of interest rates for the European Monetary Union is
available of the web site of the European Commission:
http://europa.eu.int/comm/eurostat. Click on Euro yield curve.
20
Euro Yield curve Sept 25, 2002
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
21
21
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)
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
22
22
Constant perpetuity
• Ct =C for t =1, 2, 3, .....
PV =
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
r
• Examples: Preferred stock (Stock paying a fixed dividend)
• Suppose r =10%
Yearly dividend =
50
• Market value P0?
50
P0 =
.10
= 500
• Note: expected price next year =
• Expected return =
18/11/2002
div
A.Farber & H.Pirotte
1
P1 =
50
= 500
.10
+ ( P1 − P 0 )
50 + ( 500 − 500
=
P0
500
)
= 10 %
MBA Finance – 03 Present Value
23
This is a simple and powerful formula.
Always keep in mind that this formula gives you the present value of a
perpetuity one year before the first cash flow.
In our example, the first cash flow is receive at time t=1 and the formula
gives the price at time 0.
You can combine the formula with the discount factor to compute the
present value of a deferred perpetuity, a perpetuity starting at some future
date.
As an example, suppose that you start a pension fund. You expect to pay
€10m (millions) each year to your employees when they retire. Your first
payment in due in 31 years (your employees are pretty young). What is the
present value of your pension liability if the discount rate is 8%?
First calculate the future value of the pension payments one year before the
first payment (this is in year t = 30). This value is €10m / 0.08 = €125m
Now discount it back to today by multiplying the future value by the 30year discount factor DF30 = 1/(1.08)30 = 0.0994
The present value of your pension liability is:
PV = €125m * 0.0994 =€12.42m
23
Growing perpetuity
• Ct =C1 (1+g)t-1 for t=1, 2, 3, .....
PV =
r>g
C1
r−g
• Example: Stock valuation based on:
• Next dividend div1, long term growth of dividend g
• If r = 10%, div1 = 50, g = 5%
P0 =
• Note: expected price next year =
50
= 1,000
.10 − .05
P1 =
• Expected return =
18/11/2002
52.5
= 1,050
.10 − .05
div1 + ( P1 − P0 ) 50 + (1,050 − 1,000)
=
= 10%
1,000
P0
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
24
This formula is widely used for valuing stocks. In that case, g is the growth
rate of the dividend.
Note that the growth rate should be less than the discount rate. This is
because the growth rate is constant up to infinity. The condition is required
for the geometric series to converge.
For those of you willing to have a proof, here is a short one:
PV = C1 * d + C1 (1+g) * d² + C1 (1+g)² * d³ + …
where d = 1/(1+r)
Multiply both side by (1+g).
PV (1+g) = C1 (1+g) * d + C1 (1+g)² * d² + C1 (1+g)³ * d³ + …
The right hand side is the value of a perpetuity with a discount rate
x =(1+r)/(1+g) – 1 = (r – g)/(1+g)
So:
PV(1+g) = C1 / x = C1 (1+g) / (r-g)
24
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.
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
25
25
Constant Annuity
• Ct = C for t = 1, 2, …,T
PV =
C
1
[1 −
]
r
(1 + r ) T
• 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
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
26
The present value of a constant annuity can be expressed as the annual cash
flow multiplied by an annuity factor.
PV = C * Annuity Factor
The T-year annuity factor is the sum of the first T discount factors:
Annuity Factor = DF1 + DF2 + … + DFT
This formulation is useful if you need to calculate an annuity factor when
the discount rate is not a constant.
When the discount rate is a constant, the annuity factor can be expressed as
the difference between two perpetuity: one perpetuity beginning in year 1
minus a second deferred perpetuity beginning in year T+1.
The value of the first perpetuity is C/r (250,000 in our example).
The value of the second perpetuity is C/r in year T. It’s value today is equal
to C/r * DFT (in our example: 250,000 * 0.1486)
As a consequence, the annuity factor can be expressed as:
Annuity factor = (1/r)*(1 – DFT)
In our example: Annuity factor = (1 / 0.10) * (1 – 0.1486) = 8.5136
26
Annuity Factors
Interest rate per year
# years
18/11/2002
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
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
27
27
Growing annuity
• Ct = C1 (1+g)t-1 for t = 1, 2, …, T
PV =
r?g
T
C1   1 + g  
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
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
28
This derivation of the formula for a growing annuity is similar to the
derivation of the formula for the constant annuity.
Start from a growing perpetuity the present value of which is C1/(r – g)
Subtract from this growing perpetuity the present value of a second growing
perpetuity starting in T+1. The annual cash flow at that time is C1(1+g)T.
The present value at time 0 of this second growing perpetuity is
[C1(1+g)T/(r–g)] / (1+r)T.
Note that this formula works for any g ? r. In particular, it will work for
g>r. In this case, r-g is negative but simultaneously the ratio (1+g)/(1+r) is
greater than 1 so that the difference between brackets is also negative.
28
Compounding interval
• Up to now, interest paid annually
• If n payments per year, compounded value after 1 year :
 r
1 + 
 n
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%
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
29
Here we introduce one additional complication: the compounding interval.
Up to now, we have assumed that interest are paid and reinvested once a
year. But this not always the case.
Suppose that you borrow for 3 months. The stated annual interest rate (or
the annual percentage rate) is the interest quoted by the bank. Suppose it is
6%. This means that the interest you will pay after 3 months is 6% / 4 =
1.5%
Now imagine that you want to borrow for 1 year. The bank tells you that it
will charge an annual percentage rate of 6% but that interest are payable
quarterly. This means that you will have to borrow at the end of each quarter
the principal and the interest. The future value after 1-year will be:
FV1 = (1+6%/4)4 = 1.06136
The 1-year borrowing rate is effectively 6.136%. This is the effective annual
interest rate.
29
Juggling with compounding intervals
•
•
•
•
•
18/11/2002
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
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
30
Here I want to show how to take into account the timing of cash flows when
the time interval between cash flows is less than a year.
The most obvious solution consists in changing the discount factor to take
into account the spacing of cash flows. But this should be done carefully
while keeping the values of annual cash flows unchanged.
In our example, we want to value a perpetuity of 1 per month. We calculate
an interest per month while keeping the effective annual interest rate
unchanged. We then use the formula for valuing a perpetuity. We get:
PV = (C / 12) /(rmonthly)
But this is the same as:
PV = C / (12 * rmonthly)
The product 12 * rmontly is the stated annual interest rate with monthly
compounding with the same effective annual interest rate.
Note that the stated annual rate with monthly compounding is lower than the
effective annual rate.
30
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
à 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
18/11/2002
A.Farber & H.Pirotte
MBA Finance – 03 Present Value
31
31