Financial Management
Dr. Andrea Rigamonti
andrea.rigamonti@econ.muni.cz
Lecture 3
Content:
• General evaluation principles
• Present value of some common investments
• Amortized loans
• Investment decision rules
General evaluation principles
Given the interest rate R and the number of periods n, the
value of an investment:
• in terms of dollars today is the present value (PV):
𝑃𝑉 =
𝐹𝑉
(1 + 𝑅) 𝑛
• in terms of dollars in the future is the future value (FV):
𝐹𝑉 = 𝑃𝑉 ∗ (1 + 𝑅) 𝑛
(Risky cash flows must be discounted at a rate equal to the
risk-free rate plus an appropriate risk premium.)
General evaluation principles
General process for pricing securities (in a market without
arbitrage opportunities):
1. Identify the cash flows that will be paid by the security
2. Determine the PV of the security’s cash flows
Unless the price of the security equals this present value,
there is an arbitrage opportunity:
No-Arbitrage Price = PV(All cash flows paid by the security)
General evaluation principles
It is only possible to compare or combine values at the
same point in time.
To move a cash flow C forward in time you must compound
it (compute the future value):
𝐹𝑉𝑛 = 𝐶 ∗ (1 + 𝑅) 𝑛
To find the equivalent value today of a future cash flow C
you must discount it (compute the present value):
𝑃𝑉 =
𝐶
(1 + 𝑅) 𝑛
General evaluation principles
Present Value of a Cash Flow Stream:
𝑃𝑉 = ෍
𝑛=0
𝑁
𝑃𝑉(𝐶 𝑛) = ෍
𝑛=0
𝑁
𝐶 𝑛
(1 + 𝑅) 𝑛
Future Value of a Cash Flow Stream with Present Value PV:
𝐹𝑉𝑛 = 𝑃𝑉 ∗ (1 + 𝑅) 𝑛
Present value of some common investments
A perpetuity is a stream of equal cash flows that occur
at regular intervals and last forever.
The present value of a perpetuity with payment C and
interest rate R is given by
𝑃𝑉 = ෍
𝑛=1
∞
𝐶
(1 + 𝑅) 𝑛
Because the first cash flow is in one period, C0 = 0.
It can be shown that:
𝑃𝑉(𝐶 𝑖𝑛 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦) =
𝐶
𝑅
Present value of some common investments
An annuity is a stream of N equal cash flows paid at
regular intervals (the first one at date 1). At the end of it,
you get your initial investment back.
The present value of an N-period annuity with payment
C and interest rate R is 𝑃𝑉 = σ 𝑛=1
𝑁 𝐶
(1+𝑅) 𝑛
It can be shown that the PV of an annuity is:
𝑃𝑉 = 𝐶
1
𝑅
1 −
1
(1 + 𝑅) 𝑁
The Future Value of an Annuity is:
𝐹𝑉 = 𝐶
1
𝑅
((1 + 𝑅) 𝑁
−1)
Present value of some common investments
A growing perpetuity is a stream of cash flows that occur at
regular intervals and grow at a constant rate forever. A
growing perpetuity with a first payment C and a growth rate
g has present value
𝑃𝑉 = ෍
𝑛=1
∞
𝐶(1 + 𝑔) 𝑛−1
(1 + 𝑅) 𝑛
It must be g < R, so that each successive term in the sum is
less than the previous term and the overall sum is finite.
It can be shown that:
𝑃𝑉(𝑔𝑟𝑜𝑤𝑖𝑛𝑔 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦) =
𝐶
𝑅 − 𝑔
Present value of some common investments
A growing annuity is a stream of N growing cash flows, paid
at regular intervals. The first cash flow arrives at the end of
the first period, and the first cash flow does not grow. The
present value of an N-period growing annuity with initial
cash flow C, growth rate g, and interest rate r is given by
𝑃𝑉 = 𝐶
1
𝑅 − 𝑔
1 −
1 + 𝑔
1 + 𝑅
𝑁
Because the annuity has only a finite number of terms, we
can have g > R. The formula does not work for g = R, but in
that case growth and discounting cancel out: 𝑃𝑉 = 𝐶
𝑁
1+𝑅
Amortized loans
An important application of compound interests are
amortized loans. Given a loan of amount 𝐿 that has to be
repaid in 𝑁 periods, with interest 𝑅, the payments 𝑃 are
determined from
𝐿 = ෍
𝑛=1
𝑁
𝑃
(1 + 𝑅) 𝑛
Notice that the longer the amortization period, the higher
the amount to be paid in form of interests.
Amortized loans are often use in automobile, home
mortgage, and business loans.
Amortized loans
In a standard amortized loan, fixed payments are made at
regular intervals, with a fixed interest rate. This qualifies it
as an annuity, and we can compute the amount of the
periodic payments using the annuity formula:
𝐿 = 𝑃
1
𝑅
1 −
1
(1 + 𝑅) 𝑁
Given the amount of the loan 𝐿, the number of periods 𝑁
and the interest rate 𝑅, one simply needs to solve for 𝑃:
𝑃 =
𝐿
1
𝑅
−
1
𝑅 1 + 𝑅 𝑁
Investment decision rules
Net present value (NPV): NPV = PV(benefits) – PV(costs)
If NPV > 0 the investment is profitable.
Internal rate of return (IRR): the interest rate that sets the
net present value of the cash flows equal to zero.
To compute it you just need to invert the formulas. This is
only easy with one or two periods with cash flows.
Abel–Ruffini theorem (Abel's impossibility theorem): there
is no general algebraic solution to polynomial equations of
degree five or higher!
Investment decision rules
NPV Investment Rule: take the investment with the highest
NPV. This is equivalent to receiving its NPV in cash today.
NPV profile: a graph of the project’s NPV over a range of
discount rates. Useful when there is some uncertainty
regarding the project’s cost of capital.
The difference between the cost of capital and the IRR is
the maximum estimation error in the cost of capital that
can exist without altering the original decision.
Investment decision rules
In this example, the cost of capital r is estimated = 10%
The investment remains profitable as long as the real r < 14%
Investment decision rules
The NPV rule is the most accurate and reliable, but other
tools are applied by practitioners. They often give the same
indication; if they don’t, they should be disregarded.
The internal rate of return (IRR) investment rule suggests that
if the average return on the investment opportunity (i.e. the
IRR) is greater than the return on other alternatives in the
market with equivalent risk and maturity (i.e. the project’s
cost of capital), you should undertake the investment.
IRR Investment Rule: Take any investment opportunity
where the IRR exceeds the opportunity cost of capital. Turn
down any opportunity whose IRR is less than the opportunity
cost of capital.
Investment decision rules
The IRR rule is only guaranteed to work for a stand-alone
project if all of the project’s negative cash flows precede its
positive cash flows. When the benefits of an investment
occur before the costs, the NPV is an increasing function of
the discount rate, and the IRR rule fails.
Another problematic situation occurs with multiple IRRs.
Finally, the IRR might be non-existent because the NPV is
positive for every discount rate.
Investment decision rules
Payback investment rule: you should only accept a
project if its cash flows pay back its initial investment
within a prespecified period, called payback period.
Problems with this rule:
1.It ignores the project’s cost of capital and the time
value of money;
2.it ignores cash flows after the payback period;
3.it relies on an ad hoc decision criterion (the choice of
the length of the payback period is arbitrary).
Investment decision rules
Sometimes a firm must choose just one among several
mutually exclusive possible projects.
The correct choice is to pick the project with the highest NPV.
Picking the project with the larger IRR can lead to mistakes:
when projects differ in their scale of investment, the timing of
their cash flows, or their riskiness, their IRRs cannot be
meaningfully compared.
A modified version called incremental IRR has been proposed
but it also can be problematic, and it is also more difficult to
use than the NPV, so it should not be used.