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2. The Time Value of Money
2.1 Compounding and Discounting
Capitalization (compounding, finding future values) is a process of moving a value forward in time. It yields the future value given the relevant compounding rate (return rate, interest rate, growth rate).
Actualization (discounting, finding present values) is the reverse process. When we compute present values, we move backward in time. Discounting yields the present value of a future value given the relevant discounting rate (decline rate, interest rate, reduction rate).
Compounding (or discounting) is also a process of converting flow variables (such as sales, expenditures, cash flows, all items shown in income statement or cash flow statement)) into a stock variable (value, a balance sheet item) in two steps:
- Identifying the cash flows provided by the asset
- Compounding or discounting these cash flows at the appropriate growth or discounting rate.
Table 1. Future Value and Present Value Factors
Factor Formula Method of Calculation Future value of a single sum, FVFi,n (1 + i)n (1 + i)n
Present value of a single sum, PVFi,n (^) n (1 i)
1
1
Future value of an ordinary annuity, FVFAi,n
∑ (^) =^ + − n t 1 (1 i)n^1 i
(1 +i)n^ − 1
Future value of an annuity due (1^ +^ i)∑ nt= 1 (1^ +^ i)n−^1 (1 i) i
(1 +i)n− (^1) +
Present value of an annuity, PVFAi,n (^) ∑ (^) =
n t (^1) (1 i)n
1
i
(1 i) 1 1 n
− +
Present value of an annuity due (^) ( )∑ (^) =
- (^) + n t (^1) (1 i)n 1 i^1 ( ) (1i^ i)
1 1 1 i
n
2.2 A single cash flow
The future value factor (growth factor) is the future value of $1 at interest rate i for n periods. The future value of a single sum in n years is determined by
(1) FVFi, (^) n =(1+i)n
The present value factor (discounting factor) is the present value of $1 received or paid at the end of the n th year with the rate i. Discounting is simply the reverse of compounding. The present value factor is equal to
(2) (^) i,n (^) n (1 i)
PVF
The future value (FV) is equal to the present value (PV) multiplied by the future value factor
(3) FV =PV(1+ i)n
The present value (PV) is equal to the future value multiplied by the present value factor
(4) (^) n (1 i)
PV FV
The rate (compounding or growth rate, discounting rate) is found as
PV
FV
i =n^ −
Number of periods is given by
ln(1 i)
PV
FV
ln n
Discrete Compounding
To reflect the frequency of compounding periods, two adjustments are required. First, the interest rate is converted to a per-period rate by dividing the annual rate by the number of compounding periods in a year. Second, the number of years is multiplied by the number of periods that occur each year, mn. The calculation of future value using discrete compounding is
mn
m
i FV PV (^1)
Continuous Compounding
If the compounding period approaches to zero, the future value is
(2) in
mn
m
PVe m
i FV limPV (^1) =
= ^ +
→∞
where e is Euler’s constant, which is approximately 2,72.
If you know the future value, the rate and the number of periods, the payment (ordinary annuity) is
(14) (1 i) 1
i PMT FV n
A perpetuity is a series of equal payments that continue forever (to infinity). The present value of a perpetuity received or paid at the end of each period can be found with a formula:
i (1 i)
(1 i)
(1 i)
(1 i)
(1 i)
limPMT (1 i) i
(1 i) 1 PV=limPMT n
n
n n
n
n n
n n
→∞ → ∞
or
(16) i
PV=PMT
The present value of a perpetuity received or paid at the beginning of each period can be found as follows:
(17) (1 i) i
PV =PMT +
2.4 Growing Cash Flows
The general equation used to find the future value of an n-period growing annuity at a constant rate g is shown below:
i g
(1 i) (1 g) ... PMT(1 g) (1+i) +PMT(1 g) =PMT
FV=PMT(1+i) +PMT(1 g)(1+i) +PMT(1 g)(1+i) +... n n 1
n 1 1
n 2 1 1
2 n 3 1
n 2 1
n- 1 1
− −
− −
The present value of an n-period annuity growing at a constant rate g is given by
n 1 1 i
1 g 1 i g
PMT
PV
A growing perpetuity is one in which the cash flows increase each period by constant rate, that cash flow in any period is (1+g) times larger than the previous cash flow. This process results in equation
i g
PMT
1 i
1 g 1 i g
PMT
PV lim^1
n 1 n (^) −
→∞
2.5 Uneven Cash Flow Streams
In practice most investment or financial decisions involve uneven or nonconstant cash flows.
PV
The present value of uneven cash flows is found as the sum of the present values of the individual cash flows:
(21) 11 22 nn (1 i)
CF
(1 i)
CF
(1 i)
CF
PV
NPV
In practice positive cash flows (cash inflows) and negative cash flows (cash outflows) are usually distinguished. The Net Present Value (NPV) of a stream of cash flows is the difference between the present value of the inflows and the present value of the outflows, that is
(3) NPV = PVinflows - Pvoutflows
The equation for the NPV is usually presented as follows:
(22) 11 22 nn (1 i)
NCF
(1 i)
NCF
(1 i)
NCF
NPV
Cash flows may be positive or negative. The rate is the same for all periods ( traditional approach ).
It is also possible to use different discounting rates for each period ( non-arbitrage approach ). The NPV is then calculated as
(23) (^) n n
n 2 2
2 1 1
1 (1 i )
NCF
(1 i )
NCF
(1 i)
NCF
NPV
IRR
Internal rate of return (IRR) is the discounting rate that makes NPV=0, or, equivalently, the rate that makes the present value of inflows equal to the present value of outflows.
Finding IRR means solving the following equation:
(24) 0 (1 IRR)
NCF
(1 IRR)
NCF
(1 IRR)
NCF
n
n 2
2 1
This equation is a polynomial of the n th degree. If cash flows occur over more than two periods, the IRR cannot be solved directly, and therefore the trial-and error method or interpolation method becomes necessary.
