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Time value of money
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The present value of $1,000, 100 years into the future. Curves represent constant discount rates of 2%, 3%, 5%, and 7%.
The time value of money is the greater benefit of receiving money now rather than later. It is founded on time preference.
The principle of the time value of money explains why interest is paid or earned: Interest, whether it is on a bank deposit or debt, compensates the depositor or lender for the time value of money.
It also underlies investment. Investors are willing to forgo spending their money now if they expect a favorable return on their investment in the future.
Historically in Christian societies, and in many Islamic societies today, charging any interest at all would be considered usury.
Contents
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1 History
2 Calculations
3 Formula
3.1 Future value of a present sum
3.2 Present value of a future sum
3.3 Present value of an annuity for n payment periods
3.4 Present value of a growing annuity
3.5 Present value of a perpetuity
3.6 Present value of a growing perpetuity
3.7 Future value of an annuity
3.8 Future value of a growing annuity
3.9 Formula table
4 Derivations
4.1 Annuity derivation
4.2 Perpetuity derivation
5 Examples
5.1 Example 1: Present value
5.2 Example 2: Present value of an annuity—solving for the payment amount
5.2.1 An approximate solution
5.3 Example 3: Solving for the period needed to double money
5.4 Example 4: What return is needed to double money?
5.5 Example 5: Calculate the value of a regular savings deposit in the future.
5.6 Example 6: Price/earnings (P/E) ratio
6 Continuous compounding
6.1 Examples
7 Differential equations
8 See also
9 Notes
10 References
11 External links
History[edit]
The Talmud (~500 CE) recognizes the time value of money. In Tractate Makkos page 3a the Talmud discusses a case where witnesses falsely claimed that the term of a loan was 30 days when it was actually 10 years. The false witnesses must pay the difference of the value of the loan "in a situation where he would be required to give the money back (within) thirty days..., and that same sum in a situation where he would be required to give the money back (within) 10 years...The difference is the sum that the testimony of the (false) witnesses sought to have the borrower lose; therefore, it is the sum that they must pay." [1]
The notion was later described by Martín de Azpilcueta (1491–1586) of the School of Salamanca.
Calculations[edit]
Time value of money problems involve the net value of cash flows at different points in time.
In a typical case, the variables might be: a balance (the real or nominal value of a debt or a financial asset in terms of monetary units), a periodic rate of interest, the number of periods, and a series of cash flows. (In the case of a debt, cash flows are payments against principal and interest; in the case of a financial asset, these are contributions to or withdrawals from the balance.) More generally, the cash flows may not be periodic but may be specified individually. Any of the variables may be the independent variable (the sought-for answer) in a given problem. For example, one may know that: the interest is 0.5% per period (per month, say); the number of periods is 60 (months); the initial balance (of the debt, in this case) is 25,000 units; and the final balance is 0 units. The unknown variable may be the monthly payment that the borrower must pay.
For example, £100 invested for one year, earning 5% interest, will be worth £105 after one year; therefore, £100 paid now and £105 paid exactly one year later both have the same value to a recipient who expects 5% interest assuming that inflation would be zero percent. That is, £100 invested for one year at 5% interest has a future value of £105 under the assumption that inflation would be zero percent.[2]
This principle allows for the valuation of a likely stream of income in the future, in such a way that annual incomes are discounted and then added together, thus providing a lump-sum "present value" of the entire income stream; all of the standard calculations for time value of money derive from the most basic algebraic expression for the present value of a future sum, "discounted" to the present by an amount equal to the time value of money. For example, the future value sum F V {\displaystyle FV} to be received in one year is discounted at the rate of interest r {\displaystyle r} to give the present value sum P V {\displaystyle PV} :
P V = F V ( 1 + r ) {\displaystyle PV={\frac {FV}{(1+r)}}}
Some standard calculations based on the time value of money are:
Present value: The current worth of a future sum of money or stream of cash flows, given a specified rate of return. Future cash flows are "discounted" at the discount rate; the higher the discount rate, the lower the present value of the future cash flows. Determining the appropriate discount rate is the key to valuing future cash flows properly, whether they be earnings or obligations.[3]
Present value of an annuity: An annuity is a series of equal payments or receipts that occur at evenly spaced intervals. Leases and rental payments are examples. The payments or receipts occur at the end of each period for an ordinary annuity while they occur at the beginning of each period for an annuity due.[4]
Present value of a perpetuity is an infinite and constant stream of identical cash flows.[5]
Future value: The value of an asset or cash at a specified date in the future, based on the value of that asset in the present.[6]
Future value of an annuity (FVA): The future value of a stream of payments (annuity), assuming the payments are invested at a given rate of interest.
There are several basic equations that represent the equalities listed above. The solutions may be found using (in most cases) the formulas, a financial calculator or a spreadsheet. The formulas are programmed into most financial calculators and several spreadsheet functions (such as PV, FV, RATE, NPER, and PMT).[7]
For any of the equations below, the formula may also be rearranged to determine one of the other unknowns. In the case of the standard annuity formula, however, there is no closed-form algebraic solution for the interest rate (although financial calculators and spreadsheet programs can readily determine solutions through rapid trial and error algorithms).
These equations are frequently combined for particular uses. For example, bonds can be readily priced using these equations. A typical coupon bond is composed of two types of payments: a stream of coupon payments similar to an annuity, and a lump-sum return of capital at the end of the bond's maturity—that is, a future payment. The two formulas can be combined to determine the present value of the bond.
An important note is that the interest rate i is the interest rate for the relevant period. For an annuity that makes one payment per year, i will be the annual interest rate. For an income or payment stream with a different payment schedule, the interest rate must be converted into the relevant periodic interest rate. For example, a monthly rate for a mortgage with monthly payments requires that the interest rate be divided by 12 (see the example below). See compound interest for details on converting between different periodic interest rates.
The rate of return in the calculations can be either the variable solved for, or a predefined variable that measures a discount rate, interest, inflation, rate of return, cost of equity, cost of debt or any number of other analogous concepts. The choice of the appropriate rate is critical to the exercise, and the use of an incorrect discount rate will make the results meaningless.
For calculations involving annuities, you must decide whether the payments are made at the end of each period (known as an ordinary annuity), or at the beginning of each period (known as an annuity due). If you are using a financial calculator or a spreadsheet, you can usually set it for either calculation. The following formulas are for an ordinary annuity. If you want the answer for the present value of an annuity due simply multiply the PV of an ordinary annuity by (1 + i).
Formula[edit]
The following formula use these common variables:
PV is the value at time=0 (present value)
FV is the value at time=n (future value)
A is the value of the individual payments in each compounding period
n is the number of periods (not necessarily an integer)
i is the interest rate at which the amount compounds each period
g is the growing rate of payments over each time period
Future value of a present sum[edit]
The future value (FV) formula is similar and uses the same variables.
F V = P V ⋅ ( 1 + i ) n {\displaystyle FV\ =\ PV\cdot (1+i)^{n}}
Present value of a future sum[edit]
The present value formula is the core formula for the time value of money; each of the other formulae is derived from this formula. For example, the annuity formula is the sum of a series of present value calculations.
The present value (PV) formula has four variables, each of which can be solved for:
P V = F V ( 1 + i ) n {\displaystyle PV\ =\ {\frac {FV}{(1+i)^{n}}}}
The cumulative present value of future cash flows can be calculated by summing the contributions of FVt, the value of cash flow at time t
P V = ∑ t = 1 n F V t ( 1 + i ) t {\displaystyle PV\ =\ \sum _{t=1}^{n}{\frac {FV_{t}}{(1+i)^{t}}}}
Note that this series can be summed for a given value of n, or when n is ∞.[8] This is a very general formula, which leads to several important special cases given below.
Present value of an annuity for n payment periods[edit]
In this case the cash flow values remain the same throughout the n periods. The present value of an annuity (PVA) formula has four variables, each of which can be solved for:
P V ( A ) = A i ⋅ [ 1 − 1 ( 1 + i ) n ] {\displaystyle PV(A)\,=\,{\frac {A}{i}}\cdot \left[{1-{\frac {1}{\left(1+i\right)^{n}}}}\right]}
To get the PV of an annuity due, multiply the above equation by (1 + i).
Present value of a growing annuity[edit]
In this case each cash flow grows by a factor of (1+g). Similar to the formula for an annuity, the present value of a growing annuity (PVGA) uses the same variables with the addition of g as the rate of growth of the annuity (A is the annuity payment in the first period). This is a calculation that is rarely provided for on financial calculators.
Where i ≠ g :
P V = A ( i − g ) [ 1 − ( 1 + g 1 + i ) n ] {\displaystyle PV\,=\,{A \over (i-g)}\left[1-\left({1+g \over 1+i}\right)^{n}\right]}
Where i = g :
P V = A × n 1 + i {\displaystyle PV\,=\,{A\times n \over 1+i}}
To get the PV of a growing annuity due, multiply the above equation by (1 + i).
Present value of a perpetuity[edit]
A perpetuity is payments of a set amount of money that occur on a routine basis and continues forever. When n → ∞, the PV of a perpetuity (a perpetual annuity) formula becomes simple division.
P V ( P ) = A i {\displaystyle PV(P)\ =\ {A \over i}}
Present Value of Int Factor Annuity
A = P ( 1 + r / n ) n t {\displaystyle A=P(1+r/n)^{nt}}
Example:
Investment P = $1000
Interest i = 6.90% Compounded Qtrly (4 Times in Year)
Tenure Years n = 5
= 1000 × ( 1 + 0.069 / 4 ) ( 5 y r s × 4 q t r s i n a y e a r ) = 1000 × ( 1 + 0.069 / 4 ) 20 ≈ 1407.84 {\displaystyle =1000\times (1+0.069/4)^{({5\ yrs}\ \times \ 4\ {qtrs\ in\ a\ year})}=1000\times (1+0.069/4)^{20}\approx 1407.84}
Present value of a growing perpetuity[edit]
When the perpetual annuity payment grows at a fixed rate (g) the value is theoretically determined according to the following formula. In practice, there are few securities with precise characteristics, and the application of this valuation approach is subject to various qualifications and modifications. Most importantly, it is rare to find a growing perpetual annuity with fixed rates of growth and true perpetual cash flow generation. Despite these qualifications, the general approach may be used in valuations of real estate, equities, and other assets.
This is the well known Gordon Growth model used for stock valuation.
Future value of an annuity[edit]
The future value of an annuity (FVA) formula has four variables, each of which can be solved for:
F V ( A ) = A ⋅ ( 1 + i ) n − 1 i {\displaystyle FV(A)\,=\,A\cdot {\frac {\left(1+i\right)^{n}-1}{i}}}
To get the FV of an annuity due, multiply the above equation by (1 + i).
Future value of a growing annuity[edit]
The future value of a growing annuity (FVA) formula has five variables, each of which can be solved for:
Where i ≠ g :
F V ( A ) = A ⋅ ( 1 + i ) n − ( 1 + g ) n i − g {\displaystyle FV(A)\,=\,A\cdot {\frac {\left(1+i\right)^{n}-\left(1+g\right)^{n}}{i-g}}}
Where i = g :
F V ( A ) = A ⋅ n ( 1 + i ) n − 1 {\displaystyle FV(A)\,=\,A\cdot n(1+i)^{n-1}}
Formula table[edit]
The following table summarizes the different formulas commonly used in calculating the time value of money.[9] These values are often displayed in tables where the interest rate and time are specified.
Find
Given
Formula
Future value (F) Present value (P) F = P ⋅ ( 1 + i ) n {\displaystyle F=P\cdot (1+i)^{n}}
Present value (P) Future value (F) P = F ⋅ ( 1 + i ) − n {\displaystyle P=F\cdot (1+i)^{-n}}
Repeating payment (A) Future value (F) A = F ⋅ i ( 1 + i ) n − 1 {\displaystyle A=F\cdot {\frac {i}{(1+i)^{n}-1}}}
Repeating payment (A) Present value (P) A = P ⋅ i ( 1 + i ) n ( 1 + i ) n − 1 {\displaystyle A=P\cdot {\frac {i(1+i)^{n}}{(1+i)^{n}-1}}}
Future value (F) Repeating payment (A) F = A ⋅ ( 1 + i ) n − 1 i {\displaystyle F=A\cdot {\frac {(1+i)^{n}-1}{i}}}
Present value (P) Repeating payment (A) P = A ⋅ ( 1 + i ) n − 1 i ( 1 + i ) n {\displaystyle P=A\cdot {\frac {(1+i)^{n}-1}{i(1+i)^{n}}}}
Future value (F) Gradient payment (G) F = G ⋅ ( 1 + i ) n − i n − 1 i 2 {\displaystyle F=G\cdot {\frac {(1+i)^{n}-in-1}{i^{2}}}}
Present value (P) Gradient payment (G) P = G ⋅ ( 1 + i ) n − i n − 1 i 2 ( 1 + i ) n {\displaystyle P=G\cdot {\frac {(1+i)^{n}-in-1}{i^{2}(1+i)^{n}}}}
Fixed payment (A) Gradient payment (G) A = G ⋅ [ 1 i − n ( 1 + i ) n − 1 ] {\displaystyle A=G\cdot \left[{\frac {1}{i}}-{\frac {n}{(1+i)^{n}-1}}\right]}
Future value (F) Exponentially increasing payment (D)
Increasing percentage (g)
F = D ⋅ ( 1 + g ) n − ( 1 + i ) n g − i {\displaystyle F=D\cdot {\frac {(1+g)^{n}-(1+i)^{n}}{g-i}}} (for i ≠ g)
F = D ⋅ n ( 1 + i ) n 1 + g {\displaystyle F=D\cdot {\frac {n(1+i)^{n}}{1+g}}} (for i = g)
Present value (P) Exponentially increasing payment (D)
Increasing percentage (g)
P = D ⋅ ( 1 + g 1 + i ) n − 1 g − i {\displaystyle P=D\cdot {\frac {\left({1+g \over 1+i}\right)^{n}-1}{g-i}}} (for i ≠ g)
P = D ⋅ n 1 + g {\displaystyle P=D\cdot {\frac {n}{1+g}}} (for i = g)
Notes:
A is a fixed payment amount, every period
G is a steadily increasing payment amount, that starts at G and increases by G for each subsequent period.
D is an exponentially or geometrically increasing payment amount, that starts at D and increases by a factor of (1+g) each subsequent period.
Derivations[edit]
Annuity derivation[edit]
The formula for the present value of a regular stream of future payments (an annuity) is derived from a sum of the formula for future value of a single future payment, as below, where C is the payment amount and n the period.
A single payment C at future time m has the following future value at future time n:
F V = C ( 1 + i ) n − m {\displaystyle FV\ =C(1+i)^{n-m}}
Summing over all payments from time 1 to time n, then reversing t
F V A = ∑ m = 1 n C ( 1 + i ) n − m = ∑ k = 0 n − 1 C ( 1 + i ) k {\displaystyle FVA\ =\sum _{m=1}^{n}C(1+i)^{n-m}\ =\sum _{k=0}^{n-1}C(1+i)^{k}}
Note that this is a geometric series, with the initial value being a = C, the multiplicative factor being 1 + i, with n terms. Applying the formula for geometric series, we get
F V A = C ( 1 − ( 1 + i ) n ) 1 − ( 1 + i ) = C ( 1 − ( 1 + i ) n ) − i {\displaystyle FVA\ ={\frac {C(1-(1+i)^{n})}{1-(1+i)}}\ ={\frac {C(1-(1+i)^{n})}{-i}}}
The present value of the annuity (PVA) is obtained by simply dividing by ( 1 + i ) n {\displaystyle (1+i)^{n}} :
P V A = F V A ( 1 + i ) n = C i ( 1 − 1 ( 1 + i ) n ) {\displaystyle PVA\ ={\frac {FVA}{(1+i)^{n}}}={\frac {C}{i}}\left(1-{\frac {1}{(1+i)^{n}}}\right)}
Another simple and intuitive way to derive the future value of an annuity is to consider an endowment, whose interest is paid as the annuity, and whose principal remains constant. The principal of this hypothetical endowment can be computed as that whose interest equals the annuity payment amount:
Principal × i = C {\displaystyle {\text{Principal}}\times i=C}
Principal = C / i + g o a l {\displaystyle {\text{Principal}}=C/i+{goal}}
Note that no money enters or leaves the combined system of endowment principal + accumulated annuity payments, and thus the future value of this system can be computed simply via the future value formula:
F V = P V ( 1 + i ) n {\displaystyle FV=PV(1+i)^{n}}
Initially, before any payments, the present value of the system is just the endowment principal ( P V = C / i {\displaystyle PV=C/i} ). At the end, the future value is the endowment principal (which is the same) plus the future value of the total annuity payments ( F V = C / i + F V A {\displaystyle FV=C/i+FVA} ). Plugging this back into the equation:
C i + F V A = C i ( 1 + i ) n {\displaystyle {\frac {C}{i}}+FVA={\frac {C}{i}}(1+i)^{n}}
F V A = C i [ ( 1 + i ) n − 1 ] {\displaystyle FVA={\frac {C}{i}}\left[\left(1+i\right)^{n}-1\right]}
Perpetuity derivation[edit]
Without showing the formal derivation here, the perpetuity formula is derived from the annuity formula. Specifically, the term:
( 1 − 1 ( 1 + i ) n ) {\displaystyle \left({1-{1 \over {(1+i)^{n}}}}\right)}
can be seen to approach the value of 1 as n grows larger. At infinity, it is equal to 1, leaving C i {\displaystyle {C \over i}} as the only term remaining.
Examples[edit]
Example 1: Present value[edit]
One hundred euros to be paid 1 year from now, where the expected rate of return is 5% per year, is worth in today's money:
P = F × ( P / F ) = F × 1 ( 1 + i ) n = 100 1.05 = 95.24 {\displaystyle P=F\times (P/F)=F\times {1 \over (1+i)^{n}}={\frac {100}{1.05}}=95.24}
So the present value of €100 one year from now at 5% is €95.24.
Example 2: Present value of an annuity—solving for the payment amount[edit]
Consider a 10-year mortgage where the principal amount P is $200,000 and the annual interest rate is 6%.
The number of monthly payments is
n = 10 y e a r s × 12 m o n t h s p e r y e a r = 120 m o n t h s {\displaystyle n=10{\rm {\ years}}\times 12{\rm {\ months\ per\ year}}=120{\rm {\ months}}}
and the monthly interest rate is
i = 6 % p e r y e a r 12 m o n t h s p e r y e a r = 0.5 % p e r m o n t h {\displaystyle i={6{\rm {\%\ per\ year}} \over 12{\rm {\ months\ per\ year}}}=0.5{\rm {\%\ per\ month}}}
The annuity formula for (A/P) calculates the monthly payment:
A = P × ( A / P ) = P × i ( 1 + i ) n ( 1 + i ) n − 1 = $ 200 , 000 × 0.005 ( 1.005 ) 120 ( 1.005 ) 120 − 1 {\displaystyle A\ =\ P\times \left(A/P\right)\ =\ P\times {i(1+i)^{n} \over (1+i)^{n}-1}\ =\ \$200,000\times {0.005(1.005)^{120} \over (1.005)^{120}-1}}
≈ $ 200 , 000 × 0.01110205 ≈ $ 2 , 220.41 p e r m o n t h {\displaystyle \approx \$200,000\times 0.01110205\ \approx \ \$2,220.41{\rm {\ per\ month}}}
This is considering an interest rate compounding monthly. If the interest were only to compound yearly at 6%, the monthly payment would be significantly different.
An approximate solution[edit]
For those who only want a rough idea of the mortgage payment there is a much less intimidating approximate formula here. For the numbers given above we simply compute an approximate annual repayment of 200,000*(1/n + (2/3)*i) where n=10 yrs, i=0.06. So 200,000*(1/10 + (2/3)*0.06) = 200,000*(0.1+0.04) = 200,000*0.14 = $28,000 per year, roughly, via mental arithmetic alone. Note, as this is an approximation we may ignore the subtleties of monthly compounding. Now $28,000 per year is about 28,000/12 = $2,333 per month, which approximates the true answer to within about 5% but has required only mental arithmetic.
Example 3: Solving for the period needed to double money[edit]
Consider a deposit of £100 placed at 10% (annual). How many years are needed for the value of the deposit to double to £200?
Using the algrebraic identity that if:
x = b y {\displaystyle x\ =\ b^{y}}
then
y = log  ( x ) log  ( b ) {\displaystyle y\ =\ {\log(x) \over \log(b)}}
The present value formula can be rearranged such that:
y = log  ( F V P V ) log  ( 1 + i ) = log  ( 200 100 ) log  ( 1.10 ) = 7.27 ( y e a r s ) {\displaystyle y\ =\ {\log({FV \over PV}) \over \log(1+i)}\ =\ {\log({200 \over 100}) \over \log(1.10)}\ =\ 7.27{(years)}}
This same method can be used to determine the length of time needed to increase a deposit to any particular sum, as long as the interest rate is known. For the period of time needed to double an investment, the Rule of 72 is a useful short-cut that gives a reasonable approximation of the period needed.
Example 4: What return is needed to double money?[edit]
Similarly, the present value formula can be rearranged to determine what rate of return is needed to accumulate a given amount from an investment. For example, £100 is invested today and £200 return is expected in five years; what rate of return (interest rate) does this represent?
The present value formula restated in terms of the interest rate is:
i = ( F V P V ) 1 n − 1 = ( 200 100 ) 1 5 − 1 = 2 0.20 − 1 = 0.15 = 15 % {\displaystyle i\ =\ \left({FV \over PV}\right)^{1 \over n}-1\ =\ \left({200 \over 100}\right)^{1 \over 5}-1\ =\ 2^{0.20}-1\ =\ 0.15\ =\ 15\%}
see also Rule of 72
Example 5: Calculate the value of a regular savings deposit in the future.[edit]
To calculate the future value of a stream of savings deposit in the future requires two steps, or, alternatively, combining the two steps into one large formula. First, calculate the present value of a stream of deposits of $1,000 every year for 20 years earning 7% interest:
P V A = A ⋅ 1 − 1 ( 1 + i ) n i = 1000 ⋅ 1 − 1 ( 1 + .07 ) 20 .07 = 1000 ⋅ 1 − 0.258 .07 = 1000 × 10.594 ≈ $ 10 , 594 {\displaystyle PVA\,=\,A\cdot {\frac {1-{\frac {1}{\left(1+i\right)^{n}}}}{i}}\ =\ 1000\cdot {\frac {1-{\frac {1}{\left(1+.07\right)^{20}}}}{.07}}\ =\ 1000\cdot {1-0.258 \over .07}\ =\ 1000\times 10.594\ \approx \ \$10,594}
This does not sound like very much, but remember - this is future money discounted back to its value today; it is understandably lower. To calculate the future value (at the end of the twenty-year period):
F V = P V ( 1 + i ) n = $ 10 , 594 × ( 1 + .07 ) 20 ≈ $ 10 , 594 × 3.87 = $ 40 , 995 {\displaystyle FV\ =\ PV(1+i)^{n}\ =\$10,594\times (1+.07)^{20}\ \approx \$10,594\times 3.87\ =\$40,995}
These steps can be combined into a single formula:
F V = A ⋅ 1 − 1 ( 1 + i ) n i ⋅ ( 1 + i ) n = A ⋅ ( 1 + i ) n − 1 i {\displaystyle FV\,=\,A\cdot {\frac {1-{\frac {1}{\left(1+i\right)^{n}}}}{i}}\cdot (1+i)^{n}\,=\,A\cdot {\frac {\left(1+i\right)^{n}-1}{i}}}
Example 6: Price/earnings (P/E) ratio[edit]
It is often mentioned that perpetuities, or securities with an indefinitely long maturity, are rare or unrealistic, and particularly those with a growing payment. In fact, many types of assets have characteristics that are similar to perpetuities. Examples might include income-oriented real estate, preferred shares, and even most forms of publicly traded stocks. Frequently, the terminology may be slightly different, but are based on the fundamentals of time value of money calculations. The application of this methodology is subject to various qualifications or modifications, such as the Gordon growth model.
For example, stocks are commonly noted as trading at a certain P/E ratio. The P/E ratio is easily recognized as a variation on the perpetuity or growing perpetuity formulae - save that the P/E ratio is usually cited as the inverse of the "rate" in the perpetuity formula.
If we substitute for the time being: the price of the stock for the present value; the earnings per share of the stock for the cash annuity; and, the discount rate of the stock for the interest rate, we can see that:
P E = 1 i = P V A {\displaystyle {P \over E}\ =\ {1 \over i}\ =\ {PV \over A}}
And in fact, the P/E ratio is analogous to the inverse of the interest rate (or discount rate).
1 P / E = i {\displaystyle {1 \over P/E}\ =\ i}
Of course, stocks may have increasing earnings. The formulation above does not allow for growth in earnings, but to incorporate growth, the formula can be restated as follows:
P E = 1 ( i − g ) {\displaystyle {P \over E}\ =\ {1 \over (i-g)}}
If we wish to determine the implied rate of growth (if we are given the discount rate), we may solve for g:
g = i − E P {\displaystyle g\ =\ i-{E \over P}}
Continuous compounding[edit]
Rates are sometimes converted into the continuous compound interest rate equivalent because the continuous equivalent is more convenient (for example, more easily differentiated). Each of the formulæ above may be restated in their continuous equivalents. For example, the present value at time 0 of a future payment at time t can be restated in the following way, where e is the base of the natural logarithm and r is the continuously compounded rate:
PV = FV ⋅ e − r t {\displaystyle {\text{PV}}={\text{FV}}\cdot e^{-rt}}
This can be generalized to discount rates that vary over time: instead of a constant discount rate r, one uses a function of time r(t). In that case the discount factor, and thus the present value, of a cash flow at time T is given by the integral of the continuously compounded rate r(t):
PV = FV ⋅ exp  ( − ∫ 0 T r ( t ) d t ) {\displaystyle {\text{PV}}={\text{FV}}\cdot \exp \left(-\int _{0}^{T}r(t)\,dt\right)}
Indeed, a key reason for using continuous compounding is to simplify the analysis of varying discount rates and to allow one to use the tools of calculus. Further, for interest accrued and capitalized overnight (hence compounded daily), continuous compounding is a close approximation for the actual daily compounding. More sophisticated analysis includes the use of differential equations, as detailed below.
Examples[edit]
Using continuous compounding yields the following formulas for various instruments:
Annuity
P V = A ( 1 − e − r t ) e r − 1 {\displaystyle \ PV\ =\ {A(1-e^{-rt}) \over e^{r}-1}}
Perpetuity
P V = A e r − 1 {\displaystyle \ PV\ =\ {A \over e^{r}-1}}
Growing annuity
P V = A e − g ( 1 − e − ( r − g ) t ) ( r − g ) − 1 {\displaystyle \ PV\ =\ {Ae^{-g}(1-e^{-(r-g)t}) \over ^{(r-g)}-1}}
Growing perpetuity
P V = A e − g e ( r − g ) − 1 {\displaystyle \ PV\ =\ {Ae^{-g} \over e^{(r-g)}-1}}
Annuity with continuous payments
P V = 1 − e ( − r t ) r {\displaystyle \ PV\ =\ {1-e^{(-rt)} \over r}}
These formulas assume that payment A is made in the first payment period and annuity ends at time t.[10]
Differential equations[edit]
Ordinary and partial differential equations (ODEs and PDEs) – equations involving derivatives and one (respectively, multiple) variables are ubiquitous in more advanced treatments of financial mathematics. While time value of money can be understood without using the framework of differential equations, the added sophistication sheds additional light on time value, and provides a simple introduction before considering more complicated and less familiar situations. This exposition follows (Carr & Flesaker 2006, pp. 6–7).
The fundamental change that the differential equation perspective brings is that, rather than computing a number (the present value now), one computes a function (the present value now or at any point in future). This function may then be analyzed—how does its value change over time—or compared with other functions.
Formally, the statement that "value decreases over time" is given by defining the linear differential operator L {\displaystyle {\mathcal {L}}} as:
L := − ∂ t + r ( t ) . {\displaystyle {\mathcal {L}}:=-\partial _{t}+r(t).}
This states that values decreases (−) over time (∂t) at the discount rate (r(t)). Applied to a function it yields:
L f = − ∂ t f ( t ) + r ( t ) f ( t ) . {\displaystyle {\mathcal {L}}f=-\partial _{t}f(t)+r(t)f(t).}
For an instrument whose payment stream is described by f(t), the value V(t) satisfies the inhomogeneous first-order ODE L V = f {\displaystyle {\mathcal {L}}V=f} ("inhomogeneous" is because one has f rather than 0, and "first-order" is because one has first derivatives but no higher derivatives) – this encodes the fact that when any cash flow occurs, the value of the instrument changes by the value of the cash flow (if you receive a £10 coupon, the remaining value decreases by exactly £10).
The standard technique tool in the analysis of ODEs is Green's functions, from which other solutions can be built. In terms of time value of money, the Green's function (for the time value ODE) is the value of a bond paying £1 at a single point in time u – the value of any other stream of cash flows can then be obtained by taking combinations of this basic cash flow. In mathematical terms, this instantaneous cash flow is modeled as a Dirac delta function δ u ( t ) := δ ( t − u ) . {\displaystyle \delta _{u}(t):=\delta (t-u).}
The Green's function for the value at time t of a £1 cash flow at time u is
b ( t ; u ) := H ( u − t ) ⋅ exp  ( − ∫ t u r ( v ) d v ) {\displaystyle b(t;u):=H(u-t)\cdot \exp \left(-\int _{t}^{u}r(v)\,dv\right)}
where H is the Heaviside step function – the notation " ; u {\displaystyle ;u} " is to emphasize that u is a parameter (fixed in any instance—the time when the cash flow will occur), while t is a variable (time). In other words, future cash flows are exponentially discounted (exp) by the sum (integral, ∫ {\displaystyle \textstyle {\int }} ) of the future discount rates ( ∫ t u {\displaystyle \textstyle {\int _{t}^{u}}} for future, r(v) for discount rates), while past cash flows are worth 0 ( H ( u − t ) = 1 if t < u , 0 if t > u {\displaystyle H(u-t)=1{\text{ if }}t<u,0{\text{ if }}t>u} ), because they have already occurred. Note that the value at the moment of a cash flow is not well-defined – there is a discontinuity at that point, and one can use a convention (assume cash flows have already occurred, or not already occurred), or simply not define the value at that point.
In case the discount rate is constant, r ( v ) ≡ r , {\displaystyle r(v)\equiv r,} this simplifies to
b ( t ; u ) = H ( u − t ) ⋅ e − ( u − t ) r = { e − ( u − t ) r t < u 0 t > u , {\displaystyle b(t;u)=H(u-t)\cdot e^{-(u-t)r}={\begin{cases}e^{-(u-t)r}&t<u\\0&t>u,\end{cases}}}
where ( u − t ) {\displaystyle (u-t)} is "time remaining until cash flow".
Thus for a stream of cash flows f(u) ending by time T (which can be set to T = + ∞ {\displaystyle T=+\infty } for no time horizon) the value at time t, V ( t ; T ) {\displaystyle V(t;T)} is given by combining the values of these individual cash flows:
V ( t ; T ) = ∫ t T f ( u ) b ( t ; u ) d u . {\displaystyle V(t;T)=\int _{t}^{T}f(u)b(t;u)\,du.}
This formalizes time value of money to future values of cash flows with varying discount rates, and is the basis of many formulas in financial mathematics, such as the Black–Scholes formula with varying interest rates.
See also[edit]
Actuarial science
Discounted cash flow
Earnings growth
Exponential growth
Finance
Hyperbolic discounting
Internal rate of return
Net present value
Option time value
Real versus nominal value (economics)
Notes[edit]
Jump up ^ "Makkot 3a William Davidson Talmud online".
Jump up ^ Carther, Shauna (3 December 2003). "Understanding the Time Value of Money".
Jump up ^ Staff, Investopedia (25 November 2003). "Present Value - PV".
Jump up ^ "Present Value of an Annuity".
Jump up ^ Staff, Investopedia (24 November 2003). "Perpetuity".
Jump up ^ Staff, Investopedia (23 November 2003). "Future Value - FV".
Jump up ^ Hovey, M. (2005). Spreadsheet Modelling for Finance. Frenchs Forest, N.S.W.: Pearson Education Australia.
Jump up ^ http://mathworld.wolfram.com/GeometricSeries.html Geometric Series
Jump up ^ "NCEES FE exam".
Jump up ^ "Annuities and perpetuities with continuous compounding".
References[edit]
Carr, Peter; Flesaker, Bjorn (2006), Robust Replication of Default Contingent Claims (presentation slides) (PDF), Bloomberg LP. See also Audio Presentation and paper.
Crosson, S.V., and Needles, B.E.(2008). Managerial Accounting (8th Ed). Boston: Houghton Mifflin Company.
External links[edit]
Time Value of Money hosted by the University of Arizona
Time Value of Money ebook
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values: {\displaystyle FV\ =\ PV\cdot (1+i)^{n}}

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values: Edit section: Present value of a future sum

values: {\displaystyle PV\ =\ {\frac {FV}{(1+i)^{n}}}}

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values: Edit section: Present value of an annuity for n payment periods

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values: PV(A) \,=\,\frac{A}{i} \cdot \left[ {1-\frac{1}{\left(1+i\right)^n}} \right]

values: Edit section: Present value of a growing annuity

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values: PV\,=\,{A \over (i-g)}\left[ 1- \left({1+g \over 1+i}\right)^n \right]

values: {\displaystyle PV\,=\,{A\times n \over 1+i}}

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values: {\displaystyle A=F\cdot {\frac {i}{(1+i)^{n}-1}}}

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values: {\displaystyle F=A\cdot {\frac {(1+i)^{n}-1}{i}}}

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values: {\displaystyle F=G\cdot {\frac {(1+i)^{n}-in-1}{i^{2}}}}

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values: {\displaystyle F=D\cdot {\frac {n(1+i)^{n}}{1+g}}}

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values: P=D\cdot \frac{\left({1+g \over 1+i}\right)^n-1}{g-i}

values: {\displaystyle P=D\cdot {\frac {n}{1+g}}}

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values: FVA = \frac{C}{i} \left[ \left(1+i \right)^n - 1 \right]

values: {\displaystyle \left({1-{1 \over {(1+i)^{n}}}}\right)}

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values: Edit section: Example 2: Present value of an annuity—solving for the payment amount

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values: n = 10 {\rm \ years} \times 12 {\rm \ months \ per \ year} = 120 {\rm \ months}

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values: i = { 6 {\rm \% \ per \ year} \over 12 {\rm \ months \ per \ year} } =  0.5 {\rm \% \ per \ month}

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values: Edit section: Example 3: Solving for the period needed to double money

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values: {\displaystyle y\ =\ {\log(x) \over \log(b)}}

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values: Edit section: Example 4: What return is needed to double money?

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values: FV \,=\,A\cdot\frac{1-\frac{1}{\left(1+i\right)^n}}{i} \cdot (1+i)^n \,=\,A\cdot\frac{\left(1+i\right)^n-1}{i}

values: Edit section: Example 6: Price/earnings (P/E) ratio

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values: {P \over E}  \ = \ {1 \over i} \ = \ {PV \over A }

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values: {\displaystyle {P \over E}\ =\ {1 \over (i-g)}}

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values: \text{PV}  = \text{FV}\cdot \exp\left(-\int_0^T r(t)\,dt\right)

values: {\displaystyle \ PV\ =\ {A(1-e^{-rt}) \over e^{r}-1}}

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values: {\displaystyle \ PV\ =\ {A \over e^{r}-1}}

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values: {\displaystyle \ PV\ =\ {Ae^{-g} \over e^{(r-g)}-1}}

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values: {\displaystyle \ PV\ =\ {1-e^{(-rt)} \over r}}

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values: b(t;u) := H(u-t)\cdot \exp\left(-\int_t^u r(v)\,dv\right)

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values: {\displaystyle \textstyle {\int _{t}^{u}}}

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Chadwick Boseman
From Wikipedia, the free encyclopedia
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Chadwick Boseman
Boseman in 2017
Born
Chadwick Aaron Boseman
(1977-11-29) November 29, 1977 (age 40)
or 1976/1977 (age 40–41) (sources differ)
Anderson, South Carolina, U.S.
Alma mater
Howard University, B.F.A. 2000
British American Drama Academy
Occupation
Actor, director, producer
Years active
2000–present
Chadwick Aaron Boseman[1] (born November 29, 1977[2] or 1976)[3] (sources differ) is an American actor. He is known for portraying Jackie Robinson in 42 (2013), James Brown in Get on Up (2014), and Thurgood Marshall in Marshall (2017). He has also had roles in the television series Lincoln Heights (2008) and Persons Unknown (2010), and the films The Express (2008), Draft Day (2014), and Message from the King (2016).
He has played Black Panther in the Marvel Cinematic Universe films Captain America: Civil War (2016), Black Panther (2018), and Avengers: Infinity War (2018).
Contents
[hide]
1 Early life
2 Career
3 Personal life
4 Filmography
4.1 Film
4.2 Television
5 References
6 External links
Early life
Boseman was born and raised in Anderson, South Carolina,[2] the only child of Carolyn[4] and Leroy Boseman, both African American.[5] According to Boseman, DNA testing has indicated that his ancestors were Yoruba people from Nigeria and Limba people from Sierra Leone.[6] His mother was a nurse and his father worked at a textile factory, keeping an upholstery business as well.[7] Boseman graduated from T. L. Hanna High School in 1995.[8] In his junior year, he wrote his first play, Crossroads, and staged it at the school after a classmate was shot and killed.[7] He studied at Howard University in Washington, DC, graduating in 2000 with a bachelor of fine arts in directing.[9] One of his teachers was Phylicia Rashad, who became a mentor.[7] She helped raise funds so that Boseman and some classmates could attend the Oxford Mid-Summer Program of the British American Drama Academy in London, to which they'd been accepted.[7]
Boseman wanted to write and direct, and initially began studying acting to learn how to relate to actors.[10] After he returned to the US, he graduated from New York City's Digital Film Academy.[11] He lived in Brooklyn at the start of his career.[7] Boseman worked as the drama instructor in the Schomburg Junior Scholars Program, housed at the Schomburg Center for Research in Black Culture in Harlem, New York.[1] In 2008 he moved to Los Angeles to pursue an acting career.[12]
Career
Boseman got his first television role in 2003, in an episode of Third Watch. His early work included episodes of the series Law & Order, CSI:NY, and ER.[1] He also continued to write plays, with his script for Deep Azure performed at the Congo Square Theatre Company in Chicago, Illinois; it was nominated for a 2006 Joseph Jefferson Award for New Work.[13] In 2008, he played a recurring role on the television series Lincoln Heights and appeared in his first feature film, The Express.[14] He landed a regular role in 2010 in another television series, Persons Unknown.[14]
Boseman had his first starring role in the 2013 film 42, in which he portrayed baseball pioneer and star Jackie Robinson.[14] He had been directing an off-Broadway play in East Village when he auditioned for the role.[15] He had considered giving up acting and pursuing directing full-time at the time.[16] About 25 other actors had been seriously considered for the role, but director Brian Helgeland liked Boseman's bravery and cast him after he had auditioned twice.[17][9] In 2013, Boseman also starred in the indie film The Kill Hole, which was released in theaters a few weeks before 42.[18]
In 2014, Boseman appeared opposite Kevin Costner in Draft Day, in which he played an NFL draft prospect.[19] Later that year, he starred as James Brown in Get on Up. In 2016, he starred as Thoth, a deity from Egyptian mythology, in Gods of Egypt.[20]
Boseman at the Deauville Film Festival in September 2014.
He started playing the Marvel Comics character T'Challa / Black Panther in 2016, with Captain America: Civil War being his first film in a five-picture deal with Marvel.[21][22] He headlined Black Panther in 2018,[23] which focused on his character and his home country of Wakanda, in Africa. The film opened to great anticipation, becoming one of the highest-grossing films in the United States. He reprised the role in Avengers: Infinity War, which was released in April 2018, and also in the 2019 sequel.[24]
Personal life
Boseman was raised a Christian. He was baptized, and was part of a church choir and youth group. His former pastor said that he still keeps his faith.[25] Boseman has stated that he prayed to be the Black Panther before he was cast as the character in the Marvel Cinematic Universe.[26]
Filmography
Film
Year
Title
Role
Director
Notes
2008 The Express Floyd Little Gary Fleder
2013 The Kill Hole Lt. Samuel Drake Mischa Webley
42 Jackie Robinson Brian Helgeland
2014 Draft Day Vontae Mack Ivan Reitman
Get on Up James Brown Tate Taylor
2016 Gods of Egypt Thoth Alex Proyas
Captain America: Civil War T'Challa / Black Panther Anthony & Joe Russo Nominated – Saturn Award for Best Supporting Actor in a Film[27]
Message from the King Jacob King Fabrice Du Welz Also executive producer
2017 Marshall Thurgood Marshall Reginald Hudlin Also co-producer
2018 Black Panther T'Challa / Black Panther Ryan Coogler Nominated – Saturn Award for Best Actor in a Film
Avengers: Infinity War Anthony & Joe Russo
2019 Untitled Avengers film Post-production
Television
Year
Title
Role
Notes
2003 All My Children Reggie Porter Recurring role
2003 Third Watch David Wafer Episode: "In Lieu of Johnson"
2004 Law & Order Foster Keyes Episode: "Can I Get a Witness?"
2006 CSI: NY Rondo Episode: "Heroes"
2008 ER Derek Taylor Episode: "Oh, Brother"
2008 Cold Case Dexter Collins Episode: "Street Money"
2008–2009 Lincoln Heights Nathaniel "Nate" Ray 9 episodes
2009 Lie to Me Cabe McNeil Episode: "Truth or Consequences"
2010 Persons Unknown Sergeant McNair 13 episodes
2010 The Glades Michael Richmond Episode: "Honey"
2011 Castle Chuck Russell Episode: "Poof, You're Dead"
2011 Fringe Mark Little / Cameron James Episode: "Subject 9"
2011 Detroit 1-8-7 Tommy Westin Episode: "Beaten/Cover Letter"
2011 Justified Ralph Beeman Episode: "For Blood or Money"
2018 Saturday Night Live Himself Episode: "Chadwick Boseman/Cardi B"
References
^ Jump up to: a b c Holsey, Steve (August 6, 2014). "Star On The Rise: Chadwick Boseman". Michigan Chronicle. Detroit, Michigan. Archived from the original on August 4, 2015. Retrieved April 29, 2015.
^ Jump up to: a b "Chadwick Boseman Biography: Screenwriter, Actor, Director (1977–)". Biography.com (FYI / A&E Networks. Archived from the original on April 2, 2016. Retrieved May 9, 2016.
Jump up ^ His age is given as 37, making his birth year c. 1976, at Rothman, Michael (October 29, 2014). "5 Things to Know About Chadwick Boseman". ABC News. Retrieved January 11, 2018. and Feinberg, Scott (October 29, 2014). "Ocsar Contender and New Marvel Superhero Chadwick Boseman on his Journey to Stardom". The Hollywood Reporter. Retrieved January 12, 2018. Neither source gives specific birth date.
Jump up ^ Alexander, Bryan (January 31, 2018). "The best things you missed at the 'Black Panther' party (Usher dancing on the tables!)". USA Today. Archived from the original on March 6, 2018. Retrieved March 6, 2018.
Jump up ^ Barrow, Jerry L. (February 2018). "Father Stretch My Hands: 'Black Panther' & Raising Our Sons To Shine". BET. Archived from the original on February 15, 2018. Retrieved March 6, 2018.
Jump up ^ Colbert, Stephen (May 17, 2018). "Chadwick Boseman On Bringing Humanity To 'Black Panther'" (Interview). New York City: The Late Show with Stephen Colbert.
^ Jump up to: a b c d e Eells, Josh (February 18, 2018). "The 'Black Panther' Revolution". Rolling Stone. Archived from the original on February 25, 2018.
Jump up ^ "Hanna Grad Chad Boseman Plays Jackie Robinson in "42"" (PDF). Parent Newsletter (47). T. L. Hanna High School. April 11, 2013. Archived from the original (PDF) on March 4, 2016.
^ Jump up to: a b Jenkins, Mark (April 11, 2013). "Jackie Robinson film '42' opens, starring Howard graduate Chadwick Boseman". The Washington Post. Archived from the original on May 9, 2016. Retrieved April 29, 2015.
Jump up ^ McDonald, Soraya Nadia (October 29, 2014). "Everything you need to know about Chadwick Boseman — Marvel's new superhero, Black Panther". The Washington Post. Archived from the original on May 9, 2016. Retrieved April 29, 2015.
Jump up ^ Jain, Sara McDermott (March 18, 2014). "DFA Grad Chadwick Boseman to Play James Brown in Biopic". Digital Film Academy. Archived from the original on December 22, 2017. Retrieved December 22, 2017.
Jump up ^ Rose, Steve (November 20, 2014). "I got the feelin' – Chadwick Boseman on playing James Brown". The Guardian. UK. Archived from the original on June 12, 2017. Retrieved April 29, 2015.
Jump up ^ Wood, Jennifer (October 28, 2013). "10 Actors Who Have Gone From Supporting Role to Leading Man in 2013: Chadwick Boseman". Complex. Archived from the original on October 13, 2016. Retrieved April 29, 2015.
^ Jump up to: a b c Gregory, Sean (February 21, 2013). "Going Places With Chadwick Boseman". Time. Archived from the original on March 2, 2016. Retrieved April 29, 2015.
Jump up ^ Miller, Julie (April 12, 2013). "'42' Star Chadwick Boseman on Playing Jackie Robinson, Copying His Baseball Moves, and Being Stood Up by the President". Vanity Fair. Retrieved April 29, 2015.
Jump up ^ Riley, Daniel (September 2014). "The Surprisingly Sudden Arrival of Chadwick Boseman". GQ. Archived from the original on February 18, 2018. Retrieved April 29, 2015.
Jump up ^ Herndon, Jessica (April 8, 2013). "Chadwick Boseman: The Distinguished Rise of Cinema's Next Champion". Flaunt. Archived from the original on March 7, 2018. Retrieved April 29, 2015.
Jump up ^ Yamato, Jenn (February 14, 2013). "Indie Vendetta Pic With Parallels To Christopher Dorner Saga Gets Release Date". Deadline Hollywood. Retrieved April 29, 2015.
Jump up ^ Kaye, Don (April 11, 2014). "Chadwick Boseman on 'Draft Day,' Sports Legends, and Becoming James Brown". Moviefone. Retrieved April 29, 2015.
Jump up ^ Fleming, Mike, Jr. (January 30, 2014). "Chadwick Boseman Set For Summit's 'Gods Of Egypt'". Deadline Hollywood. Archived from the original on September 25, 2017. Retrieved April 29, 2015.
Jump up ^ Strom, Marc (October 8, 2015). "Marvel Studios Phase 3 Update". Marvel Entertainment. Archived from the original on January 16, 2018. Retrieved January 12, 2016.
Jump up ^ Strom, Marc (October 28, 2014). "Chadwick Boseman to Star in Marvel's Black Panther". Marvel Entertainment. Archived from the original on October 28, 2014. Retrieved October 28, 2014.
Jump up ^ Strom, Marc (February 10, 2015). "Marvel Studios Schedules New Release Dates for 4 Films". Marvel Entertainment. Archived from the original on February 10, 2015. Retrieved February 9, 2015.
Jump up ^ "Marvel's 'Black Panther' Will Set Up 'Avengers: Infinity War,' Says Kevin Feige". The Hollywood Reporter. Retrieved February 13, 2018.
Jump up ^ Remien, Shale. "Anderson native Chadwick Boseman stars as the Black Panther: Local pastor reacts". Greenville, South Carolina: WHNS. Archived from the original on February 2018. Retrieved February 16, 2018.
Jump up ^ Watson, Elliot. "Chadwick Boseman on Black Panther and the healing power of cinema". Hunger. London, UK: Hunger Publishing Ltd. Archived from the original on February 16, 2018. Retrieved February 16, 2018.
Jump up ^ McNary, Dave (March 2, 2017). "'Rogue One,' 'Walking Dead' Lead Saturn Awards Nominations". Variety. Retrieved March 3, 2017.
External links
Wikimedia Commons has media related to Chadwick Boseman.
Chadwick Boseman on Facebook
Chadwick Boseman on Twitter
Chadwick Boseman on Instagram
Chadwick Boseman on IMDb
Borrelli, Christopher (April 11, 2013). "Robinson actor swings for the fences". Chicago Tribune.
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Johnny Depp (2007)
Emma Watson (2017)
Chadwick Boseman (2018)
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