Time Value of Money
Future Value
\[ FV_t = PV(1+r)^t \tag{1} \]
Where:
- \(FV_t\): future value at time \(t\)
- \(PV\): present value at time \(0\)
- \(r\): discount rate per period
- \(t\): number of compounding periods
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## Future Value
$$
FV_t = PV(1+r)^t \tag{1}
$$
Where:
* $FV_t$: future value at time $t$
* $PV$: present value at time $0$
* $r$: discount rate per period
* $t$: number of compounding periods Future Value with Continuous Compounding
\[ FV_t = PVe^{r t} \tag{2} \]
Where:
- \(FV_t\): future value at time \(t\)
- \(PV\): present value at time \(0\)
- \(r\): discount rate per period
- \(t\): time in continuous periods
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## Future Value with Continuous Compounding
$$
FV_t = PVe^{r t} \tag{2}
$$
Where:
* $FV_t$: future value at time $t$
* $PV$: present value at time $0$
* $r$: discount rate per period
* $t$: time in continuous periods Present Value
\[ FV_t = PV(1+r)^t \]
\[ PV = FV_t \left[ \frac{1}{(1+r)^t} \right] \]
\[ PV = FV_t (1+r)^{-t} \tag{3} \]
Where:
- \(PV\): present value at time \(0\)
- \(FV_t\): future value at time \(t\)
- \(r\): discount rate per period
- \(t\): number of compounding periods
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## Present Value
$$
FV_t = PV(1+r)^t
$$
$$
PV = FV_t \left[ \frac{1}{(1+r)^t} \right]
$$
$$
PV = FV_t (1+r)^{-t} \tag{3}
$$
Where:
* $PV$: present value at time $0$
* $FV_t$: future value at time $t$
* $r$: discount rate per period
* $t$: number of compounding periods Present Value with Continuous Compounding
\[ PV_t = FV e^{-rt} \tag{4} \]
Where:
- \(PV_t\): present value at time \(t\)
- \(FV\): future value
- \(r\):
- \(t\):
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## Present Value with Continuous Compounding
$$
PV_t = FV e^{-rt} \tag{4}
$$
Where:
* $PV_t$: present value at time $t$
* $FV$: future value
* $r$:
* $t$: Present Value of a Discount (Zero-Coupon) Bond
\[ PV(Discount Bond) = \frac{FV_t}{(1+r)^t} \tag{5} \]
Where:
- \(PV\): present value of the bond
- \(FV_t\): principal (face value) paid at maturity
- \(r\): market discount rate per period
- \(t\): number of periods to maturity
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## Present Value of a Discount (Zero-Coupon) Bond
$$
PV(Discount Bond) = \frac{FV_t}{(1+r)^t} \tag{5}
$$
Where:
* $PV$: present value of the bond
* $FV_t$: principal (face value) paid at maturity
* $r$: market discount rate per period
* $t$: number of periods to maturity Present Value of a Coupon Bond
\[ \text{PV(Coupon Bond)} \]
\[ = \frac{PMT_1}{(1+r)^1} + \frac{PMT_2}{(1+r)^2} + \dots + \frac{PMT_N + FV_N}{(1+r)^N} \tag{6} \]
Where:
- \(PV\): present value of the bond
- \(PMT_N\):
- \(FV_N\): principal repaid at maturity
- \(r\): market discount rate per period
- \(N\): number of periods to maturity
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## Present Value of a Coupon Bond
$$
\text{PV(Coupon Bond)}
$$
$$
= \frac{PMT_1}{(1+r)^1} + \frac{PMT_2}{(1+r)^2} + \dots + \frac{PMT_N + FV_N}{(1+r)^N} \tag{6}
$$
Where:
* $PV$: present value of the bond
* $PMT_N$:
* $FV_N$: principal repaid at maturity
* $r$: market discount rate per period
* $N$: number of periods to maturity Present Value of a Perpetual Bond (Perpetuity)
\[ PV_{\text{Perpetual Bond}} = \frac{PMT}{r} \tag{7} \]
Where:
- \(PV\): present value of the perpetuity
- \(PMT\): fixed periodic payment
- \(r\): discount rate per period
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## Present Value of a Perpetual Bond (Perpetuity)
$$
PV_{\text{Perpetual Bond}} = \frac{PMT}{r} \tag{7}
$$
Where:
* $PV$: present value of the perpetuity
* $PMT$: fixed periodic payment
* $r$: discount rate per period Annuity Payment Formula
\[ A = \frac{r(PV)}{1-(1+r)^{-t}} \tag{8} \]
where:
- \(A\) = periodic cash flow
- \(r\) = market interest rate per period
- \(PV\) = present value or principal amount of loan or bond
- \(t\) = number of payment periods
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## Annuity Payment Formula
$$
A = \frac{r(PV)}{1-(1+r)^{-t}} \tag{8}
$$
where:
* $A$ = periodic cash flow
* $r$ = market interest rate per period
* $PV$ = present value or principal amount of loan or bond
* $t$ = number of payment periods Present Value of Stock with Constant Dividends (Infinite Series)
\[ PV_t = \sum_{i=1}^{\infty} \frac{D_t}{(1+r)^i} \tag{9} \]
Where:
- \(PV_t\): present value of the stock at time \(t\)
- \(D_t\): constant dividend per period
- \(r\):
- \(i\): dividend payment period index
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## Present Value of Stock with Constant Dividends (Infinite Series)
$$
PV_t = \sum_{i=1}^{\infty} \frac{D_t}{(1+r)^i} \tag{9}
$$
Where:
* $PV_t$: present value of the stock at time $t$
* $D_t$: constant dividend per period
* $r$:
* $i$: dividend payment period index Present Value of Stock with Constant Dividends (Perpetuity Form)
\[ PV_t = \frac{D_t}{r} \tag{10} \]
Where:
- \(PV_t\): present value of the stock at time \(t\)
- \(D_t\): constant dividend per period
- \(r\):
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## Present Value of Stock with Constant Dividends (Perpetuity Form)
$$
PV_t = \frac{D_t}{r} \tag{10}
$$
Where:
* $PV_t$: present value of the stock at time $t$
* $D_t$: constant dividend per period Dividend Growth Formula (One Period Ahead)
\[ D_{t+1} = D_t(1+g) \tag{11} \]
Where:
- \(D_{t+1}\): dividend at time \(t+1\)
- \(D_t\): dividend at time \(t\)
- \(g\): constant dividend growth rate
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## Dividend Growth Formula (One Period Ahead)
$$
D_{t+1} = D_t(1+g) \tag{11}
$$
Where:
* $D_{t+1}$: dividend at time $t+1$
* $D_t$: dividend at time $t$
* $g$: constant dividend growth rate Dividend Growth Formula (Multiple Periods)
\[ D_{t+i} = D_t(1+g)^i \tag{12} \]
Where:
- \(D_{t+i}\): dividend at time \(t+i\)
- \(D_t\): dividend at time \(t\)
- \(g\): constant dividend growth rate
- \(i\): number of periods after time \(t\)
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## Dividend Growth Formula (Multiple Periods)
$$
D_{t+i} = D_t(1+g)^i \tag{12}
$$
Where:
* $D_{t+i}$: dividend at time $t+i$
* $D_t$: dividend at time $t$
* $g$: constant dividend growth rate
* $i$: number of periods after time $t$ Present Value of Stock with Constant Dividend Growth (Infinite Series)
\[ PV_t = \sum_{i=1}^{\infty} \frac{D_t(1+g)^i}{(1+r)^i} \tag{13} \]
Where:
- \(PV_t\): present value of the stock at time \(t\)
- \(D_t\): dividend at time \(t\)
- \(g\): constant dividend growth rate
- \(r\):
- \(i\): dividend payment period index
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## Present Value of Stock with Constant Dividend Growth (Infinite Series)
$$
PV_t = \sum_{i=1}^{\infty} \frac{D_t(1+g)^i}{(1+r)^i} \tag{13}
$$
Where:
* $PV_t$: present value of the stock at time $t$
* $D_t$: dividend at time $t$
* $g$: constant dividend growth rate
* $r$:
* $i$: dividend payment period index Constant Growth Dividend Discount Model (Gordon Growth Model)
\[ PV_t = \frac{D_t(1+g)}{r-g} = \frac{D_{t+1}}{r-g} \tag{14} \]
Where:
- \(PV_t\): present value of the stock at time \(t\)
- \(D_t\): dividend at time \(t\)
- \(D_{t+1}\): dividend at time \(t+1\)
- \(r\):
- \(g\): constant dividend growth rate
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## Constant Growth Dividend Discount Model (Gordon Growth Model)
$$
PV_t = \frac{D_t(1+g)}{r-g} = \frac{D_{t+1}}{r-g} \tag{14}
$$
Where:
* $PV_t$: present value of the stock at time $t$
* $D_t$: dividend at time $t$
* $D_{t+1}$: dividend at time $t+1$
* $r$:
* $g$: constant dividend growth rate name? Two-Stage Dividend Discount Model (General Form)
\[ PV_t = \sum_{i=1}^{n} \frac{D_t(1+g_s)^i}{(1+r)^i} + \sum_{j=n+1}^{\infty} \frac{D_{t+n}(1+g_l)^j}{(1+r)^j} \tag{15} \]
Where:
- \(PV_t\): present value of the stock at time \(t\)
- \(D_t\): dividend at time \(t\)
- \(g_s\): initial higher short-term dividend growth rate
- \(g_l\): lower long-term dividend growth rate
- \(r\):
- \(n\): number of periods of short-term growth
- \(i\): dividend period index during short-term growth
- \(j\): dividend period index during long-term growth
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## name? Two-Stage Dividend Discount Model (General Form)
$$
PV_t = \sum_{i=1}^{n} \frac{D_t(1+g_s)^i}{(1+r)^i} + \sum_{j=n+1}^{\infty} \frac{D_{t+n}(1+g_l)^j}{(1+r)^j} \tag{15}
$$
Where:
* $PV_t$: present value of the stock at time $t$
* $D_t$: dividend at time $t$
* $g_s$: initial higher short-term dividend growth rate
* $g_l$: lower long-term dividend growth rate
* $r$:
* $n$: number of periods of short-term growth
* $i$: dividend period index during short-term growth
* $j$: dividend period index during long-term growth name? Two-Stage Dividend Discount Model (Terminal Value Form)
\[ PV_t = \sum_{i=1}^{n} \frac{D_t(1+g_s)^i}{(1+r)^i} + \frac{E(S_{t+n})}{(1+r)^n} \tag{16} \]
Where:
- \(PV_t\): present value of the stock at time \(t\)
- \(D_t\): dividend at time \(t\)
- \(g_s\): short-term dividend growth rate
- \(r\):
- \(n\): number of periods of short-term growth
- \(E(S_{t+n})\): stock value of the stock in \(n\) periods
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## name? Two-Stage Dividend Discount Model (Terminal Value Form)
$$
PV_t = \sum_{i=1}^{n} \frac{D_t(1+g_s)^i}{(1+r)^i} + \frac{E(S_{t+n})}{(1+r)^n} \tag{16}
$$
Where:
* $PV_t$: present value of the stock at time $t$
* $D_t$: dividend at time $t$
* $g_s$: short-term dividend growth rate
* $r$:
* $n$: number of periods of short-term growth
* $E(S_{t+n})$: stock value of the stock in $n$ periods Terminal Value
\[ E(S_{t+n}) = \frac{D_{t+n+1}}{r-g_l} \tag{17} \]
Where:
- \(E(S_{t+n})\): stock value of the stock in \(n\) periods
- \(D_{t+n+1}\): dividend at time \(t+n+1\)
- \(r\):
- \(g_l\): long-term dividend growth rate
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## Terminal Value
$$
E(S_{t+n}) = \frac{D_{t+n+1}}{r-g_l} \tag{17}
$$
Where:
* $E(S_{t+n})$: stock value of the stock in $n$ periods
* $D_{t+n+1}$: dividend at time $t+n+1$
* $r$:
* $g_l$: long-term dividend growth rate implied periodic return earned over the life of the instrument (\(t\) periods)
\[ r = \sqrt[t]{\frac{FV_t}{PV}} - 1 = \left( \frac{FV_t}{PV} \right)^{\frac{1}{t}} - 1 \tag{18} \]
Where:
- \(r\): implied periodic return earned over the life of the instrument (\(t\) periods)
- \(FV_t\): future value at time \(t\)
- \(PV\): present value at time \(0\)
- \(t\): number of periods
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## implied periodic return earned over the life of the instrument ($t$ periods)
$$
r = \sqrt[t]{\frac{FV_t}{PV}} - 1 = \left( \frac{FV_t}{PV} \right)^{\frac{1}{t}} - 1 \tag{18}
$$
Where:
* $r$: implied periodic return earned over the life of the instrument ($t$ periods)
* $FV_t$: future value at time $t$
* $PV$: present value at time $0$
* $t$: number of periods Yield-to-Maturity Equation for a Coupon Bond
\[ PV(Coupon\ Bond) = \frac{PMT_1}{(1+r)^1} + \frac{PMT_2}{(1+r)^2} + \dots + \frac{PMT_N + FV_N}{(1+r)^N} \tag{19} \]
Where:
- \(PV\): present value (price) of the bond
- \(PMT_N\): coupon payment at period \(N\)
- \(FV_N\): principal repaid at maturity
- \(r\): yield-to-maturity per period
- \(N\): number of periods to maturity
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## Yield-to-Maturity Equation for a Coupon Bond
$$
PV(Coupon\ Bond)
= \frac{PMT_1}{(1+r)^1} + \frac{PMT_2}{(1+r)^2} + \dots + \frac{PMT_N + FV_N}{(1+r)^N} \tag{19}
$$
Where:
* $PV$: present value (price) of the bond
* $PMT_N$: coupon payment at period $N$
* $FV_N$: principal repaid at maturity
* $r$: yield-to-maturity per period
* $N$: number of periods to maturity Dividend Yield under Constant Growth
\[ r - g = \frac{D_t(1+g)}{PV_t} = \frac{D_{t+1}}{PV_t} \tag{20} \]
Where:
- \(r\): expected or required rate of return?
- \(g\): constant dividend growth rate
- \(D_{t+1}\): dividend expected at time \(t+1\)
- \(PV_t\): present value of the stock at time \(t\)
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## Dividend Yield under Constant Growth
$$
r - g = \frac{D_t(1+g)}{PV_t} = \frac{D_{t+1}}{PV_t} \tag{20}
$$
Where:
* $r$: expected or required rate of return?
* $g$: constant dividend growth rate
* $D_{t+1}$: dividend expected at time $t+1$
* $PV_t$: present value of the stock at time $t$ name? implied return on a stock given its expected dividend yield and growth
\[ r = \frac{D_t(1+g)}{PV_t} + g = \frac{D_{t+1}}{PV_t} + g \tag{21} \]
Where:
- \(r\): expected or required rate of return?
- \(D_{t+1}\): dividend expected at time \(t+1\)
- \(PV_t\): present value of the stock at time \(t\)
- \(g\): constant dividend growth rate
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## name? implied return on a stock given its expected dividend yield and growth
$$
r = \frac{D_t(1+g)}{PV_t} + g = \frac{D_{t+1}}{PV_t} + g \tag{21}
$$
Where:
* $r$: expected or required rate of return?
* $D_{t+1}$: dividend expected at time $t+1$
* $PV_t$: present value of the stock at time $t$
* $g$: constant dividend growth rate Stock’s Implied Dividend Growth Rate
\[ g = \frac{r * PV_t - D_t}{PV_t + D_t} = r - \frac{D_{t+1}}{PV_t} \tag{22} \]
Where:
- \(g\): implied constant dividend growth rate
- \(r\): expected or required rate of return?
- \(D_{t+1}\): dividend expected at time \(t+1\)
- \(PV_t\): present value of the stock at time \(t\)
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## Stock's Implied Dividend Growth Rate
$$
g = \frac{r * PV_t - D_t}{PV_t + D_t} = r - \frac{D_{t+1}}{PV_t} \tag{22}
$$
Where:
* $g$: implied constant dividend growth rate
* $r$: expected or required rate of return?
* $D_{t+1}$: dividend expected at time $t+1$
* $PV_t$: present value of the stock at time $t$ name? Price-to-Earnings Ratio
\[ \frac{PV_t}{E_t} = \frac{ \frac{D_t}{E_t} \times (1+g) }{r-g} \tag{23} \]
Where:
- \(PV_t\): price of the stock at time \(t\)
- \(E_t\): earnings per share at time \(t\)
- \(D_t\): dividend per share at time \(t\)
- \(r\): expected or required rate of return?
- \(g\): constant dividend growth rate
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## name? Price-to-Earnings Ratio
$$
\frac{PV_t}{E_t} = \frac{ \frac{D_t}{E_t} \times (1+g) }{r-g} \tag{23}
$$
Where:
* $PV_t$: price of the stock at time $t$
* $E_t$: earnings per share at time $t$
* $D_t$: dividend per share at time $t$
* $r$: expected or required rate of return?
* $g$: constant dividend growth rate Forward Price-to-Earnings Ratio
\[ \frac{PV_t}{E_{t+1}} = \frac{ \frac{D_{t+1}}{E_{t+1}} }{r - g}\tag{24} \]
Where:
- \(PV_t\): price of the stock at time \(t\)
- \(E_{t+1}\): expected earnings per share at time \(t+1\)
- \(D_{t+1}\): expected dividend per share at time \(t+1\)
- \(r\): required rate of return
- \(g\): constant dividend growth rate
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## Forward Price-to-Earnings Ratio
$$
\frac{PV_t}{E_{t+1}} = \frac{ \frac{D_{t+1}}{E_{t+1}} }{r - g}\tag{24}
$$
Where:
* $PV_t$: price of the stock at time $t$
* $E_{t+1}$: expected earnings per share at time $t+1$
* $D_{t+1}$: expected dividend per share at time $t+1$
* $r$: required rate of return
* $g$: constant dividend growth rate Cash Flow Additivity and Implied Forward Rate Relationship
\[ FV_2 = PV_0 \times (1+r_2)^2 = PV_0 \times (1+r_1)(1+F_{1,1}) \tag{25} \]
Where:
- \(FV_2\): future value in two years
- \(PV_0\): present value at time 0
- \(r_1\): one-year bond rate
- \(r_2\): two-year bond rate
- \(F_{1,1}\): the one year forward rate starting in one year
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## Cash Flow Additivity and Implied Forward Rate Relationship
$$
FV_2 = PV_0 \times (1+r_2)^2 = PV_0 \times (1+r_1)(1+F_{1,1}) \tag{25}
$$
Where:
* $FV_2$: future value in two years
* $PV_0$: present value at time 0
* $r_1$: one-year bond rate
* $r_2$: two-year bond rate
* $F_{1,1}$: the one year forward rate starting in one year To fix
confused the \(r\) definition for a few formulas wihtout realizing so i question the definition or left it empty
Some formula names need to be confirmed