Formulas: Rates and Returns

Module: M01

Required Rate of Return

r = r_{f} + Default RP + Liquidity RP + Maturity RP

r_{f} = Real risk-free rate + Inflation premium

H P R = \frac{P _{1} - P _{0} + I _{1}}{P _{0}}

H P R = (1 + R_{1}) (1 + R_{2}) \dots (1 + R_{n}) - 1

\overset{ˉ}{R} = \frac{R _{1} + R _{2} + \dots + R _{n}}{n}

\overset{ˉ}{R}_{G} = n (1 + R_{1}) (1 + R_{2}) \dots (1 + R_{n}) - 1

Also: $\overset{ˉ}{R}_{G} = n 1 + H P R - 1$

\overset{ˉ}{X}_{H} = \frac{n}{\sum _{i = 1}^{n} \frac{1}{X _{i}}}

\overset{ˉ}{X}_{H} = \frac{n}{\sum _{i = 1}^{n} \frac{1}{1 + R _{i}}} - 1

\overset{ˉ}{R}_{H} \leq \overset{ˉ}{R}_{G} \leq \overset{ˉ}{R}

Equality holds only when all $R_{i}$ are identical.

Solve for $I R R$ in:

t = 0 \sum n \frac{C F _{t}}{( 1 + I R R ) ^{t}} = 0

T W R = (1 + H P R_{1}) (1 + H P R_{2}) \dots (1 + H P R_{n}) - 1

For periods > 1 year, annualize using geometric mean.

R_{net} = R_{gross} - Fees

R_{after-tax} = R_{pretax} \times (1 - tax rate)

R_{real} = \frac{1 + R _{nominal}}{1 + Inflation} - 1

R_{L} = R_{p} + \frac{V _{B}}{V _{E}} (R_{p} - r_{D})

where $V_{B}$ = borrowed capital, $V_{E}$ = equity capital, $r_{D}$ = borrowing cost