Formulas
Every calculation in the course, typeset on one page. Each group links back to the module that explains it.
Contained metal
\[ M = T \times g \qquad \text{oz} = \frac{M_{\text{grams}}}{31.1035} \]
\(T\) is tonnage and \(g\) the grade. The starting point of every valuation.
Recovered and payable metal
\[ M_{\text{rec}} = M \times R \qquad M_{\text{pay}} = M_{\text{rec}} \times p \]
\(R\) is metallurgical recovery and \(p\) the payable share in a concentrate. Only payable metal earns revenue.
Unit conversions
\[ 1\ \text{g/t} = 1\ \text{ppm} \qquad 1\% = 10\ \text{kg/t} = 10{,}000\ \text{ppm} \qquad 1\ \text{t} = 2{,}204.62\ \text{lb} \]
Metal equivalent grade
\[ g_{\text{eq}} = g_1 + \sum_{i \ge 2} g_i \, \frac{P_i \, R_i}{P_1 \, R_1} \]
Converts by-product metals into the main metal at assumed prices \(P\) and recoveries \(R\), with grades in consistent units.
Strip ratio
\[ SR = \frac{\text{waste mined (t)}}{\text{ore mined (t)}} \]
C1 cash cost
\[ C1 = \frac{C_{\text{mine}} + C_{\text{process}} + C_{\text{freight}} + \text{TC/RC} - \text{by-product credits}}{\text{payable lb}} \]
The reference cost measure for copper.
All-in sustaining cost
\[ \text{AISC} = \frac{C_{\text{op}} + \text{royalties} + \text{sustaining capex} + \text{G\&A} + \text{sustaining exploration}}{\text{oz sold}} \]
The World Gold Council standard since 2013.
Operating leverage
\[ \frac{\Delta m}{m} = \frac{\Delta P}{P} \times \frac{P}{P - \text{AISC}} \]
A thin margin \(m\) amplifies every move in the price \(P\).
Net present value
\[ \text{NPV} = \sum_{t=1}^{N} \frac{CF_t}{(1+r)^t} \]
In real terms, prices, costs and the rate \(r\) all exclude inflation.
Internal rate of return
\[ \sum_{t=0}^{N} \frac{CF_t}{(1+\text{IRR})^t} = 0 \]
Real rate
\[ r_{\text{real}} = \frac{1 + r_{\text{nominal}}}{1 + \pi} - 1 \]
The Fisher relation, with \(\pi\) as inflation.
Net asset value
\[ \text{NAV} = \sum_{j} \text{NPV}_j + \text{cash} - \text{debt} - \text{PV}(\text{corporate G\&A}) + \text{other assets} \]
P/NAV and enterprise value
\[ \text{P/NAV} = \frac{\text{market cap}}{\text{NAV}} \qquad \text{EV} = \text{market cap} + \text{net debt} + \text{minorities} \]
EV per ounce divides EV by ounces in resource or reserve.
Debt service coverage
\[ \text{DSCR}_t = \frac{\text{CFADS}_t}{\text{interest}_t + \text{principal}_t} \]
The core covenant of a project loan.
NSR royalty
\[ \text{Royalty}_t = \rho \times \left( \text{revenue}_t - \text{smelting}_t - \text{freight}_t \right) \]
Implied cost of a stream
\[ D = \sum_{t=1}^{N} \frac{q_t \, S_t \, (1-k)}{(1+i)^t} \]
Solve for \(i\): \(D\) is the deposit, \(q_t\) the ounces delivered, \(S_t\) the spot price and \(k\) the share of spot the streamer pays.
Offers and accretion
Explained in module 9
Premium
\[ \text{Premium} = \frac{P_{\text{offer}}}{P_{\text{undisturbed}}} - 1 \]
Exchange ratio
\[ ER = \frac{(1-c)\,P_{\text{offer}}}{P_{\text{acquirer}}} \]
\(c\) is the share of the price paid in cash.
New shares and target ownership
\[ N_{\text{new}} = ER \times N_T \qquad \omega_T = \frac{N_{\text{new}}}{N_A + N_{\text{new}}} \]
NAV per share accretion
\[ \text{Accretion} = \frac{\left(\text{NAV}_A + \text{NAV}_T - \text{cash paid}\right) / \left(N_A + N_{\text{new}}\right)}{\text{NAV}_A / N_A} - 1 \]
The same logic applies to production, cash flow or reserves per share.