Block Q · Financial mathematics basics
‹ NadirAlpha financial markets handbook: overview
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Six calculation concepts carry almost all of capital market practice; whoever masters them can derive every metric instead of believing it.
195 · Compound interest, present value & future value
Future value = K x (1+r)^n; present value = future payment / (1+r)^n; discounting is the inverse of compounding.
The basis of EVERY valuation (bond, DCF, annuity).
The rule of 72 as mental arithmetic: 72/return ~ years to doubling.
Linear thinking; compounding works exponentially, above all toward the end of long periods.
196 · IRR; money-weighted vs. time-weighted return · 197 · Annuities & amortization schedules
196 · IRR; money-weighted vs. time-weighted return
IRR = the rate that sets the present value of all payments to zero (money-weighted, MWR); time-weighted (TWR) eliminates payment flows and measures the manager's performance.
TWR for manager/fund comparison; MWR/IRR for one's own wealth development and private equity.
A large contribution before a weak phase pushes MWR below TWR; the two numbers tell different stories.
Comparing PE IRRs with the TWR of liquid investments.
198 · Return calculation (discrete vs. continuous) · 112 · Sharpe ratio
197 · Annuities & amortization schedules
A constant periodic payment of interest + principal; the annuity formula links loan amount, rate, term, installment.
Financings, withdrawal plans ("how long does the wealth last at withdrawal X?").
In annuities the interest share falls, the principal share rises; the remaining debt declines slowly at first.
Computing withdrawal plans without inflation-adjusting the withdrawal.
198 · Return calculation (discrete vs. continuous)
Discrete return = P1/P0 - 1; continuous (log) return = ln(P1/P0); log returns are additive over time.
Statistics/models use log returns; reporting uses discrete ones.
-50% + 100% = 0% only in the log world; in discrete terms it takes +100% to offset -50% ("volatility drag").
Presenting the arithmetic mean of discrete returns as the achieved return; compute geometrically.
196 · IRR; money-weighted vs. time-weighted return · 110 · Volatility (historical/implied)
199 · Yield curve bootstrapping (basic idea)
Deriving the zero coupon curve (spot rates) step by step from coupon bonds/swaps; from it discount factors and forward rates.
Consistent valuation of all cashflows; understanding where "the" curve in YAS/SWPM comes from.
Forward rates are an arithmetic consequence of the curve; not a forecast (see Blocks E/F).
Equating the yield curve (YTM) with the spot curve.
200 · Option pricing theory basics (Black-Scholes, put-call parity)
Black-Scholes prices options via replication (hedge portfolio); drivers: strike, spot, time, rate, VOL; put-call parity links call, put, spot and the present value of the strike.
A fair-value feel for all option structures (certificates!); vega/theta intuition.
Direction is NOT priced in; what is traded is volatility; parity violations = arbitrage or data errors.
Calling options "cheap" because the premium is small; what matters is implied vol vs. expectation.
61 · Options (call/put, basics) · 110 · Volatility (historical/implied)