Foundations
Core Concepts
The groundwork for the rest of the site: interest and time value, day counts, building yield curves, and basic probability and statistics.
Introductory5 pagesPractitioner4 pagesAdvanced2 pages
1Interest and the Time Value of Money
- IntroductoryCompounding and DiscountingIf you deposit 100 yen for three years at 2%, you end up with 106 yen under simple interest, which pays interest only on the principal, and 106.12 yen with annual compounding.
- IntroductoryPresent Value and Discount FactorsAt an interest rate of 2%, the present value of 102 yen received in one year is 100 yen.
- PractitionerZero Rates and Forward RatesA spot rate is the compound yield on a discount bond that pays no interest before maturity.
2Dates and Day Counts
- IntroductoryDay Count ConventionsInterest is the principal times the rate times "days in the period ÷ days in a year," and the days in the period include the start date but not the end date.
- IntroductoryBusiness Day Conventions and Holiday CalendarsIn Japan, banks are closed on Sundays, Saturdays, holidays under the Act on National Holidays, and from December 31 to January 3.
3Yield Curves
- PractitionerBootstrapping a Yield CurveSwap fixed rates do not show discount factors directly, so you solve for them in order of maturity, starting with the shortest.
- PractitionerThe Multi-Curve FrameworkDiscounting uses an OIS curve, projection uses a curve for each reference rate and tenor, and the projection curve is solved from the condition that market swaps have zero value.
- AdvancedCurve InterpolationThe quantity you interpolate is the discount factor, its logarithm, or the zero rate, and the quantity you choose sets the shape of the forward rate between the points.
4Mathematical Preliminaries
- IntroductoryProbability and Statistics BasicsIf a price will be 103 yen, 101 yen, or 95 yen with probabilities of 25%, 50%, and 25%, the expected value is 100 yen and the standard deviation is 3 yen.
- PractitionerRandom Walks and Brownian MotionA symmetric random walk moving +1 or −1 at each step has mean 0 and variance n after n steps, so its standard deviation after 100 steps is 10.
- AdvancedIntroduction to Stochastic Calculus (Ito's Lemma)A stochastic differential equation writes the small movement of a value as a drift term plus a dW term, and writing it in integral form brings in the Ito integral.