Chapter 1 Continuous Compounding P Siy To accompany Financial Modeling, 4th Edition Simon Benninga MIT Press, 2014 Effective Annual Rate (EAR) πΈπ΄π = 1 + ππππ π π -1 With: EAR the effective annual rate rnom some nominal annual rate n the number of compounding Periods Rnom/n = periodic rate FM3 Financial modeling hints 2 TI BA II Rate Conversion ο Note the keys for conversion between nominal and effective on the TI BA II… β¦ 2ND ICONV FM3 Financial modeling hints 3 Excel Rate Conversion ο Calculate the effective rate β¦ EFFECT(nominal_rate, npery) ο Calculate the nominal rate β¦ NOMINAL(effect_rate, npery) FM3 Financial modeling hints 4 EAR Review Problem ο If Capital One Quotes an APR of 24.9% what is the effective rate? FM3 Financial modeling hints 5 EAR Example ο If Ally Bank quotes 1.60% APY on a 5-year certificate of deposit that is compounded daily, speculate whether this is the APR or some other rate… FM3 Financial modeling hints 6 Euler’s Identity lim 1 π→∞ 1 π + π = π 1 = 2.71828… = “Euler Number” Euler is pronounced? Appears similar to the EAR calculation… 7 Extending Euler’s Identity to Compounding ππππ lim 1 + π→∞ π ο π = ππ So we can apply Euler’s to a solve for a continuously compounded rate! FM3 Financial modeling hints 8 Continuous Compounding πΈπ΄π = 1 + ππππ π π -1 Therefore in a continuous compounded environment, (n -> ∞) substituting the Euler result: πΈπ΄π = π ππππ - 1 rnom also described as the stated annual rate EAR here is the effective rate given continuous compounding FM3 Financial modeling hints 9 Continuous Compounding Result πΈπ΄π = π ππππ - 1 Or 1+ ππππ = π ππππ So a nominal annual rate is converted to an annual effective rate given continuous compounding FM3 Financial modeling hints 10 Continuous Compounding Continuous compounding is used throughout Benninga’s text ο Implicit in portfolio theory and options pricing ο We will see why… ο FM3 Financial modeling hints 11 Useful Exponential Identities FM3 Financial modeling hints 12 Natural Logs in Excel ο E to a power β¦ EXP(number) ο Natural logarithm β¦ LN(number) ο For the TI BA II? FM3 Financial modeling hints 13 Continuous Compounding Example ο Given an annual (nominal) rate of 10% what is the effective rate given continuous compounding? β¦ Higher / Lower FM3 Financial modeling hints 14 Converting to Equivalent Continuously Compounded Rates ο Conversely, we need to calculate the continuously compounded rate (the stated annual rate with continuous compounding) corresponding to an effective annual rate ο Applies to stock prices! FM3 Financial modeling hints 15 Continuous Rate Conversion cont. ο How to solve given: β¦ πΈπ΄π = Refπ = π ππππ -1 β¦ I.e. solve for rnom FM3 Financial modeling hints 16 Continuous Rate Conversion ο For example, given an effective rate of 10% continuously compounded, what is the stated annual rate? β¦ Higher / Lower FM3 Financial modeling hints 17 Discrete (Stock Price) Returns ο Provide the formula for a (discrete) holding period return herein referred to as a “discrete return” for stock prices FM3 Financial modeling hints 18 Discrete Returns (cont.) Typically use Yahoo dividend-adjusted prices ο Returns are ALWAYS quoted in % ο β¦ NOT dollars! β¦ NOT decimal! ο Provide returns with at least 1 basis point precision (BPS) (1/100 of 1% or 0.01%) FM3 Financial modeling hints 19 Discrete Returns in Excel ο Easier to input in Excel, discrete returns always take the form β¦ HPR = π1 π0 −1 ο Why? β¦ Note the percentage (operation) is accomplished via formatting in Excel, not multiplication FM3 Financial modeling hints 20 Continuous Returns Usage We will take (stock price) holding period return(s) assuming a continuous compounding assumption, herein referred to as a “continuous return” ο Benninga uses continuous returns throughout Financial Modeling ο β¦ As do investment professionals performing statistical analysis of investment returns… FM3 Financial modeling hints 21 Example 1: Applying Continuous Returns to Stock Prices ο Given stock prices P0 = $1,000 and P1 = $1,200 find the percentage return… β¦ Wait, based on annual compounding? β¦ Or two compounding periods per year? β¦ What about as N –> ∞ ? ο So it depends on the compounding assumption! FM3 Financial modeling hints 22 Example 2: Discrete versus Continuous Returns for Stocks ο If P1 = $100 and P0 = $90, find the discrete and continuous (holding period) return(s) FM3 Financial modeling hints 23 Discrete versus Continuous Return Example 2 Cont. Likewise, if P2 = $90 and P1 = $100, find the discrete and continuous (holding period) return(s) ο Find the average of returns between P0 and P2 ο State the advantage(s) of continuous returns ο FM3 Financial modeling hints 24 Addendum FUTURE VALUE AND PRESENT VALUE WITH CONTINUOUS COMPOUNDING FM3 Financial modeling hints 25 Future Value with Continuous Compounding ο We know β¦ πΉπ = ππ ∗ 1 + π ο π In a continuous compounded world β¦ πΉπ = ππ ∗ π π ο So $100 invested for one year at 10% continuously compounded yields… FM3 Financial modeling hints 26 Multi-Year Future Value ο With ο ο Then ο ο πΉπ = ππ ∗ (π π )π πΉπ = ππ ∗ π π∗π So $100 invested for two years at 10% continuously compounded yields… FM3 Financial modeling hints 27 Fluctuating Rate Future Value ο With ο ο Then for varying rates, say for two years ο ο πΉπ = ππ ∗ π π1 ∗ π π2 And therefore ο ο πΉπ = ππ ∗ π π πΉπ = ππ ∗ π π1 +π2 So $100 invested for two years at 10% the first year and 11% the second year continuously compounded yields… FM3 Financial modeling hints 28 Present Value with Continuous Discounting ο We know β¦ πΉπ = ππ ∗ 1 + π β¦ ππ = ο π πΉπ 1+π π And continuously compounded future value is thus β¦ πΉπ = ππ ∗ π π∗π ο So β¦ ππ = πΉπ π π∗π = πΉπ ∗ π −π∗π FM3 Financial modeling hints 29 Continuous Discounting ο Note that the continuously discounted PV is always β¦More / Less than a discretely compounded PV FM3 Financial modeling hints 30
