CFA Level 1 Quantitative Methods

Perpetuity

A perpetuity is a level series of payments made into perpetuity. PV = A/r

Annuity due

An annuity due has the first payment due this year. PV = A + PV(ordinary annuity)

Ordinary annuity - PV & FV

An ordinary annuity is a level series of payments made within a finite period of time, with the first payment made one year in the future. FV = A*{[(1+r)^N - 1]/r}, where A is the annuity payment PV = A*[1-(1/(1+r)^N]/r

Future Value of a sum of money in a finite period of time

FV(t=N) = PV*(1+r)^N, where N is the number of compounding periods

FV of a sum of money that is continuously compounding

FV(t=N) = PV*e^(N*r)

Holding period return (HPR)

HPR is the return that an investor earns over a period of holding the investment/portfolio. HPR = (P1 - P0 + D1)/P0, where P1 = price received at the end of the holding period P0 = price initially paid / amount invested D1 = cash paid out by investment at the end of the holding period

Internal Rate of Return (IRR) and the IRR rule

IRR is the interest rate that makes NPV=0. To obtain IRR, PV inflows = PV outflows. The IRR rule states that if the IRR is above the existing discount rate (i.e. the weighted average cost of capital), make the investment

Treasury bills - how they work

Investors pay face value - discount, and receive face value @ maturity. Quoted on a bank discount basis (i.e. annualizes the discount as a percentage of FV, based on a 360 day year

Net Present Value and the NPV rule

NPV = PV inflows - PV outflows The NPV rule states that you should make the investment if NPV>0. If NPV=0, no value is added to the shareholders and you should be indifferent about the investment.

Conditional probability

P(A | B) = probability of A given that B occurs P(A | B) = P(AB)/P(B), P(B) not= 0

Effective annual interest (EAR)

Stated annual interest is not equal to EAR because of compounding of interest. EAR = [(1+r)^N] - 1 Continuous compounding: EAR = (e^r) - 1

Conflict between NPV & IRR rules

The IRR rule is problematic when the size & scale of the projects differ; or when the timing of the cash flows differ. When the 2 rules conflict, just follow the NPV rule. Why? The IRR rule assumes that the CFs are reinvested at IRR. This, however, won't be realistic especially if the IRR is very high. The NPV calculation, on the other hand, uses an externally determined cost of capital. This is more realistic and economically relevant.

Money-weighted rate of return - definition and its drawbacks

The money-weighted rate of return is essentially an IRR. Obtain it the same way as an IRR. To find the HPR over a number of years, calculate the HPR for each year and average it out. The drawback of the HPR calculated as an IRR is that the client normally decides how to invest. This measure can't therefore accurately measure the performance of the portfolio.

Time-weighted rate of return - definition and calculation

This measures the compound rate of growth of the sum initially invested over a period of time. This is a better gauge of performance of the portfolio. To obtain the time-weighted rate of return: 1) Price the portfolio prior to any significant change (either withdrawal/addition) in funds and break the evaluation period into sub-periods according to dates of cash inflows & outflows. 2) Calculate the HPR for each sub-period. 3) Compound HPRs to obtain an annual rate of return for the year. Take the geometric mean of all the HPRs. [(1+HPR1)*(1+HPR2)...*(1+HPR(t=N))]^(1/N) - 1 = time-weighted rate of return

Compound annual growth rate (CAGR)

Used to average growth rates over a period of time. CAGR = (FV/PV)^(1/N) - 1

Interest rate, r

r = Real risk-free interest rate + Inflation premium + Default risk premium + Liquidity premium + Maturity premium Note that nominal risk-free rate (RFR) = Real risk-free rate + Inflation