Library/CFA (Chartered Financial Analyst)/SchweserNotes 2022 Level I CFA Book 1: Quantitative Methods and Economics/Reading 1: The Time Value of Money

Reading 1: The Time Value of Money

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Interest Rates and Components5 min

Interest rates serve as the exchange rate between current and future consumption. They represent the opportunity cost of spending money now rather than investing it. The rate of return required by an investor is the sum of the real risk-free rate and compensation for various risks. The real risk-free rate reflects the pure time value of money assuming zero inflation. To determine the nominal risk-free rate (like T-bill rates), an inflation premium is added. For risky assets, additional premiums are required: a default risk premium for the possibility of non-payment, a liquidity risk premium for the difficulty of converting the asset to cash quickly without loss of value, and a maturity risk premium for the increased volatility associated with longer-term investments.

Key Points

Interest rates can be interpreted as discount rates, opportunity costs, or required rates of return.
Nominal risk-free rate = Real risk-free rate + Expected inflation rate.
Required interest rate = Nominal risk-free rate + Default risk premium + Liquidity risk premium + Maturity risk premium.

Effective Annual Rate (EAR) and Compounding5 min

Financial institutions often quote stated annual interest rates with specific compounding frequencies (e.g., monthly, quarterly). However, the actual return earned is the Effective Annual Rate (EAR), which accounts for the effect of interest earning interest. As the compounding frequency increases (from annual to semiannual to monthly to daily), the EAR increases for a given stated rate. Comparing investments is only valid when using the EAR. The calculation involves determining the periodic rate (stated rate divided by compounding periods per year) and using the formula EAR = (1 + periodic rate)^m - 1.

Key Points

EAR represents the annual rate of return actually realized.
Periodic rate = Stated annual rate / m (where m is compounding periods per year).
EAR increases as the frequency of compounding increases.
EAR = (1 + periodic rate)^m - 1.

Future and Present Value of Single Sums5 min

The Future Value (FV) of a single sum calculates what a deposit today will grow to by a specific future date given a compound interest rate. The formula is FV = PV(1 + I/Y)^N. Conversely, the Present Value (PV) determines the value today of a cash flow expected in the future, effectively discounting it back to time zero. The PV formula is PV = FV / (1 + I/Y)^N. These calculations are the building blocks for more complex TVM problems. The factor (1 + I/Y)^N is the future value factor, while 1 / (1 + I/Y)^N is the discount factor.

Key Points

Compounding moves cash flows forward in time to find FV.
Discounting moves cash flows backward in time to find PV.
PV and FV are inversely related; higher discount rates yield lower PVs.
N represents the total number of compounding periods, not necessarily years.

Annuities and Perpetuities6 min

Annuities are streams of equal cash flows at regular intervals. An ordinary annuity has payments at the end of each period, while an annuity due has payments at the beginning. Because annuity due payments are received sooner, they are worth more; specifically, the value of an annuity due is the value of an ordinary annuity multiplied by (1 + I/Y). A perpetuity is a unique type of annuity with infinite life (N = infinity), such as preferred stock dividends. The PV of a perpetuity is simply the periodic payment divided by the interest rate (PMT / I/Y). Solving for other variables like the number of periods (N) or the rate (I/Y) requires iterative calculation or the use of financial calculators.

Key Points

Ordinary Annuity: Payments at end of period (e.g., mortgage).
Annuity Due: Payments at beginning of period (e.g., rent, tuition).
PV of Annuity Due = PV of Ordinary Annuity * (1 + periodic rate).
PV of Perpetuity = Payment / Rate.

Uneven Cash Flows and Additivity4 min

Many financial investments do not follow a strict annuity pattern. For uneven cash flow streams, the PV or FV is calculated by summing the PV or FV of each individual cash flow. This utilizes the Cash Flow Additivity Principle, which states that the value of a stream of cash flows is equal to the sum of the values of its components. Financial calculators can handle these streams using cash flow worksheets to calculate Net Present Value (NPV). Time lines are critical tools for organizing these problems, identifying exactly when inflows and outflows occur to ensure the correct number of discounting or compounding periods is applied.

Key Points

Uneven cash flows are treated as series of single sums.
Cash Flow Additivity Principle allows summing separate PVs.
Time lines help prevent timing errors (e.g., confusing t=0 with t=1).
Net Present Value (NPV) aggregates the PV of unequal flows.

Questions

Question 1

An interest rate is best described as which of the following?

Reading 1: The Time Value of Money

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