Module 1

Time value of money

Establishes the equivalence between cash flows occurring on different dates

Interest rate (r)

Interest rates (or rates of return) represent the relationship between cash flows occurring at different points in time. They can be interpreted in three key ways:

1. Required rate of return – The minimum return an investor expects to earn for accepting the risk of an investment.

2. Discount rate – Often used interchangeably with interest rate, it’s the rate used to calculate the present value of future cash flows.

3. Opportunity cost – The potential return an investor forgoes when choosing one option over another. For example, if $9,500 today is equivalent to $10,000 in one year (implying a 5.26% interest rate), choosing to spend the $9,500 now instead of investing it means giving up that 5.26% return. This foregone return is the opportunity cost of current consumption.

Usually, interest rate is approximated annually, meaning a quoted 3 percent interest rate on a 90-day government debt is the annualized rate and not the actual interest rate over the 90 days

Determinants of Interest Rates

Real risk-free interest rate + inflation premium + Default risk premium + Liquidity premium + Maturity premium

Real risk free interest rate

The single-period interest rate for a completely risk-free security if no inflation were expected (we subtract the inflation %)

Inflation premium

Compensates investor for the expected inflation and reflects the average rate of inflation through the maturity of the debt.

Inflation

Inflation is the general rise in prices of goods and services over time, which reduces the purchasing power of money — the amount of goods and services one can buy with it.

Default risk premium

Compensates investor for the possibility that borrower will fail to make a promised payment at the contracted time & amount

Liquidity premium

If an asset is hard to sell quickly without losing value, investors will expect a higher return to compensate for that inconvenience or risk.
That extra return is the liquidity premium.

Example:

• A government bond is very liquid — you can sell it quickly.

• A rare real estate property is illiquid — it might take months to sell.

If both offer a 5% return, investors would prefer the liquid bond.
So to attract buyers, the real estate might need to offer 7% — the extra 2% is the liquidity premium.

Maturity premium

Long-term investments are riskier because:

• There's more uncertainty about interest rates, inflation, and economic conditions.

• Your money is tied up for longer.

So, to compensate for this extra risk and time, investors expect a higher return — that's the maturity premium.

Example:

• A 2-year bond might yield 3%.

• A 10-year bond might yield 5%.

The extra 2% is partly due to the maturity premium — a reward for locking in your money longer and facing more uncertainty.

Nominal risk-free rate

The nominal risk-free rate combines:

1. The real risk-free rate (compensation for deferring consumption), and

2. The inflation premium (compensation for expected loss of purchasing power).

Nominal risk-free rate = Real risk-free rate + Inflation premium

Holding Period Return

The return earned from holding an asset for a single specified period.

Holding Period Return Formula

R = ((P₁ - P₀) + I₁) / P₀

P₁: Price after

P₀: Price before

I₁: Income made from holding (e.g. dividends)

Arithmetic mean

Definition:
The simple average of values — add them all up and divide by the number of values.

When to use:
Use when calculating average values over one period or when observations are independent.
→ e.g., average test scores, one-year investment returns.

Geometric mean

Definition:
The average rate of return that shows compounded growth over multiple periods.

When to use:
Use for multi-period investment returns or anything involving compounding.
→ e.g., average return over 5 years.

Harmonic mean

Definition:
A special average used when values are expressed as rates or ratios.

When to use:
Use when averaging rates (like P/E ratios or speeds), especially when each item gets equal investment.

Also a good measure of mean, especially when outliers are present that can skew the data.
→ e.g., averaging P/E ratios across stocks with the same dollar investment.

What central tendency method (mean) will you use to calculate average buying price of a stock, provided that ever time the investor invested equal amounts?

We will use the harmonic mean

Money-weighted return

Accounts for the money invested and provides the investor with information on the actual return she earns on her investment. Amount invested is a cash outflow from the investor’s perspective (should be cash inflow for the bank or wherever the investment is going), and the amount returned or withdrawn by the investor, or the money that remains at the end of an investment cycle, is cash inflow for the investor.

Calculations are similar to the internal rate of return.

Internal rate of return

The discount rate at which the sum of present values of cash flows will equal zero (NPV = 0).

Annualized return

To compare returns that may have occurred during different periods, such as weekly, monthly or even quarterly — to’s most convenient to annualize these returns.

How to annualize any return

To annualize any return shorter than a year, compound the return of the period by the number of period in a year. A monthly return is compounded by 12 times, a weekly by 52, a quarterly by 4 and daily by 365.

R_annual = (1 + R_period)^c - 1

C: compounding factir

Continuously Compounded Return

A return calculated by assuming that compounding happens at every instant (continuously) instead of at discrete intervals like daily, monthly, or annually.

How it works (conceptually):

• Based on natural logarithms (ln).

• It’s the return you'd get if returns were reinvested infinitely often.

Gross return

A return earned prior to deduction for management expenses, custodial fees, taxes, or any other expenses that are not directly related to the generation of returns, but rather related to the management and administration of an investment.

Net return

A return that has accounted for all managerial and administrative expenses, which reduces an investors return. This is the amount the investor actually receives.

Pre-tax and After-tax nominal return

All return measures, unless stated otherwise are pre-tax nominal return, they do not account for the taxes paid.

Leveraged return

When an investor invests money with the help of a loan.

If return of portfolio is less than rate of debt, leveraged return is lowered. You ideally want the R_p to be greater than r_D

Net present value

Net Present Value (NPV) is a financial metric used to evaluate the profitability of an investment by calculating the difference between the present values of cash inflows and outflows over time

• If NPV > 0: The project adds value → ACCEPT it.

• If NPV < 0: The project destroys value → REJECT it.

• If NPV = 0: The project earns exactly the required return, meaning you should accept it.

Does gross return include trading expenses or are they already deducted?

Trading expenses are already deducted from gross return, so if in a question it indicates trading expenses and gross return, don’t subtract treading expenses from gross return, as it has already been done.