Reading 1

Interest rates measure the time value of money, although risk differences in financial securities lead to differences in their equilibrium interest rates. Equilibrium interest rates are the required rate of return for a particular investment, in the sense that the market rate of return is the return that investors and savers require to get them to willingly lend their funds. Interest rates are also referred to as discount rates and, in fact, the terms are often used interchangeably. If an individual can borrow funds at an interest rate of 10%, then that individual should discount payments to be made in the future at that rate to get their equivalent value in current dollars or other currencies. Finally, we can also view interest rates as the opportunity cost of current consumption. If the market rate of interest on 1-year securities is 5%, earning an additional 5% is the opportunity forgone when current consumption is chosen rather than saving (postponing consumption).

The real risk-free rate of interest is a theoretical rate on a single-period loan that contains no expectation of inflation and zero probability of default. What the real risk-free rate represents in economic terms is time preference, the degree to which current consumption is preferred to equal future consumption.

When we speak of a real rate of return, we are referring to an investor's increase in purchasing power (after adjusting for inflation). Because expected inflation in future periods is not zero, the rates we observe on U.S. Treasury bills (T-bills), for example, are essentially risk-free rates, but not real rates of return. T-bill rates are nominal risk-free rates because they contain an inflation premium. This is the relation:

\[(1+\text{nominal risk-free rate}) = (1+\text{real risk-free rate})(1+\text{expected inflation rate})\]

Often, including in many parts of the CFA curriculum, this relation is approximated as follows:

\[\text{nominal risk-free rate} \approx \text{real risk-free rate} + \text{expected inflation rate}\]

Securities may have one or more types of risk, and each added risk increases the required rate of return. These types of risks are as follows:

• Default risk. This is the risk that a borrower will not make the promised payments in a timely manner.
• Liquidity risk. This is the risk of receiving less than fair value for an investment if it must be sold quickly for cash.
• Maturity risk. As we will see in the Fixed Income topic area, the prices of longer-term bonds are more volatile than those of shorter-term bonds. Longer-maturity bonds have more maturity risk than shorter-term bonds and require a maturity risk premium.

Each of these risk factors is associated with a risk premium that we add to the nominal risk-free rate to adjust for greater default risk, less liquidity, and longer maturity relative to a liquid, short-term, default risk-free rate such as that on T-bills. We can write the following:

\[\text{nominal rate of interest} = \text{real risk-free rate} + \text{inflation premium} + \text{default risk premium} + \text{liquidity premium} + \text{maturity premium}\]

Holding period return (HPR) is simply the percentage increase in the value of an investment over a given period:

\[\text{holding period return} = \dfrac{\text{end-of-period value}}{\text{beginning-of-period value}} - 1\]

For example, a stock that pays a dividend during a holding period has an HPR for that period equal to:

\[\dfrac{P_t + \text{Div}_t}{P_0} - 1, \text{ or } \dfrac{P_t - P_0 + \text{Div}_t}{P_0}\]

If a stock is valued at €20 at the beginning of the period, pays €1 in dividends over the period, and at the end of the period is valued at €22, the HPR is:

\[\text{HPR} = (22 + 1)/20 - 1 = 0.15 = 15\%\]

Returns over multiple periods reflect compounding. For example, given HPRs for Years 1, 2, and 3, the HPR for the entire three-year period is:

\[\text{HPR} = (1+\text{HPR}_{\text{Year 1}})(1+\text{HPR}_{\text{Year 2}})(1+\text{HPR}_{\text{Year 3}}) - 1\]

Later in this reading, we will see that a return over multiple years is typically stated as an annualized return rather than an HPR.

Average Returns

The arithmetic mean return is the simple average of a series of periodic returns. It has the statistical property of being an unbiased estimator of the true mean of the underlying distribution of returns:

\[\text{arithmetic mean return} = \dfrac{R_1 + R_2 + R_3 + \cdots + R_n}{n}\]

The geometric mean return is a compound rate. When periodic rates of return vary from period to period, the geometric mean return will have a value less than the arithmetic mean return:

\[\text{geometric mean return} = \sqrt[n]{(1+R_1)\times(1+R_2)\times(1+R_3)\times\cdots\times(1+R_n)} - 1\]

For example, for returns \(R_t\) over three annual periods, the geometric mean return is calculated as the following example shows.

In the previous example, the geometric mean results in an annual rate of return because the holding periods were years. If the holding periods are other than years, the geometric mean is not the same as the annual return. The root for the geometric mean is the number of periods, while the root for the annual return is the number of years.

A harmonic mean is used for certain computations, such as the average cost of shares purchased over time. The harmonic mean is calculated as \(\overline{X}_H = \dfrac{N}{\sum_{i=1}^{N}\frac{1}{X_i}}\), where there are \(N\) values of \(X_i\).

The previous example illustrates the interpretation of the harmonic mean in its most common application. Note that the average price paid per share ($8.93) is less than the arithmetic average of the share prices, which is \(\dfrac{8+9+10}{3} = 9\).

We can only calculate a harmonic mean of positive numbers. For a set of returns that includes negative numbers, we can treat them the same way we did with geometric means, using (1 + return) for each period, then subtracting 1 from the result.

The relationship among arithmetic, geometric, and harmonic means can be stated as follows:

\[\text{arithmetic mean} \times \text{harmonic mean} = (\text{geometric mean})^2\]

For values that are not all equal, harmonic mean < geometric mean < arithmetic mean. This mathematical fact is the basis for the claimed benefit of purchasing the same money amount of mutual fund shares each month or each week. Some refer to this practice as cost averaging.

Measures of average return can be affected by outliers, which are unusual observations in a dataset. Two of the methods for dealing with outliers are a trimmed mean and a winsorized mean. We will examine these in our reading on Statistical Measures of Asset Returns.

Appropriate uses for the various return measures are as follows:

• Arithmetic mean. Include all values, including outliers.
• Geometric mean. Compound the rate of returns over multiple periods.
• Harmonic mean. Calculate the average share cost from periodic purchases in a fixed money amount.
• Trimmed or winsorized mean. Decrease the effect of outliers.

The money-weighted return applies the concept of the internal rate of return (IRR) to investment portfolios. An IRR is the interest rate at which a series of cash inflows and outflows sum to zero when discounted to their present value. That is, they have a net present value (NPV) of zero. The IRR and NPV are built-in functions on financial calculators that CFA Institute permits candidates to use for the exam.

The money-weighted rate of return is defined as the IRR on a portfolio, taking into account all cash inflows and outflows. The beginning value of the account is an inflow, as are all deposits into the account. All withdrawals from the account are outflows, as is the ending value.

Time-weighted rate of return measures compound growth and is the rate at which $1 compounds over a specified performance horizon. Time-weighting is the process of averaging a set of values over time. The annual time-weighted return for an investment may be computed by performing the following steps:

Step 1: Value the portfolio immediately preceding significant additions or withdrawals. Form subperiods over the evaluation period that correspond to the dates of deposits and withdrawals.

Step 2: Compute the holding period return (HPR) of the portfolio for each subperiod.

Step 3: Compute the product of (1 + HPR) for each subperiod to obtain a total return for the entire measurement period \([(1 + \text{HPR}_1) \times (1 + \text{HPR}_2) \dots (1 + \text{HPR}_n)] - 1\). If the total investment period is greater than one year, you must take the geometric mean of the measurement period return to find the annual time-weighted rate of return.

The time-weighted rate of return is not affected by the timing of cash inflows and outflows. In the investment management industry, time-weighted return is the preferred method of performance measurement because portfolio managers typically do not control the timing of deposits to and withdrawals from the accounts they manage.

In the preceding examples, the time-weighted rate of return for the portfolio was 15.84%, while the money-weighted rate of return for the same portfolio was 13.86%. The results are different because the money-weighted rate of return gave a larger weight to the Year 2 HPR, which was 10%, versus the 22% HPR for Year 1. This is because there was more money in the account at the beginning of the second period.

If funds are contributed to an investment portfolio just before a period of relatively poor portfolio performance, the money-weighted rate of return will tend to be lower than the time-weighted rate of return. On the other hand, if funds are contributed to a portfolio at a favorable time (just before a period of relatively high returns), the money-weighted rate of return will be higher than the time-weighted rate of return. The use of the time-weighted return removes these distortions, and thus provides a better measure of a manager's ability to select investments over the period. If the manager has complete control over money flows into and out of an account, the money-weighted rate of return would be the more appropriate performance measure.

Interest rates and market returns are typically stated as annualized returns, regardless of the actual length of the time period over which they occur. To annualize an HPR that is realized over a specific number of days, use the following formula:

\[\text{annualized return} = (1 + \text{HPR})^{365/\text{days}} - 1\]

In time value of money calculations (which we will address in more detail in our reading on The Time Value of Money in Finance), more frequent compounding has an impact on future value and present value computations. Specifically, because an increase in the frequency of compounding increases the effective interest rate, it also increases the future value of a given cash flow and decreases the present value of a given cash flow.

This is the general formula for the present value of a future cash flow:

\[PV = FV_N \left(1 + \dfrac{r}{m}\right)^{-mN}\]

where:

• r = quoted annual interest rate
• N = number of years
• m = compounding periods per year

The mathematical limit of shortening the compounding period is known as continuous compounding. Given an HPR, we can use the natural logarithm (ln, or LN on your financial calculator) to state its associated continuously compounded return:

\[R_{CC} = \ln(1+\text{HPR}) = \ln\!\left(\dfrac{\text{ending value}}{\text{beginning value}}\right)\]

Notice that because the calculation is based on 1 plus the HPR, we can also perform it directly from the price relative. The price relative is just the end-of-period value divided by the beginning-of-period value.

A useful property of continuously compounded rates of return is that they are additive for multiple periods. That is, a continuously compounded return from t = 0 to t = 2 is the sum of the continuously compounded return from t = 0 to t = 1 and the continuously compounded return from t = 1 to t = 2.

Gross return refers to the total return on a security portfolio before deducting fees for the management and administration of the investment account. Net return refers to the return after these fees have been deducted. Commissions on trades and other costs that are necessary to generate the investment returns are deducted in both gross and net return measures.

Pretax nominal return refers to the return before paying taxes. Dividend income, interest income, short-term capital gains, and long-term capital gains may all be taxed at different rates. After-tax nominal return refers to the return after the tax liability is deducted.

Real return is nominal return adjusted for inflation. Consider an investor who earns a nominal return of 7% over a year when inflation is 2%. The investor's approximate real return is simply 7 − 2 = 5%. The investor's exact real return is slightly lower: 1.07 / 1.02 − 1 = 0.049 = 4.9%.

Using the components of an interest rate we defined earlier, we can state a real return as follows:

\[(1 + \text{real return}) = \dfrac{(1 + \text{nominal risk-free rate})(1 + \text{risk premium})}{(1 + \text{inflation premium})}\]

Real return measures the increase in an investor's purchasing power—how much more goods she can purchase at the end of one year due to the increase in the value of her investments. If she invests $1,000 and earns a nominal return of 7%, she will have $1,070 at the end of the year. If the price of the goods she consumes has gone up 2%, from $1.00 to $1.02, she will be able to consume $1,070 / 1.02 = 1,049 units. She has given up consuming 1,000 units today, but instead, she is able to purchase 1,049 units at the end of one year. Her purchasing power has gone up 4.9%; this is her real return.

A leveraged return refers to a return to an investor that is a multiple of the return on the underlying asset. The leveraged return is calculated as the gain or loss on the investment as a percentage of an investor's cash investment. An investment in a derivative security, such as a futures contract, produces a leveraged return because the cash deposited is only a fraction of the value of the assets underlying the futures contract. Leveraged investments in real estate are common: investors pay only a portion of a property's cost in cash and borrow the rest.

To illustrate the effect of leverage on returns, consider a fund that can invest the amount \(V_0\) without leverage, and earn the rate of return r. The fund's unleveraged return (as a money amount) is simply r × \(V_0\). Now let's say this fund can borrow the amount \(V_B\) at an interest rate of \(r_B\), and earn r by investing the proceeds. The fund's leveraged return (again as a money amount), after subtracting the interest cost, then becomes \(r \times (V_0 + V_B) - (r_B \times V_B)\).

Thus, stated as a rate of return on the initial value of \(V_0\), the leveraged rate of return is as follows:

\[\text{leveraged return} = \dfrac{r(V_0 + V_B) - r_B V_B}{V_0}\]