Module 3: Managing Investment Risk and Return

Question 1

Differentiate between “interest rate,” “nominal interest rate” and “real interest rate.” Provide examples.

Answer 1

An “interest rate” is a promised rate of return that can be denominated in a specific currency, for example Canadian dollars, over some time period. When we say the interest rate is 5%, we must specify both the unit of account and the time period.

The “nominal interest rate” (R) is the growth rate of your money. Conventional fixed income investments such as bank certificates of deposit promise a nominal rate of interest. For example, if you invest $1000 at a 10% nominal interest rate, in one year you can expect to earn $100 in interest.

What you can buy today relative to what you could buy a year ago is the real return on your investment. Part of your interest earnings may have been offset by a reduction in the purchasing power of the dollars you receive at the end of your investment time period.

The “real interest rate” (r) is the growth rate of your purchasing power and its determination takes into account inflation. The real interest rate is the nominal rate reduced by the loss of purchasing power resulting from inflation. It is possible to approximate the real rate of return by using the following: r ≈ R – i, where i equals inflation rate.

For example, assume a bank loans you $200,000 to purchase a house at a 3% at a time when the inflation rate is 2%. The nominal rate is 3% and the real interest rate 1% (3% – 2%).

Question 2

Outline the fundamental factors that determine the levels of nominal and real interest rates.

Answer 2

Fundamental factors that determine the levels of interest rates include the following:

(a) The supply of funds from savers
(b) Demands by business for funds to finance physical investments in plant, equipment and inventories (real assets or capital formation)
(c) The government’s net supply and/or demand for funds as modified by actions of the monetary authority. The government and the central bank (Bank of Canada) can shift the supply and demand for funds through fiscal and monetary policies.
(d) The expected rate of inflation.

Question 3

Assume one year ago you deposited $500 in a one-year time deposit guaranteeing a 10% rate of return, and inflation since that time has been 3%. Explain how your purchasing power has been affected.

Answer 3

A rate of inflation of 3% means that the purchasing power of money has been reduced by 3% in the past year. That is, the value of each dollar invested has depreciated by 3% in terms of the goods they can now buy. Using the real interest rate approximation rule, you are left with a net increase in purchasing power of about 7% (which is 10% − 3%).

Question 4

One year ago, you deposited $1,000 in a one-year bank deposit guaranteeing a 7% rate of return. The inflation rate was 2%. Calculate the real interest rate using both the approximation rule and the real rate of interest formula. Explain the variation.

Answer 4

Using the real interest rate approximation rule:

r ≈ R - i

Where:

r = Real interest rate
R = Nominal interest rate
i = Inflation rate
r ≈ .07 - .02

≈ .05 or 5%

Using the real interest rate formula:

Where:

r = (R−i) / (1+i)

Where:

r = Real interest rate
R = Nominal interest rate
i = Inflation rate
r = .07−.021+.02

= .051.02

= .049 or 4.9%

The approximation rule indicates a slightly lower loss of purchasing power due to inflation than the formula. The approximation rule is more exact when inflation rates are low and perfectly exact for continuously compounded interest rates.

Question 5

Assume you earned 5% on your defined contribution (DC) pension plan account investments this past year. The inflation rate was 2%. Calculate the real interest rate using both the approximation rule and the real interest rate formula and compare both to the nominal rate.

Answer 5

Using the real interest rate approximation rule:

r ≈ R - i

Where:

r = Real interest rate
R = Nominal interest rate
i = Inflation rate
r ≈ .05 - .02

≈ .03 or 3%

Using the real interest rate formula:

r = R−i / 1+i

Where:

r = Real interest rate
R = Nominal interest rate
i = Inflation rate
r = .05−.02 / 1+.02

= .031.02

= .0294 or 2.94%

The real interest rate formula shows a rate of 2.94%, compared to 3% when using the real interest approximation rule, meaning that using the real interest rate formula shows a greater loss of purchasing power. However, because the inflation rate is low, the two rates are close.

The real interest rate is significantly lower than the 5% nominal rate earned because 2% inflation represents a fairly high portion of the nominal interest rate.

Question 6

Assume over the past year your defined benefit (DB) pension plan earned a nominal rate of return of 8% on its investments. The inflation rate was 4%. Calculate the real interest rate on plan assets using both the approximation rule formula and the real interest rate formula and compare both to the nominal rate.

Answer 6

Using the real interest rate approximation rule:

r ≈ R - i

Where:

r = Real interest rate
R = Nominal interest rate
i = Inflation rate
r ≈ .08 - .04

≈ .04 or 4%

Using the real interest rate formula:

r = R−i1+i

Where:

r = Real interest rate
R = Nominal interest rate
i = Inflation rate
r = .08−.041+.04

= .041.04

= .03846 or 3.85%

The real interest rate formula shows a rate of 3.85%, compared to 4% when using the real interest approximation rule, meaning that using the real interest rate formula shows a greater loss of purchasing power.

The real interest is significantly lower than the 8% nominal rate earned on your DB pension plan.

Question 7

Define the equilibrium real rate of interest and identify how the graph below is affected by the economic factors listed below the graph.

Answer 7

(a) Businesses increase their capital spending to acquire and/or upgrade buildings, machinery and equipment in order to increase their capacity or to meet growing demand for their services.

(b) Households are induced to save more because of increased uncertainty about their future Canada Pension Plan (CPP) benefits.

(c) The Bank of Canada sells Canada Treasury bonds to reduce the supply of money (i.e., the public pays for these bonds with its holdings of currency and bank deposits, directly reducing the amount of money in circulation).

The equilibrium rate of interest is found at the point of the intersection of the supply and demand curves that represent the quantity of funds and the real rate of interest. As such it is at the point where the two curves in the above graph cross. The economic factors listed above affect the equilibrium rate of interest as follows:

(a) Demand for funds has increased, shifting the demand curve to the right and increasing the equilibrium real rate of interest.
(b) Supply of funds has increased, shifting the supply curve to the right and reducing the equilibrium real rate of interest.
(c) Supply of funds has reduced, shifting the supply curve to the left, resulting in an increase in the equilibrium real rate of interest. The action by the Bank of Canada can also increase the demand for funds, shifting the demand curve to the right. The same result occurs—an increase in the equilibrium real rate of interest.

Question 8

Describe the Fisher hypothesis and its significance.

Answer 8

The Fisher hypothesis states that the nominal rate of interest should increase in step with the expected rate of inflation. This is because investors should be concerned with their real returns—the increase in their purchasing power—and expect higher nominal interest rates when inflation is higher. A higher nominal rate is needed to maintain the expected real return offered by an investment.

The implication of the Fisher hypothesis is that when real interest rates are stable, changes in nominal interest rates should predict changes in inflation rates. The hypothesis is difficult to definitively test, and data do not strongly support it.

However, nominal interest rates seem to predict inflation as well as alternative methods.

Question 9

Assume the real interest rate on an investment you are considering is 5% and the expected inflation rate is 5% over the same period. Calculate the nominal interest rate using the Fisher hypothesis.

Answer 9

Using the Fisher hypothesis:

R = r + E(i)

Where:

R = Nominal interest rate
r = Real interest rate
E(i) = Expected inflation rate
R = .05 + .05

= .10 or 10%

The nominal interest rate is 10%.

Question 10

Describe the key component of a valid comparison between returns on investments and outline why this component must be included.

Answer 10

To make a valid comparison between returns achieved on different investments, it is required that a rate of return for some common time period is available.

Typically, the longer the time horizon of the investment, the greater the amount of return received. Simply comparing the total value of returns received is not a valid comparison if the investments involved have been held for different time periods.

Question 11

Describe “annual percentage rate” (APR). Provide an example.

Answer 11

APR is the annualized rate on short-term investments (where the holding period is less than one year) and is often reported using simple rather than compound interest. For example, the APR for a credit card reporting 2% per month on your statement is 24% (2% x 12 months).

Question 12

Describe “effective annual rate” (EAR). Provide an example.

Answer 12

EAR is the annual rate of interest actually earned on an investment, calculated using compound rather than simple interest. It is used to compare different financial products that calculate annual interest with different compounding periods, e.g., daily, weekly, monthly, quarterly or semiannually. Candidates are not expected to calculate the EAR; however, to illustrate, the EAR for a credit card charging 2% per month = (1 + .02)^12 - 1 = 1.268 - 1 = 26.8%.

Question 13

Explain how the frequency of compounding impacts the difference between the APR and EAR. Provide an example.

Answer 13

An investment whose interest is compounded annually will have an EAR that is equal to its APR. Because APR quotes a yearly percentage rate regardless of compounding, when interest is compounded more frequently than annually, EAR is higher than the corresponding APR. For example, if the interest on the same investment was compounded quarterly, the EAR would then be higher than the APR because the investor would have the opportunity to reinvest the quarterly interest payments.

The difference between APR and EAR grows with the frequency of compounding. This is because with more frequent compounding, the investor earns interest on any interest paid or credited at the end of each compounding period during the year.

For example, compare the APR of 12% to the EAR on a $100 investment with 12% interest compounded semiannually. At the end of the year the investment is worth $112.36 (calculated as $100 × 1.06 × 1.06), and the EAR is 12.36%.

Question 14

Identify factors impacting an investment’s holding period return (HPR)

Answer 14

HPR is the total return received from holding an investment (i.e., income plus changes in its value) over a specified period of time. HPR depends on three factors:

(a) The difference between the asset price when it was purchased and the price at the end of the holding period

(b) Any income in the form of dividends over the holding period

(c) Any income in the form of interest over the holding period.

Question 15

Assume you are considering investing in a stock index fund. The fund currently sells for $100 per share, and your time horizon is one year. You expect a $5 annual cash dividend to be paid at the end of the holding period. The projected stock price one year from now is estimated to be $115 per share. Calculate your expected HPR

Answer 15

Using the HPR formula:

HPR = E(nding price of investment−Beginning price+Income)/Beginning price

= $115−$100+$5$100

= $20$100

= .20 or 20%

Your expected HPR is 20%.

Question 16

Assume you are considering investing in a stock index fund. The fund currently sells for $100 per share, and your time horizon is one year. You expect the cash dividend during the year to be $2.50. Instead of paying dividends at the end of the holding period, as was the case in Learning Outcome 3.2, dividends are paid semiannually. The stock price one year from now is projected to be $115 per share. Explain the impact of semiannual dividend payments on the HPR.

Answer 16

The HPR of this investment is the same as the HPR in the example in Learning Outcome 3.2 regardless of whether interest is paid once a year, twice a year or in any other frequency. The formula for HPR ignores reinvestment income earned on the dividends paid at six months and the end of the holding period.

Question 17

Assume you purchased a bond 45 days ago for $750. You received $20 in interest and sold the bond for $740. Using the HPR formula, calculate your HPR.

Answer 17

Using the HPR formula:

HPR = (Ending price of investment−Beginning price+Income) / (Beginning price)

= $740−$750+$20$750

= $10$750

= 0.01333 or 1.33%

Your HPR is 1.33%.

Question 18

Assume you have $5,000 to invest for the next year and are considering the following alternatives:

(a) A money market fund with an average maturity of 30 days offering a current yield of 6% per year

(b) A one-year savings deposit at a bank offering an interest rate of 7.5%

Explain how the HPR of each alternative will be affected if interest rates change in the year.

Answer 18

(a) Money market fund. The 6% yield holds for the first 30 days. Your HPR on the money market fund depends on the 30-day interest rates that apply as the maturing securities roll over at the end of each month over the full year.

(b) Savings deposit. The one-year savings deposit will provide a 7.5% HPR for the year. If the rate on money market instruments is expected to rise significantly above the current yield of 6% and stay there for a significant part of the year, then the money market fund could result in a higher HPR for the year than the savings deposit.

Question 19

Calculate the expected return for an investment you are considering, using the data
about market conditions and investment returns shown below.

Answer 19

State of the Market

Probability (%)

HPR (%)

Excellent

18%

25%

Good

64%

18%

Poor

18%

8%

Using the expected return formula:

E(r) = ∑s
p(s)r(s)

Where:

E(r) = Expected return
∑s
= Sum of each scenario
p(s) = Probability of each scenario
r(s) = HPR of each scenario
E(r) = (.18 x .25) + (.64 x .18) + (.18 x .08)

= .045 + .1152 + .0144

= .1746 or 17.46%

The expected return on your investment is 17.46%.

Question 20

Discuss the relevance of standard deviation and variance to investment risk.

Answer 20

The standard deviation of the rate of return is used to measure risk in the stock market or investment portfolios. The underlying assumption is that the majority of price activity follows a pattern known as a “normal distribution.” When a “normal distribution” exists, approximately 68% of all values fall within one standard deviation (either above or below) of the mean return of the entire data set, and values are within (again either above or below) two standard deviations approximately 95% of the time.

When observations in the data set are very close to the average, or mean, the standard deviation will be low, and the portfolio being measured can be characterized as “low risk.” When observations in the data set are spread over a wide range of values, the standard deviation will be high and the portfolio being measured can be categorized as “high risk,” or at a minimum, higher risk. Applying this measure to returns achieved by investment portfolios can be used to identify the relationship between the risk levels of the portfolios being measured.

Variance is a measure that can express uncertainty in a single number, and it is based on the standard deviation. To prevent the possibility of positive deviations cancelling out negative deviations, it is calculated by squaring the standard deviation from the mean. It is identified as σ2.

Question 21

Assume an investment has a standard deviation of 3.19% and an expected return of 19.75%. Describe the expected range of returns, assuming that a normal distribution exists.

Answer 21

Assuming a normal distribution, there is a 68.26% probability that the actual return will fall within 3.19% (one standard deviation) of the expected return of 19.75%. In other words, it is 68.26% likely that the return will fall within 16.56% and 22.94%. There is a 31.74% chance the actual return would fall outside this range—and a 15.87% chance the actual return will be more or less than 16.56%. If the standard deviation were lower, more of the expected returns would be clustered closer to the mean; if it were higher, the expected returns would be more widely dispersed

Question 22

Define the terms “risk-free rate,” “risk premium” and “excess return.”

Answer 22

The “risk-free rate” is the interest rate that you can earn with certainty by leaving money in risk-free assets such as T-bill, money market funds, or the bank. The “risk premium” is the expected return on an investment in excess of the risk-free rate (i.e., returns provided on risk-free securities.) The risk premium compensates for the risk of an investment. For example, if the risk-free rate is 5% per year and the expected return on an investment is 8.75%, then its risk premium is 3.75%. The difference between the actual rate of return on a risky asset and the risk-free rate is the “excess return.” Therefore, the risk premium is the expected excess return.

Question 23

Explain how risk aversion affects investment decision making.

Answer 23

The degree to which investors are willing to commit funds to risky investments depends on their level of risk aversion.

Risk preference falls on a continuum, with risk-averse falling to the far left of the continuum, risk-neutral in the middle and risk-loving falling to the far right.

Investors who are on the right of the continuum (risk-loving) are more willing to accept greater risk to obtain higher returns than investors who are on the left of the continuum (risk-averse). Risk-neutral investors find the level of risk irrelevant and consider only the expected return of prospective investments.

Investors are risk-averse in the sense that if the risk premium were zero, they would not invest any money in stocks. In theory, there must always be a positive risk premium on stocks in order to induce risk-averse investors to hold the existing supply of stocks instead of placing all their money in risk-free assets.

Question 24

Describe the Sharpe ratio and explain how it assists in the assessment of an investment manager.

Answer 24

The Sharpe ratio is widely used to evaluate the performance of investment managers. It identifies the relationship between the excess returns achieved by the manager and the amount of risk associated with the investment portfolio being assessed. As such it is a measure for calculating risk-adjusted return.

The Sharpe ratio tells an investor what portion of a portfolio’s performance is associated with risk-taking using the standard deviation as the measure of risk. It isolates the expected excess return that a risky portfolio could be expected to generate per unit of portfolio return variability.

The numerator of the Sharpe ratio identifies the amount of investment return achieved by the portfolio that exceeds the return available from a risk-free portfolio. The denominator measures the risk of the portfolio, or expected volatility, using the standard deviation to measure that risk.

Generally:

The higher the Sharpe ratio, the more return the investor is receiving for each unit of risk carried (i.e., the more attractive the risk-adjusted return).
The lower the Sharpe ratio, the more risk the investor is shouldering to earn additional returns (i.e., the less attractive the risk-adjusted return).
A ratio of 1 or better is considered good; 2 or better is very good, and 3 or better is considered excellent.
A portfolio of risk-free assets would have a Sharpe ratio of zero.
Because it uses only two portfolio measures (mean and standard deviation of returns) the Sharpe ratio is suitable to apply as a measurement of investment strategies that have normal expected return distributions. It is not suitable for measuring investments that are expected to have asymmetric returns.

Question 25

A portfolio expects a return of 7%, a standard deviation of 5%, when the return on risk-free assets is 4%. Calculate its Sharpe ratio and comment on the attractiveness of this portfolio in terms of its risk-adjusted rate of return.

Answer 25

Using the Sharpe ratio formula:

rp−rfσp

Where:

rp = Return of portfolio
rf = Risk-free rate
σp = Standard deviation of portfolio return
Sharpe ratio = (.07−.04)/.05

= .03/.05

Sharpe ratio = .60

This Sharpe ratio shows that an investor will receive only 0.6 units of return for each unit of risk. Because the Sharpe ratio is less than 1, this portfolio will not offer an attractive risk-adjusted rate of return.

Question 26

A portfolio expects a return of 10%, a standard deviation of 4% when the return on risk-free assets is 6%. Calculate its Sharpe ratio and comment on this portfolio’s attractiveness in terms of its risk-adjusted return

Answer 26

Using the Sharpe ratio formula:

(rp−rf)/σp

Where:

rp = Return of portfolio
rf = Risk-free rate
σp = Standard deviation of portfolio return
Sharpe ratio = .10−.06.04

= .04/.04

= 1.0

This Sharpe ratio shows that an investor will receive 1.0 units of return for each unit of risk. Because the Sharpe ratio 1.0 is considered good, this portfolio will offer an attractive risk-adjusted rate of return.

Question 27

Explain why Portfolio A, with an expected Sharpe ratio of 1.0 is preferred over Portfolio B, with an expected Sharpe ratio of 0.50.

Answer 27

The Sharpe ratio tells an investor what portion of a portfolio’s performance is associated with risk-taking. It isolates the expected excess return that a risky portfolio can be expected to generate per unit of portfolio return vatiability.an investor receives from holding a riskier asset. Generally, the higher the value of the Sharpe ratio, the more attractive the risk-adjusted return (i.e., the more return the investor receives per unit of risk shouldered).

Therefore, Portfolio A is preferred over Portfolio B. In practice, a ratio of 1 or better is considered good; 2 or better is very good, and 3 or better is considered excellent.

Question 28

Explain why the concept of normal distribution is significant for investment management decision making.

Answer 28

Investment management is much easier when expected rates of return can be represented by the normal distribution for four reasons:

(1) The normal distribution is symmetrical, allowing standard deviation to be the single required measure of risk.

(2) The normal distribution can be characterized as stable.

(3) Only two parameters (mean returns and standard deviation) need to be used to obtain the probabilities of future scenarios.

(4) A normal distribution means that a single correlation coefficient can be used to summarize the statistical relationship between returns.