FIN3024 - Lecture Week 3 - Students

BSc (Hons) Accounting & FinanceInvestment Management FIN3024Week 3 - Risk and Return

Interest rates and financial assets

Bond Yields

BLR

FD Rates

Equity Prices

Interest rates affects the required return on other financial assets

Real vs Nominal Interest RatesNominal Interest Rate (R):▪ The growth rate of your money.▪ Includes an inflation (i) component.▪ Approximation: R ≈ r + i.

Real Interest Rate (r):▪ The growth rate of your purchasing power.

Exact calculation: derived from

Approximate then calculate the exact real interest rate:▪ Malaysia’s 2013 inflation rate▪ Maybank 12-month FD rate

iiRr

1 i

Rr

111

Equilibrium Real Rate of Interest

▪ If r is low, households will not be willing to save (low supply) and would want to borrow (creating high demand).

▪ If r is high, households will save (increasing supply) and will not wish to borrow (lowering demand).

▪ The market will be at equilibrium where demand = supply at point E.▪ The government and the central bank can shift the supply and demand

curves to the left or right. In this example, the government has increased the demand for funds (by budget deficit) and causes the equilibrium point to move from E to E’. The result is a higher real interest rate.

EE’

Gov’t

Supply

Demand

r

Amount of borrowed funds

equilibrium r

Equilibrium funds lent

Effective vs Nominal Interest Rates▪ Nominal in this sense, is different to the “nominal” used in

comparison to real interest rates.

When interest rates are quoted, they should state:▪ The horizon (per annum, per month, etc). If no specific horizon is

stated, assume that it is per annum.▪ The compounding frequency.

▪ If you want to take a loan, a bank may quote an interest rate of say, 8% per annum. The horizon is annual but this does not mean that the compounding frequency is also annually. In fact, the banks often compound interest monthly or quarterly or semi-annually.

▪ The quoted annual rate of 8% is a nominal interest rate.▪ The effective rate is the actual interest rate you will pay and has to

be calculated.

Example▪ If I have $100 invested at 8% per annum, how much do I accumulate if

interest is compounded: annually, semi-annually, quarterly, monthly?

▪ Annually At the end of the year I have: 100*(1+0.08) = 108.00

▪ Semi-annuallyEvery 6 months I earn 8%/2 = 4%. At the end of the year I have: 100*(1+0.04)(1+0.04) = 108.16 OR: 100*(1+0.04)2 = 108.16

▪ Quarterly Every quarter I earn 8%/4 = 2%. At the end of the year I have: 100*(1.02)(1.02)(1.02)(1.02) = 108.24 OR: 100*(1.02)4 = 108.24

▪ Monthly:Every month I earn 8%/12 = 0.67%. At the end of the year I have: 100*(1.0067)12 = 108.30

Effective Interest Rate▪ We can use the previous calculations to determine the

effective interest rates. This is the measure of how much interest was actually earned.

▪ Annual compounding: 8% p.a. effective▪ Semi-annual compounding: 8.16% p.a. effective▪ Quarterly compounding: 8.24% p.a. effective▪ Monthly compounding: 8.30% p.a. effective▪ To convert interest rates from nominal to effective, use:

Where m is the compounding frequency, i is the nominal Interest rate and i is the effective interest rate per annum.

11

m

miEAR

Example▪ Maybank’s 1 month Fixed Deposit rate is 3% p.a. ▪ Assuming you deposited $10,000 into the FD above, and you

rollover the interest every month up to a year. How much will you have?

Month Amount0 10,0001 10,0252 10,050.063 10,075.194 10,100.385 10,125.636 10,150.947 10,176.328 10,201.769 10,227.26

10 10,252.8311 10,278.4612 10,304.16

Your annual return would be 3.0416%, which is the EAR for a nominal rate compounded monthly

3% per year so it is 0.25% per month.(3%/12 = 0.25%)

Maybank FD rateMaybank FD Rates

▪ What would be the EAR of 1,3,6 and 12 months Maybank Fixed Deposit rate, assuming that you rollover your interest earned for a year?

EAR vs APRIf we are given possible investment returns in the form of various nominal and effective rates, how can we decide which is best?

APR (Annual Percentage Rate) is an annualised simple interest rate. We have met this before in a different form: APR = I the nominal rate per annum. The comparison between APRs would not be useful because their compounding frequency would be different.

EAR (Effective Annual Rate) is typically used as a standardised rate to express all investment returns over one year. It is the percentage increase in funds over 1 year. We have met this before: EAR = i, the effective rate per annum.

The terminology used in everyday finance is APR and EAR.

Accumulating▪ If we have a principal of P, the accumulation after one year

is given by:

▪ If we accumulate for a number of years, n:

m

miP

1

mn

miP

1

ExamplesFind:(a) The simple interest on $1,000 for 2 years at 10% p.a.(b) The simple interest on $1,000 for 2 years at 1% per month.(c) The compound interest on $1,000 for 4 years at an effective

rate of 12% p.a.(d) The compound interest on $1,000 for 3 years at 10%

compounded semi-annually.(e) The accumulation of $1,000 if it is invested at 5%

compounded quarterly for 2 years.(f) The accumulation of $1,000 if it is invested at 8%

compounded monthly for 5 years.

Continuous Compounding▪ Without calculation, which is the better investment: $100

compounded annually at 12% p.a. or compounded monthly at 12%?

▪ The more frequently your interest is compounded, the higher the return. The extreme limit of compounding frequency is continuous compounding. This is like compounding every split-second. As the number of times compounding gets bigger, the smaller the affect (or the affect diminishes).

▪ Example: P=100,000 ; i=8%p.a. ;n=1– m=2 100,000(1.08/2)2x1 = 108,160– m=4 100,000(1.08/4)4x1 = 108,243– m=12 100,000(1.08/12)12x1 = 108,300– m=365 100,000(1.08/365)365x1 = 108,327– m=8,760 100,000(1.08/8,760)8,760x1 = 108,328– m=525,600 100,000(1.08/525,600)525,600x1 = 108,328

83572710.5

Continuous Compounding▪ In other words, we should be more familiar with the exponential

function (eX) on your calculator.

▪ Example– i=8%p.a. – n=1– m=continuous compounding

EAR = (e0.08)1-1 = 8.33%

1)( nieEAR

Discounting ▪ The opposite of accumulating is discounting. If I have an

accumulated value of S, I can calculate the principal, P by using:

Find the present value of $8,000 due in 5 years at :(a) an effective rate of 7% per annum(b) 7% compounded monthly(c) 7% compounded semi annually(d) 7% compounded daily(e) 7% compounded continuously

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n eSmiSiSP

ReturnsConsider a share bought for $100, paid dividends of $4 at the end of the year and worth $110 at the end of the year.▪ The holding period is how long the asset is held or how long

the return period is. In this example it is one year.▪ The dividend yield is the %return from dividends. 4/100 = 4%.▪ The capital gain is the return on the share value. 10/100 =

10%.▪ The holding period return (HPR) is the total return over the

investment period: 14/100 = 14%.▪ HPR = Capital Gain + Dividend Yield

.icehareBeginningS

idDividendPaicehareBeginningSiceeEndingSharHPRPrPrPr

Using Historical Data

▪ Generally, maximising your return is thought of as the ultimate goal of investment.

▪ Minimising investment risk is just as important. ▪ How do we predict future levels of risk and return? We can

use historical data as an estimate. ▪ There is no guarantee that the historical record exhibits the

worst (and best) that could occur in the future.▪ More on using historical data later.

Expected Returns▪ Future share price is uncertain, dividends are uncertain so HPR is

uncertain also. The possible outcomes are assigned probabilities.▪ The expected return is calculated as an “average” of what the

return might be.

▪ A share is now priced at $70. Calculate the expected return of the stock if you are given the following probability distribution:

▪ (0.25x97.25) + (0.5x81.50) + (0.2x65.20) + (0.05x40.10) = 80.11

ii

irprrE )(

Outlook Excellent Good Poor Crash

Probability 0.25 0.5 0.2 0.05

Yr–end Price 97.25 81.50 65.20 40.10

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irnrrE 1)(

Standard Deviation▪ The standard deviation (sd or std dev) of the rate of return is a

measure of the return’s uncertainty or risk.▪ It is a measure of the deviations from E(r).▪ σ (Sigma, lower case) is used to represent sd.

If different probabilities If all probabilities are same

▪ sd is the square root of the variance (var).▪ var = σ2 or σ = √var▪ σ2 = 0.25(97.25-80.11)2 + 0.5(81.5-80.11)2 + 0.2(65.2-80.11)2 +

0.05(40.10-80.11)2 = 446▪ σ = 14.10

22 i

ii rrp

222 1 rrn i

i 222 rrpi

ii

22 1 i

i rrn

Excess Returns & Risk Premiums▪ Risk-free rate (rf) is the rate earned on risk-free assets like T-Bills.▪ The risk premium is the expected rate of return above rf risk

premium = E(r) – rf

▪ The excess return is the actual rate of return above rf excess return = r – rf

▪ Risk Appetite refers to the degree of risk that investors are willing to undertake.

▪ Risk Aversion means an unwillingness to undertake excessive risk. A risk averse investor will only invest in a risky asset (eg. stocks) if there is a risk premium to compensate for the additional risk.

▪ Investors are assumed to be risk averse.

Sharpe RatioHow do we measure the trade-off between reward and risk?Which asset would you choose to invest in, x or y, if:a) E(rx) = 10%, σx = 9% and E(ry) = 10%, σy = 6%. b) E(rx) = 12%, σx = 8% and E(ry) = 10%, σy = 8%. c) E(rx) = 9%, σx = 7% and E(ry) = 12%, σy = 9%.

The Sharpe Ratio is a measure of reward to volatility. The higher the Sharpe Ratio, the higher the return to risk ratio for that asset.

▪ Suppose for part (c) above, that rf = 5%, which asset would you choose?▪ Asset X: Sharpe Ratio = (0.09-0.05)/0.07 = 0.57▪ Asset Y: Sharpe Ratio = (0.12-0.05)/0.09 = 0.78

turnsExcessofSDemiumRiskoSharpeRatiRe___

Pr_

Time Series Analysis▪ We can use historic time series data on securities to calculate

their past return and standard deviation.

▪ Arithmetic average: This method is useful to estimate what the expected return may be by averaging the past returns. For all data, we assign equal probabilities so if there are n observations, we give each one a probability of 1/n. We calculate the expected return as follows:

▪ This, however, does not give us an accurate reflection of the actual return over the past period.

n

s

n

s

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Geometric (Time-Weighted) Average

▪ The geometric average focuses on the holding period returns (HPR) of the security.

▪ First the terminal value is calculated: TV = (1+r1) x (1+r2) x (1+r3) x … x (1+rn)

▪ The geometric average, g, is calculated from:

▪ g is called the time-weighted average return as each past return receives an equal weight in the process of averaging.

nn

ii

n rTVg

1

1

1

)1()1(

Example

▪ Calculate the arithmetic and geometric averages of the following rates of return:

▪ Arithmetic Avg: E(r) = (1/5)(-0.1189-0.2210+0.2869+0.1088+0.0491) = 0.02098 = 2.1%

▪ Geometric avg: (1 + g) = ((1-0.1189)(1-0.2210) (1+0.2869) (1+0.1088) (1+0.0491))1/5 = 1.0054Therefore g = .0054 = 0.54%

▪ What accounts for the differences in the two results?

Period

2001 2002 2003 2004 2005

HPR -0.1189

-0.2210

0.2869 0.1088 0.0491

Year HPR Amount2000 10,0002001 -0.1189 8,811

2002 -0.221 6,863

2003 0.2869 8,832

2004 0.1088 9,7942005 0.0491 10,274

Geometric Example

▪ Return = (274/10,000)/5 = 0.0054 = 0.54%

▪ Same result as using the formula in previous slide.

Practice Questions▪ Over a period of 10 years, a stock had an end-of-year price of

54, 52, 61, 57, 58, 58, 57, 56, 55, 60. Calculate the average annual return on the stock. Calculate the std. dev. of the returns on this stock.

▪ The stock prices at the end of each of five weeks are: 20.10, 20.08, 20.05, 20.20, 20.60. Estimate the weekly rate of return and volatility of the stock.

▪ The end-of-month stock prices at the end of each of 6 months are: 33.50, 32.08, 35.69, 33.20, 34.50, 32.25. Estimate:

1. the monthly rate of return and volatility of the stock. 2. the annual rate of return and volatility of the stock.

Risk = Volatility = Std Dev▪ In finance, risk is volatility: basically how much a stock price

varies. In relation to your statistics knowledge, this risk is the standard deviation.

▪ In this comparison, which stock has a higher risk?

.

Gold vs Silver▪ By simply observing the graph, which appears riskier?

▪ So which would you expect to have the higher return?▪ What can you say about the correlation?

Gold or S&P 500?Which was riskier over the past year: Gold or S&P 500?

Does this relate to their volatility over the past 10 years?Has their correlation been similar to this also?

Interest Rates vs Return on StocksExampleMalaysia 3 month T-bill rate 3%Sunway REIT Annual Dividend 8.9 sen per shareClosing Price RM 1.32 per share Dividend Yield 0.089/1.32 = 6.74% dividend yieldSunway REIT pays 3.74% extra yield in return for risk premium.

T-bill rate increased to 3.20% (+20 basis points)Investors require higher return for risk premium.Investors require Sunway REIT dividend yields to be 6.94%, because risk free rate increase. Assuming dvd remains constant:0.089/x = 6.94% x = RM 1.28 per share

Interest Rates vs Return on StocksExampleMalaysia 3 month T-bill increased to 3.2%Sunway REIT Annual Dividend 8.9 sen per shareClosing PriceRM 1.32 per share Dividend Yield 0.089/1.32 = 6.74% dividend yield

Sunway REIT pays 3.74% extra yield in return for risk premium.