Which of the following equations is NOT equal to the total return of an asset?
Let ##p_0## be the current price, ##p_1## the expected price in one year and ##c_1## the expected income in one year.
All except answer d are mathematically equal to the total required return.
###\begin{aligned} r_\text{total} &= \dfrac{c_1+p_1-p_0}{p_0} \\ &= \dfrac{c_1}{p_0} + \dfrac{p_1-p_0}{p_0} \\ &= \dfrac{c_1}{p_0} + \dfrac{p_1}{p_0} - 1 \\ &= \dfrac{c_1+p_1}{p_0} - 1 \\ \end{aligned}###Total cash flows can be broken into income and capital cash flows.
What is the name given to the cash flow generated from selling shares at a higher price than they were bought?
A capital gain is earned when an asset price rises. The capital gain is the sale price at the end ##(P_1)## less the buy price at the start ##(P_0)##.
###\text{Capital gain} = P_1 - P_0###The capital return is the capital gain scaled by the buy price ##(P_0)##.
###\text{Capital return} = r_\text{capital} = \dfrac{P_1 - P_0}{P_0} = \dfrac{P_1}{P_0} - 1###Question 543 price gains and returns over time, IRR, NPV, income and capital returns, effective return
For an asset price to triple every 5 years, what must be the expected future capital return, given as an effective annual rate?
There's no mention of income cash flows such as dividends or rent so we'll ignore them. The capital return ##(r_\text{cap})## leading to the price rise can be calculated using the 'present value of a single cash flow' formula.
###P_0 = \dfrac{P_{5}}{(1+r_\text{cap})^{5}} ###For the price to triple in 5 years, then ##P_{5}## will be triple ##P_0##, so ##P_{5} = 3P_0##. Substitute this into the above equation and solve for the capital return.
###P_0 = \dfrac{3P_{0}}{(1+r_\text{cap})^{5}} ### ###\begin{aligned} (1+r_\text{cap})^{5} &= \dfrac{3P_{0}}{P_0} \\ &= 3 \\ \end{aligned}### ###1+r_\text{cap} = 3^{1/5} ### ###\begin{aligned} r_\text{cap} &= 3^{1/5} - 1 \\ &= 0.24573094 \\ \end{aligned}###A newly floated farming company is financed with senior bonds, junior bonds, cumulative non-voting preferred stock and common stock. The new company has no retained profits and due to floods it was unable to record any revenues this year, leading to a loss. The firm is not bankrupt yet since it still has substantial contributed equity (same as paid-up capital).
On which securities must it pay interest or dividend payments in this terrible financial year?
Coupon and principal payments on debt must be paid or else the firm will be forced to declare bankruptcy.
In most countries, corporate law only allows dividends on preferred stock and common stock to be paid out of retained profits. This prevents firms from issuing shares and simply paying shareholder's money back to them as dividends without actually investing in productive assets and earning any cash or profits, a common feature of Ponzi schemes.
Question 452 limited liability, expected and historical returns
What is the lowest and highest expected share price and expected return from owning shares in a company over a finite period of time?
Let the current share price be ##p_0##, the expected future share price be ##p_1##, the expected future dividend be ##d_1## and the expected return be ##r##. Define the expected return as:
##r=\dfrac{p_1-p_0+d_1}{p_0} ##
The answer choices are stated using inequalities. As an example, the first answer choice "(a) ##0≤p<∞## and ##0≤r< 1##", states that the share price must be larger than or equal to zero and less than positive infinity, and that the return must be larger than or equal to zero and less than one.
Due to limited liability, shares in companies will never have a negative price. Therefore the lowest possible share price is zero. The highest possible share price is infinity, there's no limit to how much the shares could be worth. Therefore: ###0≤p<∞###
Returns are a function of price. The highest possible return is also infinite similar to price, but the lowest possible return will occur at the lowest possible future price of zero, so substituting ##p_1 = 0## and ##d_1 = 0## into the return formula: ###\begin{aligned} r &= \dfrac{p_1-p_0+d_1}{p_0} \\ r_\text{min} &= \dfrac{0-p_0+0}{p_0} \\ &= -1 \\ \end{aligned}###
So the lowest possible return is -1 which is -100%. This makes sense since you can't lose more than what you invest into the share. Therefore the bounds of returns are:
##-1≤r<∞##The below screenshot of Microsoft's (MSFT) details were taken from the Google Finance website on 28 Nov 2014. Some information has been deliberately blanked out.
What was MSFT's market capitalisation of equity?
The market capitalisation of equity ##(E)## equals the number of shares ##(n)## multiplied by the market share price ##(p)##.
###\begin{aligned} E &= n.p \\ &= 8.24b \times 47.81 \\ &= 393.95b \\ \end{aligned}###Question 524 risk, expected and historical returns, bankruptcy or insolvency, capital structure, corporate financial decision theory, limited liability
Which of the following statements is NOT correct?
Stocks are more risky than debt because in the event of bankruptcy, stock holders have a residual claim on the firm's assets and are only paid out if there's anything left over after the debt holders are paid. Debt holders have first claim on the firm's assets so in the event of bankruptcy they're paid out first and thus have a higher chance of being paid when the firm's assets are liquidated.
Firms' expected future stock returns are always higher than their expected future debt returns since stocks are more risky than debt and deserve a higher return.
However, firms' past realised stock returns can be higher or lower than their past realised debt returns. This is because the firm may have had a few bad years where profits were lower than expected and the share price fell, but the debt can still be repaid so its price stayed the same. In this case the past realised returns on the shares were negative, but the returns on debt were zero.
Question 992 inflation, real and nominal returns and cash flows
You currently have $100 in the bank which pays a 10% pa interest rate.
Oranges currently cost $1 each at the shop and inflation is 5% pa which is the expected growth rate in the orange price.
This information is summarised in the table below, with some parts missing that correspond to the answer options. All rates are given as effective annual rates. Note that when payments are not specified as real, as in this question, they're conventionally assumed to be nominal.
| Wealth in Dollars and Oranges | ||||
| Time (year) | Bank account wealth ($) | Orange price ($) | Wealth in oranges | |
| 0 | 100 | 1 | 100 | |
| 1 | 110 | 1.05 | (a) | |
| 2 | (b) | (c) | (d) | |
Which of the following statements is NOT correct? Your:
The real interest rate is exactly equal to 4.7619048% pa using the Fisher formula:
###\begin{aligned} 1+r_\text{real} &= \dfrac{1+r_\text{nominal}}{1+r_\text{inflation}} \\ &= \dfrac{1+0.1}{1+0.05} \\ \end{aligned}### ###r_\text{real} = 0.047619048###You can see that this is the rate at which your wealth grows by in real terms in oranges.
The real interest rate is approximately equal to 5% pa:
###\begin{aligned} r_\text{real} &\approx r_\text{nominal} - r_\text{inflation} \\ &\approx 0.1 - 0.05 \\ &\approx 0.05 \\ \end{aligned}###But this is just a rough approximation.
The below table shows all missing values:
| Wealth in Dollars and Oranges | ||||
| Time (year) | Bank account wealth ($) | Orange price ($) | Wealth in oranges | |
| 0 | 100 | 1 | 100 | |
| 1 | 110 | 1.05 | 104.7619048 | |
| 2 | 121 | 1.1025 | 109.7505669 | |
Question 578 inflation, real and nominal returns and cash flows
Which of the following statements about inflation is NOT correct?
Interest rates at the bank are almost always quoted as nominal rates, not real rates. When a bank advertises an interest rate of say 5% pa on deposits, you would always assume that this is a nominal rate. Therefore it has not been reduced by inflation. The real rate would be lower.
Question 576 inflation, real and nominal returns and cash flows
What is the present value of a nominal payment of $1,000 in 4 years? The nominal discount rate is 8% pa and the inflation rate is 2% pa.
The nominal cash flow can be discounted by the nominal discount rate.
###\begin{aligned} V_0 &= \dfrac{V_\text{4, nominal}}{(1+r_\text{nominal})^4} \\ &= \dfrac{1,000}{(1+0.08)^4} \\ &= 735.0298528 \end{aligned}###Question 522 income and capital returns, real and nominal returns and cash flows, inflation, real estate
A residential investment property has an expected nominal total return of 6% pa and nominal capital return of 2.5% pa. Inflation is expected to be 2.5% pa.
All of the above are effective nominal rates and investors believe that they will stay the same in perpetuity.
What are the property's expected real total, capital and income returns?
The answer choices below are given in the same order.
The nominal total return and capital return are given, therefore the nominal income return can be calculated.
###r_\text{nominal, total} = r_\text{nominal, income} + r_\text{nominal, capital} ### ###0.06 = r_\text{nominal, income} + 0.025 ### ###\begin{aligned} r_\text{nominal, income} &= 0.06 - 0.025\\ &= 0.035 \\ \end{aligned}###The Fisher equation can be used to convert nominal rates to real rates. The exact version is:
###1+r_\text{real} = \dfrac{1+r_\text{nominal}}{1+r_\text{inflation}}###The approximation is:
###r_\text{real} \approx r_\text{nominal} - r_\text{inflation}###But the Fisher equation only applies to the total and capital returns, not the income return. This is obvious when considering the approximation of the Fisher equation. If inflation is subtracted from both the nominal capital and income returns, then since the total return is the sum of these two, inflation will be subtracted twice from the total return which is wrong.
Method 1: Fisher equation on total and capital returns
Work out the total and capital returns using the Fisher equation, then calculate the difference which is the income return.
To find the real total return:
###1+r_\text{real, total} = \dfrac{1+r_\text{nominal, total}}{1+r_\text{inflation}}### ###1+r_\text{real, total} = \dfrac{1+0.06}{1+0.025}### ###r_\text{real, total} = \dfrac{1+0.06}{1+0.025}-1 = 0.034146341###To find the real capital return:
###1+r_\text{real, capital} = \dfrac{1+r_\text{nominal, capital}}{1+r_\text{inflation}}### ###1+r_\text{real, capital} = \dfrac{1+0.025}{1+0.025}### ###r_\text{real, capital} = \dfrac{1+0.025}{1+0.025}-1 = 0###To find the real income return:
###r_\text{real, total} = r_\text{real, income} + r_\text{real, capital} ### ###0.034146341= r_\text{real, income} + 0### ###\begin{aligned} r_\text{real, income} &= 0.034146341\\ \end{aligned}###Method 2: Convert nominal cash flows to real cash flows
Discount all future nominal cash flows by inflation to get the real cash flows then calculate the real rates of return.
###\begin{aligned} r_\text{nominal, total} &= r_\text{nominal, income} + r_\text{nominal, capital} \\ &= \dfrac{C_\text{1, nominal}}{P_0} + \dfrac{P_\text{1, nominal}-P_0}{P_0} \\ \end{aligned}\\ \begin{aligned} r_\text{real, total} &= r_\text{real, income} + r_\text{real, capital} \\ &= \dfrac{C_\text{1, real}}{P_0} + \dfrac{P_\text{1, real}-P_0}{P_0} \\ &= \dfrac{ \left( \dfrac{C_\text{1, nominal}}{(1+r_\text{inflation})^1} \right) }{P_0} + \dfrac{\left( \dfrac{P_\text{1, nominal}}{(1+r_\text{inflation})^1} \right)-P_0}{P_0} \\ \end{aligned}\\###If the price now were, say, $1 then the nominal income cash flow in one period would be $0.035 which is the nominal income return times the price now. The nominal price in one period would be $1.025 ##(=1(1+0.025)^1)## which is the price now grown by the nominal capital return. Note that the price now ##(P_0)## is not affected by inflation. Substituting these and inflation into the above equation, the real returns can be calculated:
###\begin{aligned} r_\text{real, total} &= \dfrac{ \left( \dfrac{0.035}{(1+0.025)^1} \right) }{1} + \dfrac{\left( \dfrac{1.025}{(1+0.025)^1} \right)-1}{1} \\ &= 0.03414634146 + 0 \\ &= 0.03414634146 \\ \end{aligned}###So the real total return is 3.41%, the real capital return is 0% and the real income return is 3.41%.
Thanks to Shahzada for his corrections to this solution.
Question 523 income and capital returns, real and nominal returns and cash flows, inflation
A low-growth mature stock has an expected nominal total return of 6% pa and nominal capital return of 2% pa. Inflation is expected to be 3% pa.
All of the above are effective nominal rates and investors believe that they will stay the same in perpetuity.
What are the stock's expected real total, capital and income returns?
The answer choices below are given in the same order.
The nominal total return and capital return are given, therefore the nominal income return can be calculated.
###r_\text{nominal, total} = r_\text{nominal, income} + r_\text{nominal, capital} ### ###0.06 = r_\text{nominal, income} + 0.02 ### ###\begin{aligned} r_\text{nominal, income} &= 0.06 - 0.02\\ &= 0.04 \\ \end{aligned}###The Fisher equation can be used to convert nominal rates to real rates. The exact version is:
###1+r_\text{real} = \dfrac{1+r_\text{nominal}}{1+r_\text{inflation}}###The approximation is:
###r_\text{real} \approx r_\text{nominal} - r_\text{inflation}###But the Fisher equation only applies to the total and capital returns, not the income return. This is obvious when considering the approximation of the Fisher equation. If inflation is subtracted from both the nominal capital and income returns, then since the total return is the sum of these two, inflation will be subtracted twice from the total return which is wrong.
Method 1: Fisher equation on total and capital returns
Work out the total and capital returns using the Fisher equation, then calculate the difference which is the income return.
To find the real total return:
###1+r_\text{real, total} = \dfrac{1+r_\text{nominal, total}}{1+r_\text{inflation}}### ###1+r_\text{real, total} = \dfrac{1+0.06}{1+0.03}### ###r_\text{real, total} = \dfrac{1+0.06}{1+0.03}-1 = 0.029126214###To find the real capital return:
###1+r_\text{real, capital} = \dfrac{1+r_\text{nominal, capital}}{1+r_\text{inflation}}### ###1+r_\text{real, capital} = \dfrac{1+0.02}{1+0.03}### ###r_\text{real, capital} = \dfrac{1+0.02}{1+0.03}-1 = -0.009708738###To find the real income return:
###r_\text{real, total} = r_\text{real, income} + r_\text{real, capital} ### ###0.029126214= r_\text{real, income} - 0.009708738### ###\begin{aligned} r_\text{real, income} &= 0.029126214 + 0.009708738\\ &= 0.038834951 \\ \end{aligned}###Method 2: Convert nominal cash flows to real cash flows
Discount all future nominal cash flows by inflation to get the real cash flows then calculate the real rates of return.
###\begin{aligned} r_\text{nominal, total} &= r_\text{nominal, income} + r_\text{nominal, capital} \\ &= \dfrac{C_\text{1, nominal}}{P_0} + \dfrac{P_\text{1, nominal}-P_0}{P_0} \\ \end{aligned}\\ \begin{aligned} r_\text{real, total} &= r_\text{real, income} + r_\text{real, capital} \\ &= \dfrac{C_\text{1, real}}{P_0} + \dfrac{P_\text{1, real}-P_0}{P_0} \\ &= \dfrac{ \left( \dfrac{C_\text{1, nominal}}{(1+r_\text{inflation})^1} \right) }{P_0} + \dfrac{\left( \dfrac{P_\text{1, nominal}}{(1+r_\text{inflation})^1} \right)-P_0}{P_0} \\ \end{aligned}\\###If the price now were, say, $1 then the nominal income cash flow in one period would be $0.04 which is the nominal income return times the price now. The nominal price in one period would be $1.02 ##(=1(1+0.02)^1)## which is the price now grown by the nominal capital return. Note that the price now ##(P_0)## is not affected by inflation. Substituting these and inflation into the above equation, the real returns can be calculated:
###\begin{aligned} r_\text{real, total} &= \dfrac{ \left( \dfrac{0.04}{(1+0.03)^1} \right) }{1} + \dfrac{\left( \dfrac{1.02}{(1+0.03)^1} \right)-1}{1} \\ &= 0.038834951 - 0.009708738 \\ &= 0.029126214 \\ \end{aligned}###So the real total return is 2.91%, the real capital return is -0.97% and the real income return is 3.88%.
Question 727 inflation, real and nominal returns and cash flows
The Australian Federal Government lends money to domestic students to pay for their university education. This is known as the Higher Education Contribution Scheme (HECS). The nominal interest rate on the HECS loan is set equal to the consumer price index (CPI) inflation rate. The interest is capitalised every year, which means that the interest is added to the principal. The interest and principal does not need to be repaid by students until they finish study and begin working.
Which of the following statements about HECS loans is NOT correct?
Actually, the low interest rate on the HECS loan advantages all domestic students. The low interest rate makes education cheaper, it's a subsidy for every student.
Economically rational rich (or poor) students would not voluntarily pay off their HECS debt early if they can invest their excess funds at a rate above the low HECS interest rate. For example, say the inflation rate was currently 2% and the bank paid 3% on deposits. The HECS debt interest rate would also be 2% and the student would be 1% pa (=3%-2%) better off depositing their excess funds in the bank rather than paying off their HECS debt. Therefore nobody is advantaged by paying off their debt sooner and avoiding HECS interest.
Question 739 real and nominal returns and cash flows, inflation
There are a number of different formulas involving real and nominal returns and cash flows. Which one of the following formulas is NOT correct? All returns are effective annual rates. Note that the symbol ##\approx## means 'approximately equal to'.
Answer c is the false statement. The real future value of a nominal future payment can be found by shrinking the nominal value by the inflation rate which requires division by ##(1+r_\text{inflation})^t##, not multiplication.
###V_\text{t,real} = \dfrac{V_\text{t,nominal}}{(1+r_\text{inflation})^t}###This makes sense since the real value of a future payment is always less than the nominal value when inflation is positive.
All other statements are true. Answer a is the approximate Fisher equation while answer b is the exact Fisher equation.
Note that a present value ##(V_0)## is both real and nominal, there's no distinction. This is because positive inflation only makes nominal future values bigger than real future values after some time has elapsed. Since there's no time for inflation to have an effect because the values are at the present, then the real present value is equal to the nominal present value. So ##V_\text{0} = V_\text{0, real} = V_\text{0, nominal}##.