Question 278 inflation, real and nominal returns and cash flows
Imagine that the interest rate on your savings account was 1% per year and inflation was 2% per year.
Inflation is the proportional increase in price levels. An inflation rate of 2% means that a product that costs $10 now will cost $10.20 (=10(1+0.02)1) in one year.
If you have $1,000 in the bank right now, you can buy 100 (=1,000/10) products.
The bank interest rate is 1% so $1,000 in the bank will grow to be $1,010 (=1,000(1+0.01)1) in one year. Product prices are $10.20 at this time, so we can only buy 99 (or 99.02 =1,010/10.20) products rather than 100 products before.
Economist's method
An economist would say that the higher inflation rate has eroded our buying power.
A short cut to doing the calculations above is to find the real return using the Fisher equation,
###\begin{aligned} 1+r_\text{real} &= \frac{1+r_\text{nominal}}{1+r_\text{inflation}} \\ &= \frac{1+0.01}{1+0.02} \\ \end{aligned}###
###\begin{aligned} r_\text{real} &= \frac{1+0.01}{1+0.02} -1 \\ &= 0.990196078 -1 \\ &= -0.009803922 = -0.9803922\% \\ \end{aligned}###
This says that our real return is negative, so our wealth buys 0.98% less after one year, so instead of buying 100 products we can only buy 99.02 (=100(1-0.0098)) products in one year.
Note that the exact Fisher equation can be approximated:
###\begin{aligned} r_\text{real} &\approx r_\text{nominal} - r_\text{inflation} \\ &= 0.01 -0.02 \\ &= -0.01 = -1\%\\ \end{aligned}###
Commentary
This question was used in the '2004 Health and Retirement Survey' of Americans over the age of 50. The survey results were as follows:
- 75.2% of respondents answered it correctly,
- 13.4% were incorrect,
- 9.9% answered "don't know" and
- 1.5% refused to answer.
This question tests knowledge of inflation and was used in the research paper 'Financial Literacy and Planning: Implications for Retirement Wellbeing' by Annamaria Lusardi and Olivia S. Mitchell in 2011.
Question 295 inflation, real and nominal returns and cash flows, NPV
When valuing assets using discounted cash flow (net present value) methods, it is important to consider inflation. To properly deal with inflation:
(I) Discount nominal cash flows by nominal discount rates.
(II) Discount nominal cash flows by real discount rates.
(III) Discount real cash flows by nominal discount rates.
(IV) Discount real cash flows by real discount rates.
Which of the above statements is or are correct?
Nominal cash flows can be discounted using nominal discount rates. Also, real cash flows can be discounted using real discount rates. Both will give the same asset price.
###C_\text{0} = \dfrac{C_\text{t, nominal}}{(1+r_\text{nominal})^t} = \dfrac{C_\text{t, real}}{(1+r_\text{real})^t}###If the cash flows are nominal and the discount rate is real or vice-versa, it's usually easier to convert the discount rate to a nominal or real rate using the Fisher equation, and then discount the cash flows to arrive at the correct price.
###1+r_\text{real} = \dfrac{1+r_\text{nominal}}{1+r_\text{inflation}}###Cash flows can also be converted from nominal to real or vice versa using the inflation rate.
###C_\text{t, real} = \dfrac{C_\text{t, nominal}}{(1+r_\text{inflation})^t}###In the 'Austin Powers' series of movies, the character Dr. Evil threatens to destroy the world unless the United Nations pays him a ransom (video 1, video 2). Dr. Evil makes the threat on two separate occasions:
- In 1969 he demands a ransom of $1 million (=10^6), and again;
- In 1997 he demands a ransom of $100 billion (=10^11).
If Dr. Evil's demands are equivalent in real terms, in other words $1 million will buy the same basket of goods in 1969 as $100 billion would in 1997, what was the implied inflation rate over the 28 years from 1969 to 1997?
The answer choices below are given as effective annual rates:
To find the inflation rate that makes $1 million grow into $100 billion in 28 years, use the 'present value of a single cash flow' equation:
###V_{0} = \dfrac{V_{t}}{(1+r_\text{inflation})^t} ### ###V_{1969} = \dfrac{V_{1997}}{(1+r_\text{inflation})^{28}} ### ###1,000,000 = \dfrac{100,000,000,000}{(1+r_\text{inflation})^{28}} ### ###(1+r_\text{inflation})^{28} = \dfrac{100,000,000,000}{1,000,000}### ###1+r_\text{inflation} = \left( \dfrac{100,000,000,000}{1,000,000} \right)^{1/28}### ###\begin{aligned} r_\text{inflation} &= (100,000)^{1/28} - 1 \\ &= 0.508591 \\ \end{aligned}###That's an unreasonably high inflation rate of more than 50% pa! This may indicate that in real terms, Dr. Evil's demands were much higher in 1997 compared to 1969.
Question 353 income and capital returns, inflation, real and nominal returns and cash flows, real estate
A residential investment property has an expected nominal total return of 6% pa and nominal capital return of 3% pa.
Inflation is expected to be 2% pa. All rates are given as effective annual rates.
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} + 0.03 ### ###\begin{aligned} r_\text{nominal, income} &= 0.06 - 0.03\\ &= 0.03 \\ \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+0.02}### ###r_\text{real, total} = \dfrac{1+0.06}{1+0.02}-1 = 0.039215686 ###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+0.03}{1+0.02}### ###r_\text{real, capital} = \dfrac{1+0.03}{1+0.02}-1 = 0.009803922 ###To find the real income return:
###r_\text{real, total} = r_\text{real, income} + r_\text{real, capital} ### ###0.039215686 = r_\text{real, income} + 0.009803922 ### ###\begin{aligned} r_\text{real, income} &= 0.039215686 - 0.009803922 \\ &= 0.029411765 \\ \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 $0.03 which is the nominal income return times the price now. The nominal price in one period would be $1.03 ##(=1(1+0.03)^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{0.03}{(1+0.02)^1} \right) }{1} + \dfrac{\left( \dfrac{1.03}{(1+0.02)^1} \right)-1}{1} \\ &= 0.029411765 + 0.009803922 \\ &= 0.039215686 \\ \end{aligned}###So the real total return is 3.92%, the real capital return is 0.98% and the real income return is 2.94%.
Question 363 income and capital returns, inflation, real and nominal returns and cash flows, real estate
A residential investment property has an expected nominal total return of 8% pa and nominal capital return of 3% pa.
Inflation is expected to be 2% pa. All rates are given as effective annual rates.
What are the property's expected real total, capital and income returns? The answer choices below are given in the same order.
The Fisher equation can be used to calculate nominal and 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}###The only problem is that the Fisher equation above 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 the capital return and the income return, then since the total return is the sum of the capital and income return, 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+0.08}{1+0.02}### ###r_\text{real, total} = \dfrac{1+0.08}{1+0.02}-1 = 0.058823529 ###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+0.03}{1+0.02}### ###r_\text{real, capital} = \dfrac{1+0.03}{1+0.02}-1 = 0.009803922###To find the real income return:
###\begin{aligned} r_\text{real, total} &= r_\text{real, income} + r_\text{real, capital} \\ 0.058823529&= r_\text{real, income} + 0.009803922 \\ \end{aligned}### ###\begin{aligned} r_\text{real, income} &= 0.058823529 - 0.009803922 \\ &= 0.049019608\\ \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 $0.05 (nominal income return times price now) and the nominal price in one period would be $1.03 (nominal capital return times price now). Note that the price now 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{0.05}{(1+0.02)^1} \right) }{1} + \dfrac{\left( \dfrac{1.03}{(1+0.02)^1} \right)-1}{1} \\ &= 0.049019608 + 0.009803922 \\ &= 0.058823529\\ \end{aligned}###So the real total return is 5.88%, the real capital return is 0.98% and the real income return is 4.90%.
Question 407 income and capital returns, inflation, real and nominal returns and cash flows
A stock has a real expected total return of 7% pa and a real expected capital return of 2% pa.
Inflation is expected to be 2% pa. All rates are given as effective annual rates.
What is the nominal expected total return, capital return and dividend yield? The answers below are given in the same order.
The Fisher equation can be used to calculate nominal and 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}###The only problem is that 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 added to the real capital and income returns, then since the real total return is the sum of these two, inflation will be added twice to 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 nominal total return:
###1+r_\text{real, total} = \dfrac{1+r_\text{nominal, total}}{1+r_\text{inflation}}### ###1+0.07 = \dfrac{1+r_\text{nominal, total}}{1+0.02} ### ###r_\text{nominal, total} = (1+0.07)(1+0.02)-1 = 0.0914 ###To find the nominal capital return:
###1+r_\text{real, capital} = \dfrac{1+r_\text{nominal, capital}}{1+r_\text{inflation}}### ###1+0.02 = \dfrac{1+r_\text{nominal, capital}}{1+0.02} ### ###r_\text{nominal, capital} = (1+0.02)(1+0.02)-1 = 0.0404 ###To find the real income return:
###\begin{aligned} r_\text{nominal, total} &= r_\text{nominal, income} + r_\text{nominal, capital} \\ 0.0914 &= r_\text{nominal, income} + 0.0404 \\ \end{aligned}### ###\begin{aligned} r_\text{nominal, income} &= 0.0914 - 0.0404 \\ &= 0.051 \\ \end{aligned}###Method 2: Convert nominal cash flows to real cash flows
Grow all future real cash flows by inflation to get the nominal cash flows then calculate the nominal 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} \\ &= \dfrac{C_\text{1, real}.(1+r_\text{inflation})^1}{P_0} + \dfrac{P_\text{1, real}.(1+r_\text{inflation})^1-P_0}{P_0} \\ \end{aligned}###If the price now were, say, $1 then the nominal income cash flow in one period would be $0.05 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{nominal, total} &= \dfrac{C_\text{1, real}.(1+r_\text{inflation})^1}{P_0} + \dfrac{P_\text{1, real}.(1+r_\text{inflation})^1-P_0}{P_0} \\ &= \dfrac{0.05 \times (1+0.02)^1}{1} + \dfrac{1.02 \times (1+0.02)^1-1}{1} \\ &= 0.051 + 0.0404 \\ &= 0.0914 \\ \end{aligned}###So the real total return is 9.14%, the real capital return is 4.04% and the real income return is 5.1%.
Question 155 inflation, real and nominal returns and cash flows, Loan, effective rate conversion
You are a banker about to grant a 2 year loan to a customer. The loan's principal and interest will be repaid in a single payment at maturity, sometimes called a zero-coupon loan, discount loan or bullet loan.
You require a real return of 6% pa over the two years, given as an effective annual rate. Inflation is expected to be 2% this year and 4% next year, both given as effective annual rates.
You judge that the customer can afford to pay back $1,000,000 in 2 years, given as a nominal cash flow. How much should you lend to her right now?
Use the 'term structure of interest rates' or 'expectations hypothesis' equation to find the expected inflation rate over the next 2 years:
###(1+r_{0 \rightarrow 2 \text{ eff annual}} )^2 = (1+r_{0 \rightarrow 1 \text{ eff annual}})(1+r_{1 \rightarrow 2 \text{ eff annual}}) ### ###(1+r_{0 \rightarrow 2 \text{ eff annual}})^2 = (1+0.02)(1+0.04) ### ###\begin{aligned} r_{0 \rightarrow 2 \text{ eff annual}} &= ((1+0.02)(1+0.04))^{1/2}-1 \\ &= 0.029951455 \\ \end{aligned}###Use the Fisher equation to convert the required real return into a nominal return:
###1+r_{0 \rightarrow 2 \text{ eff annual, real}} = \dfrac{1+r_{0 \rightarrow 2 \text{ eff annual, nominal}}}{1+r_{0 \rightarrow 2 \text{ eff annual, inflation}}}### ###1+0.06 = \dfrac{1+r_{0 \rightarrow 2 \text{ eff annual, nominal}}}{1+0.029951455}### ###\begin{aligned} r_{0 \rightarrow 2 \text{ eff annual, nominal}} &= (1+0.06)(1+0.029951455) -1 \\ &= 0.091748542 \\ \end{aligned}###To find the present value of the $1 million in 2 years that will be lent now,
###\begin{aligned} V_0 &= \dfrac{V_\text{2, nominal}}{(1+r_{0 \rightarrow 2 \text{ eff annual, nominal}})^2} \\ &= \dfrac{1,000,000}{(1+0.091748542)^2} \\ &= 838,986.086 \\ \end{aligned}###Question 744 income and capital returns, real and nominal returns and cash flows, inflation
If someone says "my shares rose by 10% last year", what do you assume that they mean? The effective annual:
The statement 'my shares rose by 10% last year' suggests that the nominal capital return was 10% pa over the last year. This is because a rising share price is a positive capital return. Nothing is said about the stock's dividend yield.
By default you would assume that all returns are nominal unless stated otherwise, so this is a nominal return and not a real return.
Since the return was in the past, it's a historical return, not an expected future return.
Question 732 real and nominal returns and cash flows, inflation, income and capital returns
An investor bought a bond for $100 (at t=0) and one year later it paid its annual coupon of $1 (at t=1). Just after the coupon was paid, the bond price was $100.50 (at t=1). Inflation over the past year (from t=0 to t=1) was 3% pa, given as an effective annual rate.
Which of the following statements is NOT correct? The bond investment produced a:
All statements are true except for (e). This is because the current stock price of $100 should not be discounted by the inflation rate since it is a value now that is both real and nominal, there's no need to convert it to real. Only the nominal stock price in one year of $100.50 should be discounted by the inflation rate to convert it into a real value in one year.
###\begin{aligned} P_\text{T real} &= P_\text{T nominal}/(1+r_\text{inflation})^T \\ \end{aligned}### ###\begin{aligned} r_\text{real capital} &= \dfrac{P_\text{1 real} - P_0}{P_0} \\ &= \dfrac{P_\text{1 nominal}/(1+r_\text{inflation})^1 - P_0}{P_0} \\ &= \dfrac{100.5/(1+0.03)^1 - 100}{100} \\ &= -0.024271845 \\ \end{aligned}###An alternative method to find the real capital return is to use the exact Fisher equation which gives the same solution.
Question 745 real and nominal returns and cash flows, inflation, income and capital returns
If the nominal gold price is expected to increase at the same rate as inflation which is 3% pa, which of the following statements is NOT correct?
The real income return of gold is zero since it pays nothing. No interest, dividends or rent. The nominal income yield on gold is also zero. Income yields are generally not affected much by inflation, unlike capital returns and total returns. One way to calculate the real income return from the nominal income return is:
###\begin{aligned} r_\text{real income} &= \dfrac{C_\text{1,real}}{P_0} \\ &= \dfrac{C_\text{1,nominal}/(1+r_\text{inflation})^1}{P_0} \\ &= \dfrac{\left(\dfrac{C_\text{1,nominal}}{P_0}\right)}{(1+r_\text{inflation})^1} \\ &= \dfrac{r_\text{nominal income}}{1+r_\text{inflation}} \\ &= \dfrac{0}{1+0.03} \\ &= 0 \\ \end{aligned}###