Fight Finance

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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

 

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Real wealth at t=2 is $97.9477702 or 97.9477702 apples. To calculate this, we can find the real interest rate and grow current wealth by that real rate to find the real wealth at time 2.

The real interest rate is exactly equal to -1.0314342% pa using the Fisher formula:

###\begin{aligned} 1+r_\text{real} &= \dfrac{1+r_\text{nominal}}{1+r_\text{inflation}} \\ &= \dfrac{1+0.0075}{1+0.018} \\ \end{aligned}### ###r_\text{real} = -0.010314342###

Real wealth at t=2 is then the current (t=0) wealth grown by the real interest rate over 2 years:

###\begin{aligned} \text{RealWealth}_2 &= \text{Wealth}_0(1+r_\text{real})^2 \\ &= \text{100}(1+-0.010314342)^2 \\ &= 97.9477702 \text{ apples or dollars}\\ \end{aligned}###

Alternatively, real wealth at t=2 can be found by finding how many apples you can buy at this time, so divide nominal wealth by the nominal price of apples at time 2:

###\begin{aligned} \text{RealWealth}_2 &= \dfrac{\text{NominalWealth}_2}{\text{ApplePrice}_2} \\ &= \dfrac{\text{Wealth}_0(1+r_\text{nominal})^2}{\text{ApplePrice}_0(1+r_\text{inflation})^2} \\ &= \dfrac{100(1+0.0075)^2}{1(1+0.018)^2} \\ &= \dfrac{101.505625}{1.036324} \\ &= 97.9477702 \text{ apples or dollars}\\ \end{aligned}###

The below table shows all missing values:

Wealth in Dollars and Apples
Time (year)	Bank account wealth ($)	Apple price ($)	Wealth in apples
0	100	1	100
1	100.75	1.018	98.96856582
2	101.505625	1.036324	97.9477702

 

Commentary

Note that the negative real interest rate in early 2020 is unusual. When the economy is growing well and inflation is high within the RBA's 2 to 3% pa target band, real rates are usually positive to entice investors to save (lend or buy debt) rather than borrow (sell debt). However, in early 2020, the Australian central bank was afraid of a recession so they kept the overnight money market rate very low at 0.75% pa to encourage consumers to borrow and spend rather than save. The central bank is implementing expansionary monetary policy by keeping the interest rate so low since it encourages higher consumption and higher economic growth.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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}###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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%.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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%.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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%.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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%.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Notes and coins pay no income and therefore have zero nominal and real income returns. Since $100 of notes and coins are worth $100 now and will still be worth $100 in one year, the nominal capital return is zero. The nominal total return is the sum of the nominal income and capital return therefore it's also zero.

However, $100 of notes and coins will buy less goods and services in one year than it will now if inflation is positive because the price of things will increase. Therefore the real capital return and real total returns of notes and coins are negative.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

No explanation provided.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The nominal cash flow cannot be discounted by a real discount rate.

Method 1: Discount the nominal cash flow by the nominal discount rate

The real discount rate can be converted into a nominal rate using the Fisher equation.

###1+r_\text{real} = \dfrac{1+r_\text{nominal}}{1+r_\text{inflation}}### ###1+0.1 = \dfrac{1+r_\text{nominal}}{1+0.03}### ###\begin{aligned} r_\text{nominal} &= (1+0.1)(1+0.03) -1 \\ &= 0.133333 \\ \end{aligned}###

Now discount the nominal payment by the nominal discount rate.

###\begin{aligned} V_0 &= \dfrac{V_\text{5, nominal}}{(1+r_\text{nominal})^5} \\ &= \dfrac{100}{(1+0.133333)^5} \\ &= 53.56121877 \end{aligned}###

Method 2: Discount the real cash flow by the real discount rate

The nominal cash flow can be converted into a real cash flow by discounting using the inflation rate.

###\begin{aligned} V_\text{5, real} &= \dfrac{V_\text{5, nominal}}{(1+r_\mathbf{inflation})^5} \\ &= \dfrac{100}{(1+0.03)^5} \\ &= 86.26087844 \end{aligned}###

Now discount the real payment by the real discount rate.

###\begin{aligned} V_0 &= \dfrac{V_\text{5, real}}{(1+r_\text{real})^5} \\ &= \dfrac{86.26087844}{(1+0.1)^5} \\ &= 53.56121877 \end{aligned}###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

If all goods and services' nominal prices grow by the same inflation rate, then their real prices will be unchanged or constant.

For example, say apples and oranges originally cost $1 each, so one apple is worth one orange. Then a year later, they cost $1.03 each since inflation was 3% pa. Since one apple is still worth one orange, the real price of apples and oranges is the same. It's only the nominal price (in dollars) that has changed.

Therefore only nominal prices increase by inflation. Real prices are unaffected.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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}###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The real cash flow cannot be discounted by a nominal discount rate.

Method 1: Discount the real cash flow by the real discount rate

The nominal discount rate can be converted into a real rate using the Fisher equation.

###1+r_\text{real} = \dfrac{1+r_\text{nominal}}{1+r_\text{inflation}}### ###1+r_\text{real} = \dfrac{1+0.07}{1+0.04}### ###\begin{aligned} r_\text{real} &= \dfrac{1+0.07}{1+0.04} -1 \\ &= 0.028846154 \\ \end{aligned}###

Now discount the real payment by the real discount rate.

###\begin{aligned} V_0 &= \dfrac{V_\text{2, real}}{(1+r_\text{real})^2} \\ &= \dfrac{500}{(1+0.028846154)^2} \\ &= 472.3556643 \\ \end{aligned}###

Method 2: Discount the nominal cash flow by the nominal discount rate

The real cash flow can be converted into a nominal cash flow by growing using the inflation rate:

###V_\text{2, real} = \dfrac{V_\text{2, nominal}}{(1+r_\mathbf{inflation})^2} ### ###500 = \dfrac{V_\text{2, nominal}}{(1+0.04)^2} ### ###\begin{aligned} V_\text{2, nominal} &= 500 \times (1+0.04)^2 \\ &= 540.8 \\ \end{aligned}###

Now discount the nominal payment by the nominal discount rate.

###\begin{aligned} V_0 &= \dfrac{V_\text{2, nominal}}{(1+r_\text{nominal})^2} \\ &= \dfrac{540.8 }{(1+0.07)^2} \\ &= 472.3556643 \\ \end{aligned}###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

A (nominal) payment of $105 in one year is worth $100 in real terms today, not $105. This is because in one year oranges and apples will cost $1.05 each, so $105 in one year will only buy 100 (=$105/$1.05) apples or oranges. Therefore the real value of a nominal payment of $105 in one year is only worth 100 apples or oranges which is worth $100 in real terms today.

Another way that the real value of a future nominal payment can be found is to discount it by the inflation rate:

###V_\text{T, real} = \dfrac{V_\text{T, nominal}}{(1+r_\text{inflation})^T} ### ###\begin{aligned} V_\text{1, real} &= \dfrac{V_\text{1, nominal}}{(1+r_\text{inflation})^1} \\ &= \dfrac{105}{(1+0.05)^1} \\ &= 100 \\ \end{aligned}###

Economists call real values 'constant prices' values. This is because real values pretend that prices are constant and don't change. Inflation reduces the real value of money, it erodes the buying power of nominal wealth. That's why real values are always lower than nominal values when inflation is positive.

Economists call nominal values 'current prices' values. This is because current prices are those actually observed at the current time. Over time, (current) prices rise when inflation is positive.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

No explanation provided.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

No explanation provided.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

No explanation provided.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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}##.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Answer d is incorrect because it wrongly includes the dividend at time zero ##C_0## when actually that dividend was just paid (past tense) so if you buy the stock now you will not receive that dividend. That past dividend needs to be subtracted from the stock price, so this is the adjustment required:

###P_0=\dfrac{C_0 (1+r_\text{real})^1}{r_\text{real} - g_\text{real}} - C_0###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

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}###

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Of the surveyed managers, 68.2% (=29.8%+38.4%) value their projects correctly, which includes those:

• 29.8% who discount nominal cash flows by nominal hurdle rates; and
• 38.4% who discount real cash flows by real hurdle rates, which is also fine.