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 marketing expense in note 2 is a sunk cost that should be ignored since it can not be recouped. Even though it will lead to higher amortisation expense in the future, and thus lower taxes, this will happen regardless of whether the project goes ahead or not, so it should not affect the decision to accept or reject the project. This is why it should be left out of the NPV (and NI) calculation.

To find the break even quantity (Q) when Net Income (NI) equals zero, make an expression for NI with the quantity Q as an unknown variable, then solve for Q:

###\begin{aligned} NI &= (Q(P – VC) – FC – Depr – IntExp)(1-t_c) \\ 0 &= (Q(10 - 6) – 2m – 1m - 0)(1 - 0.3) \\ &= (Q(10 - 6) – 2m – 1m - 0)(1 - 0.3) \\ &= 4Q – 3m \\ Q &= 3m/4 \\ &= 0.75m = 750,000 \text{ units of sales} \\ \end{aligned}###

To find the break even quantity (Q) when Net Present Value (NPV) equals zero, make an expression for NPV with the quantity Q as an unknown variable, then solve for Q:

###NPV = PV(FFCF) ### ###\begin{aligned} 0 =& \text{FFCF at t=0 to build factory, no sales made so CapEx only} \\ & +\text{Present value of annuity of FFCF's at end of every year, from t=1 to 10} \\ & +\text{Present value of additional FFCF at t=10 due to sale of inventory} \\ =& -CapEx \\ & +\left( (Q(P – VC) – FC – Depr – IntExp)(1-t_c)+Depr-CapEx-\Delta NWC +IntExp \right) \times (\text{annuity factor 10yr, 10%}) \\ & +\dfrac{\Delta NWC \text{ at t=10}}{(1+r)^{10}} \\ =& -10m \\ & +((Q(10 – 6) – 2m – 1m – 0)(1-0.3)+1m -0 -0.1m +0) \times \frac{1}{0.1}\left(1-\dfrac{1}{(1+0.1)^{10}}\right) \\ & + \dfrac{10 \times 0.1m}{(1+0.1)^{10}} \\ =& -10m \\ & +((4Q – 3m) \times 0.7+0.9m) \times 6.144567106 \\ & + 0.385543289m \\ \end{aligned}### ### (4Q – 3m) \times 0.7+0.9m = \dfrac{(10m - 0.385543289m)}{6.144567106} ### ### 4Q – 3m = \frac{10m - 0.385543289m}{6.144567106 \times 0.7} - \dfrac{0.9m}{0.7} ### ###\begin{aligned} Q &= \frac{10m - 0.385543289m}{6.144567106 \times 0.7 \times 4} - \dfrac{0.9m}{0.7 \times 4} +\frac{3m}{4} \\ &= 0.987395912m \\ &= 987,396 \text{ units of sales} \\ \end{aligned}###

It's shocking how different the accounting (Net Income) and finance (Net Present Value) break-even sales figures are. Of course the finance NPV break-even figure is more accurate since it reflects the time value of money.

Note that in real business, the calculation of break-even sales can be solved automatically without using algebra using a computer spreadsheet such as MS Excel and the 'goal seek' or 'solver' functions.

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

The shares are expected to be worth $110 in one year, and the loan will be worth $107. So there is a positive expected cash flow of $3 in one year.

###\begin{aligned} V_1 &= V_\text{1, shares} - V_\text{1, loan} \\ &= V_\text{0, shares}(1+r_\text{shares})^1 - V_\text{0, loan}(1+r_\text{loan})^1 \\ &= 100(1+0.1)^1 - 100(1+0.07)^1 \\ &= 110 - 107 \\ &= 3 \\ \end{aligned} ###

Most people then discount the future value of $3 to get a present value of either $2.8037 or $2.7273 depending on whether they use a discount rate of 7 or 10% respectively. But this approach is wrong. The problem becomes apparent when trying to justify the use of one discount rate over another to find the present value of the $3. Should it be 10% or 7% or an average? Unfortunately this way of thinking was flawed from the beginning when the share's and loan's cash flows were added together because they have different risks and should be discounted by different required returns.

The way to analyse this question is to consider buying the shares and selling the loan separately. Note that 'borrowing' is the same thing as 'selling' a loan.

Since the shares are fairly priced, the NPV of buying them is zero. Similarly for the fairly priced loan, the NPV of selling it must be zero. So the NPV of the two transactions is zero plus zero which equals zero.

Alternatively, a more mathematical way of looking at it is that the expected returns of the fairly priced shares and loan are exactly equal to their respective discount rates. So they cancel out as follows:

###\begin{aligned} V_1 &= V_\text{1, shares} - V_\text{1, loan} \\ V_0 &= \frac{V_\text{1, shares}}{(1+r_\text{shares})^1} - \frac{V_\text{1, loan}}{(1+r_\text{loan})^1} \\ &= \frac{V_\text{0, shares}(1+r_\text{shares})^1}{(1+r_\text{shares})^1} - \frac{V_\text{0, loan}(1+r_\text{loan})^1}{(1+r_\text{loan})^1} \\ &= \frac{100(1+0.1)^1}{(1+0.1)^1} - \frac{100(1+0.07)^1}{(1+0.07)^1} \\ &= \frac{110}{(1+0.1)^1} - \frac{107}{(1+0.07)^1} \\ &= 100 - 100 \\ &= 0 \\ \end{aligned} ###

It seems nonsensical that there is a positive expected cash flow of $3 in one year, yet the NPV is zero. The reason why this scenario occurs in theory and in real life is that the expected value of the shares is $110 in one year but it could be a lot less. The loan, on the other hand, will definitely have $107 owing. In the worst case, after one year the shares become worthless (price = 0) and $107 is owed on the loan.

The expected gain of $3 is deserved for taking on the stock's higher level of systematic risk compared with the loan. Investors who suffer higher systematic risk deserve a higher return.

Other interesting view points about this scenario:

• In a risk-neutral world, all assets earn the risk-free rate thus there would be no positive expected future cash flow of $3. But in a risk-averse world, the $3 is compensation for taking on systematic risk.
• The principal of no-arbitrage says that in an efficient market it should be impossible to make unlimited risk-free gains. The portfolio of shares funded by the loan requires no capital so its payoff is unlimited, but the $3 expected gain is not risk-free. Thus the principal of no-(risk-free)-arbitrage holds.
• Banks prefer to lend with some form of security which has a value of more than the loan. The shares have the same value as the loan so they are unlikely to provide sufficient security. In the real world, margin loans on shares generally have a maximum debt-to-assets ratio of 0.7. Residential real estate lenders prefer borrowers to contribute a deposit of 20% of the house price, which equates to a debt-to-assets ratio of 0.8.
• An interesting line of research is the 'Kelly Criterion' and the 'Growth Optimal Portfolio'. The Kelly Criterion is widely known in the gambling literature and is used to calculate the optimal proportion of wealth to wager on a risky bet when the odds are in your favour. The Kelly criterion maximises the growth rate of wealth. It can also be applied to financial decisions such as this if the investor prefers to maximise her expected growth rate of wealth rather than her utility function which takes return and volatility into account.

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

This question appears to be simple: the underlying asset's price will rise so buy call options which should have a positive payoff at maturity.

But it's not that simple. Since you and the whole market agree that the milk price will rise by 40%, sellers of call options expect to have to pay the buyers of call options at maturity since the call options are likely to be 'in the money' when the milk price exceeds the call option's strike price.

Therefore sellers will sell their call options for a high price (at the start) to compensate them for the loss they're likely to suffer at maturity (at the end). Using this argument, the NPV of selling and buying call and put options should be zero. There is no positive NPV strategy.

This illustrates the concept that there is 'no free lunch' in an efficient market where everyone has the same information and expectations.

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

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}###

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

The inflation rate in this question is a red herring, it's not needed to answer the question. Since the loan is interest-only, the amount borrowed at the start will be the same as the amount to be paid at maturity. Therefore the nominal capital return must be zero, since there is no increase in the nominal value of the loan, measured just after each interest payment. Therefore the nominal capital return on the loan is zero.

To summarise:

• The nominal capital return will be 0%. The real capital return will be about -2%.
• The nominal income return will be 6% which is the interest rate. The real income return will be also be about 6%, or just a touch less.
• The nominal total return will be 6%. The real total return will be about 4%.

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.

Since the cash flows are real but our discount rate is nominal, we need to convert the nominal discount rate to a real rate. Using the exact Fisher equation,

###\begin{aligned} 1+r_\text{real} &= \frac{1+r_\text{nomimal}}{1+r_\text{inflation}} \\ r_\text{real} &= \frac{1+r_\text{nomimal}}{1+r_\text{inflation}} -1 \\ &= \frac{1+0.21}{1+0.1} -1 \\ &= 0.1 \\ \end{aligned} ###

Now just discount the cash flows using two annuity equations.

###\begin{aligned} V_0 &= -C_\text{0, 1, 2}.\frac{1}{r}\left(1 - \frac{1}{(1+r)^{3}} \right).(1+r)^{1} + C_\text{3, 4, ..., 12}.\frac{1}{r}\left(1 - \frac{1}{(1+r)^{10}} \right).\frac{1}{(1+r)^{2}} \\ &= -100m \times \frac{1}{0.1} \times \left(1 - \frac{1}{(1+0.1)^{3}} \right) \times (1+0.1)^{1} + 50m \times \frac{1}{0.1} \times \left(1 - \frac{1}{(1+0.1)^{10}} \right) \times \frac{1}{(1+0.1)^{2}} \\ &= -100m \times 2.48685199 \times 1.1 + 50m \times 6.14456711 \times 0.82644628 \\ &= -273.553719m + 253.9077316m \\ &= -19.64598737m \\ &= -19,645,987.37 \\ \end{aligned} ###

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

Future cash flows and returns are important.

Owners of assets such as shares are entitled to the future cash flows only, not the past cash flows which have already been paid. This is why asset prices are the present value of future cash flows.

To calculate the present value of future dividends, the dividend discount model must use the future expected return ##r## and growth rate ##g## of the market price of equity.

Of course the future is impossible to predict. Often the best guide to the future is the past, so in practice the actual historical return and growth rate are used as a proxy for what's expected in the future.

Market prices are important.

In finance, current market prices are always more important and relevant than old historical cost book prices. The market price of a share is the price that it trades for every day on the stock exchange. It's the price that a buyer will actually pay to buy the share.

When the share was first bought, the market price and book price were the same. But after that, the book price never changes while the market price goes up and down every day. Therefore the book price is old and out of date. Generally it is not the same as the current market price, unless by coincidence.

Owners equity recorded by an accountant in the firm's balance sheet is the sum of the shareholders' equity (also called contributed equity), retained profits and reserves such as asset revaluation reserve. This is often very different to the market price of equity. If the firm has been successful in the past, usually the market price of equity will be much higher than the book price.

Equity returns calculated from book prices are also therefore not very useful to determine value. They reflect the past, not the future. Therefore accounting ratios such as ROE (Net Income/Owners Equity) and ROA (Net Income/Total Assets) are not very useful for pricing stocks. But they are a reasonable guide to past performance.

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

Since the project has a positive NPV of $200m, the value of the firm's assets (V) must increase by $200m. Assuming that the debt can be fully paid off, this positive $200m NPV will all accrue to the equityholders since they have a residual claim on the firm's assets.

The project will be funded by issuing bonds which will add $600m to debt liabilities (D) and cash or equipment assets (V).

Therefore, ##ΔV=800m, ΔD = 600m, ΔE=200m##.