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

###\begin{aligned} P_\text{0, bill} =& \frac{F_d}{1 + r_\text{simple} \times \frac{d}{365}} \\ =& \frac{1,000,000}{1 + 0.025 \times \frac{30}{365}} \\ =& 997,949.419 \\ \end{aligned} ###

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

When pricing bonds or stocks or any asset, the future cash flows and discount rates are the only things that are important. So the 2% yield is the discount rate and there are 29.5 years left which is 59 six-month periods. Note that since the bond was issued at par, its initial yield and coupon rate must have been equal. Since it's a fixed coupon bond, the coupon rate will never change so it will still be 1.7% into the future.

Using the bond price equation:

###\begin{aligned} p_\text{0, bond} &= \text{PV(annuity of coupons)} + \text{PV(face value)} \\ &= \text{coupon} \times \frac{1}{r_\text{eff}}\left(1 - \frac{1}{(1+r_\text{eff})^{T}} \right) + \frac{\text{face}}{(1+r_\text{eff})^{T}} \\ &= \frac{100 \times 0.017}{2} \times \frac{1}{0.02/2}\left(1 - \frac{1}{(1+0.02/2)^{2 \times 29.5}} \right) + \frac{100}{(1+0.02/2)^{2 \times 29.5}} \\ &= 0.85 \times 44.4045887902863 + 55.5954112097137 \\ &= 37.7439004717433 + 55.5954112097137 \\ &= 93.3393116814571 \\ \end{aligned} ###

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

All answers are mathematically equivalent except (e).

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.

An APR is a discretely compounding annual rate that compounds multiple times per year.

An APR compounding once per year is an effective annual rate.

An effective rate is also a discretely compounding rate but it compounds only once per period. The time period is not necessarily annual, it can be monthly, daily, two years, or any time.

Therefore answer (c) is incorrect. An effective monthly rate is a monthly rate compounding per month.

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

Using the Cash Flow From Assets Equation,

### CFFA = NI + Depr - CapEx - \Delta NWC + IntExp ###

Capital expenditure (CapEx) can be calculated as the change in Net Fixed Assets (NFA) plus depreciation. Note that NFA is the same thing as the carrying amount of property, plant and equipment (PPE).

###\begin{aligned} CapEx &= PPE_\text{now} - PPE_\text{before} + Depr \\ &= 420 - 400 + 20 \\ &= 40 \\ \end{aligned}###

CapEx is positive, so the firm must have bought more capital assets than it sold.

To find the change in net working capital (##\Delta NWC##), take the difference between the NWC now and before. Note that current assets includes inventory, trade debtors and rent paid in advance. Current liabilities only includes trade creditors in this instance.

###\begin{aligned} \Delta NWC &= CA_\text{now} - CL_\text{now} - (CA_\text{before} - CL_\text{before}) \\ &= (60+19+3-10) - (50+6+2-8) \\ &= 72 - 50 \\ &= 22 \\ \end{aligned}###

Now just substitute the values:

###\begin{aligned} CFFA &= NI + Depr - CapEx - \Delta NWC + IntExp \\ &= 63 + 20 - 40 - 22 + 19 \\ &= 40 \\ \end{aligned}###

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

To find the standard deviation there are 2 steps. First we need to find the historical average return ##(\bar{r})##, then use the variance formula to find the historical sample variance ##(\hat{\sigma}^2)##and square root it to get a standard deviation ##(\hat{\sigma})##..

###\begin{aligned} \bar{r} &= \frac{r_{0 \rightarrow 1} + r_{1 \rightarrow 2} + r_{2 \rightarrow 3} + ... +r_{T-1 \rightarrow T}}{T} \\ &= \frac{0.3 + 0.02 + -0.2 + 0.4}{4} = 0.13 \\ \end{aligned} ###

###\begin{aligned} \hat{\sigma}^2 &= \dfrac{ \displaystyle\sum\limits_{t=1}^T{\left( \left( r_{(t-1)\rightarrow t} - \bar{r} \right)^2 \right)} }{T-1}\\ &= \frac{\left( \begin{aligned} &{(0.3-0.13)^2} + \\ &{(0.02-0.13)^2} + \\ &{(-0.2-0.13)^2} + \\ &{(0.4-0.13)^2} \\ \end{aligned} \right)\\ }{4-1} \\ &= 0.074266667 \\ \hat{\sigma} &= \left( 0.074266667\right)^{1/2} \\ &= 0.272519112 \\ \end{aligned} ###

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

When the correlation between two assets' returns increases, if one stock has a negative return, it is more likely that the other stock will also have a negative return. This means that the asset returns will not cancel each other out as often, so there will be less diversification and greater portfolio variance and volatility (standard deviation).

The higher the correlation, the higher the covariance. This is because the covariance between two stocks' returns is a function of correlation:

### cov(r_A, r_B) = corr(r_A, r_B).std(r_A).std(r_B)###

Or in mathematical notation with Greek symbols: ### \sigma_{A,B} = \rho_{A,B}.\sigma_A.\sigma_B###

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

An investor who short-sells an asset is 'short' that asset, not long.

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.

An increase in IntExp decreases NI and increases CFFA. This is very counter-intuitive, but it's because IntExp reduces taxes since it is subtracted from pre-tax income.

But since IntExp is a 'financing cash flow' which has nothing to do with the cash flow from the assets, it is added back in CFFA, and its only lingering effect is the reduction in taxes.

This is the so-called 'interest tax shield' effect of having debt and therefore interest expense.

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

The implicit rate discounts the bigger cost on day 60 so that it equals the cheaper cost on day 5. Or another way of looking at it: the implicit rate grows the cheaper cost at day 5 forward to equal the bigger cost at day 60.

Note that we pay on the last day possible whether we want the 2% discount (day 5) or not (day 60) so that we can hang onto our cash for as long as possible and collect extra interest. For example we would not pay on day 40 because we could pay on day 60 and put the cash in the bank to collect interest for those extra 20 days.

### V_{5} = \frac{V_{60}}{(1+r_\text{eff,annual})^{(60-5)/365}} ###

But the discount is 2% if paid by ##t=5##, so ###V_{5} = V_{60}(1-0.02)### Substituting into the above, then:

###V_{60}(1-0.02) = \frac{V_{60}}{(1+r_\text{eff,annual})^{(60-5)/365}} ### ###1-0.02 = \frac{1}{(1+r_\text{eff,annual})^{(60-5)/365}} ### ###(1+r_\text{eff,annual})^{(60-5)/365} = \frac{1}{1-0.02} ### ###1+r_\text{eff,annual} = \left(\frac{1}{1-0.02}\right)^{365/(60-5)} ###

###\begin{aligned} r_\text{eff,annual} =& \left(\frac{1}{1-0.02}\right)^{365/55} - 1 \\ =& 0.143475733 \\ \end{aligned} ###

So the implicit annual effective rate is 14.3%. By letting us pay later, the wholesaler is effectively giving us a loan. But this loan is not free, since he charges more (since there's no 2% saving) if we pay later. The implicit rate is the effective lending rate that he is granting to us. It is the cost of borrowing from him by paying on day 60 rather than day 5.

If we borrowed using a bank overdraft we'd pay 10% pa which is cheaper than the 14.3% from the wholesaler. Therefore it would be better to borrow from the bank and pay the wholesaler early on day 5.

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

Find the value of each interest-only mortgage loan:

###\begin{aligned} V_\text{before} &= \frac{C_\text{1, monthly}}{r_\text{eff mthly} - g_\text{eff mthly}} \\ &= \frac{2,000}{\left( \dfrac{0.0474}{12} - 0 \right)} \\ &= 506,329.1139 \\ \end{aligned}###

###\begin{aligned} V_\text{after} &= \frac{C_\text{1, monthly}}{r_\text{eff mthly} - g_\text{eff mthly}} \\ &= \frac{2,000}{\left( \dfrac{0.0449}{12} - 0 \right)} \\ &= 534,521.1581 \\ \end{aligned}###

###\begin{aligned} \text{Proportional increase} &= \frac{V_\text{after}-V_\text{before}}{V_\text{before}} \\ &= \frac{534,521.1581 - 506,329.1139}{506,329.1139} \\ &= 0.055679287 \\ &\approx 5.6\% \\ \end{aligned}###

Note that the answer is 0.029547 or 2.9547% if the mortgage loans are both fully amortising. Thanks to Shahzada for providing that solution.

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

This is an equivalent annual cost question. There are two steps, first find the NPV of costs over the life of each car, then spread these costs over the life using the annuity formula to find the equivalent annual cost.

Carlos buys the new car that he keeps for 4 years and Edwin buys the older car which he keeps for 11 years. First find the NPV of the costs of each car.
Note that a revenue is a negative cost. If that's confusing, then find the equivalent annual benefit, so costs are negative and revenues are positive which is arguably more intuitive. The numerical answers will be the same, but the signs will be switched around.

###\begin{aligned} V_\text{0, Carlos} &= C_0 - \frac{C_4}{(1+0.1)^4} \\ &= 40,000 - \frac{20,000}{(1+0.1)^4} \\ &= 26,339.73089 \\ \end{aligned} ###

###\begin{aligned} V_\text{0, Edwin} &= C_0 + \frac{C_\text{0,1,...10}}{r} \left(1-\frac{1}{(1+r)^{11}} \right)(1+r)^1 - \frac{C_{11}}{(1+r)^{11}} \\ &= 20,000 + \frac{1,000}{0.1} \left(1-\frac{1}{(1+0.1)^{11}} \right)(1+0.1)^1 - \frac{2,000}{(1+0.1)^{11}} \\ &= 26,443.57931 \\ \end{aligned} ###

Now spread the costs over the life of each car using the annuity formula. This will give the 'equivalent annual cost'.

###V_\text{0, Carlos} = \frac{C_\text{EAC Carlos}}{r} \left(1-\frac{1}{(1+r)^{T}} \right) ### ###26,339.73089 = \frac{C_\text{EAC Carlos}}{0.1} \left(1-\frac{1}{(1+0.1)^{4}} \right) ### ###C_\text{EAC Carlos} = 8,309.416074 ###

Now for Edwin:

###V_\text{0, Edwin} = \frac{C_\text{EAC Edwin}}{r} \left(1-\frac{1}{(1+r)^{T}} \right) ### ###26,443.57931 = \frac{C_\text{EAC Edwin}}{0.1} \left(1-\frac{1}{(1+0.1)^{11}} \right) ### ###C_\text{EAC Edwin} = 4,071.336556 ###

To find how much larger Carlos' equivalent annual cost is compared with Edwin:

###\begin{aligned} C_\text{difference} &= C_\text{EAC Carlos} - C_\text{EAC Edwin} \\ &= 8,309.416074 - 4,071.336556 \\ &= 4,238.079518 \\ \end{aligned} ###

Therefore Carlos spends $4,238 more than Edwin per year, as an equivalent end-of-year cost.

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

Since most of the cash flows are weekly, it's easier to work with effective weekly rates than annual ones so convert the effective annual rate to an effective weekly rate:

###\begin{aligned} r_\text{eff wkly} &= (1+r_\text{eff yrly})^{1/52}-1 \\ &= (1+0.098)^{1/52}-1 \\ &= 0.001799508 \\ &\approx 0.0018 \\ \end{aligned} ###

Since all discount rates and cash flows are real, there is no need to do any conversions using inflation.

University fees will be ##4 \times $1,277 = $5,108## per semester, paid at t=0 for the first semester and again at t=19 weeks for the second semester.
The present value of university fees for one year is:

###\begin{aligned} V_\text{0, annual fee} &= (\text{First semester cost now}) + \frac{(\text{Second semester cost in 19 weeks})}{(1+0.001799508)^{19}} \\ &= 4 \times 1,277 + \frac{4 \times 1,277}{(1+0.001799508)^{19}} \\ &= 10,044.45768 \\ \end{aligned} ###

But as well as this explicit annual cost there is also the implicit opportunity cost which is that students can't work full-time while they are studying full-time. Students can still work during the summer holidays, but they can't work from t=0 to 38 weeks.
At $20/hr and 35hrs/wk they miss out on $700/wk paid in arrears. The present value of this annual opportunity cost is:

###\begin{aligned} V_\text{0, annual wages foregone} &= \frac{C_\text{wage 1,2,..38}}{r} \left(1-\frac{1}{(1+r)^{38}} \right) \\ &= \frac{700}{0.001799508} \left(1-\frac{1}{(1+0.001799508)^{38}} \right) \\ &= 25,688.58368 \\ \end{aligned} ###

Adding up the present value of the annual explicit and implicit costs:

###\begin{aligned} V_\text{0, annual cost} &= V_\text{0, annual fee} + V_\text{0, annual wages foregone} \\ &= 10,044.45768 + 25,688.58368 \\ &= 35,733.04136 \\ \end{aligned} ###

The annual costs are expected to be constant every year, so the present value of the costs over the whole 3 year degree is:

###\begin{aligned} V_\text{0, 3yr cost} &= \frac{V_\text{0,1,2, annual cost}}{r_\text{eff yrly}} \left(1-\frac{1}{(1+r_\text{eff yrly})^{3}} \right)(1+r_\text{eff yrly})^1 \\ &= \frac{35,733.04136}{0.098} \left(1-\frac{1}{(1+0.098)^{3}} \right)(1+0.098)^1 \\ &= $97,915.91465 \\ \end{aligned} ###