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

This is an equivalent annual cost question since the jet and yacht last for different amounts of time.

###\begin{aligned} V_\text{0, jet, all costs} &= -\text{PurchaseCost} -\text{MaintenanceCosts} +\text{SaleRevenue} \\ &= -C_0-\frac{C_\text{monthly}}{r_\text{eff monthly}} \left(1-\frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) + \frac{C_T}{(1+r_\text{eff monthly})^{T_\text{months}}} \\ &= -6m-\frac{0.012m}{0.00797414} \left(1-\frac{1}{(1+0.00797414)^{12 \times12}} \right) + \frac{6m}{(1+0.00797414)^{12\times12}} \\ &= -5.113583224m \\ \end{aligned} ###

###\begin{aligned} V_\text{0, yacht, all costs} &= -C_0-\frac{C_\text{monthly}}{r_\text{eff monthly}} \left(1-\frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) + \frac{C_T}{(1+r_\text{eff monthly})^{T_\text{months}}} \\ &= -4m-\frac{0.02m}{0.00797414} \left(1-\frac{1}{(1+0.00797414)^{20 \times12}} \right) + \frac{4m}{(1+0.00797414)^{20\times12}} \\ &= -5.540718655m \\ \end{aligned} ###

Although the jet appears cheaper because it has a lower present value of costs, we have to recognise that the jet has a shorter life than the yacht, so of course the present value of its costs will be less. We need to use the annuity formula to spread the costs over each project's life so we can get an equivalent annual cost.

For the jet,

###V_\text{0, jet, all costs} = \frac{C_\text{EAC jet}}{r_\text{eff annual}} \left(1-\frac{1}{(1+r_\text{eff annual})^{T_\text{years}}} \right) ### ###-5.113583224m = \frac{C_\text{EAC jet}}{0.1} \left(1-\frac{1}{(1+0.1)^{12}} \right) ### ###C_\text{EAC jet} = -0.750486426m ###

For the yacht,

###V_\text{0, yacht, all costs} = \frac{C_\text{EAC yacht}}{r_\text{eff annual}} \left(1-\frac{1}{(1+r_\text{eff annual})^{T_\text{years}}} \right) ### ###-5.540718655m = \frac{C_\text{EAC yacht}}{0.1} \left(1-\frac{1}{(1+0.1)^{20}} \right) ### ###C_\text{EAC yacht} = -0.650810734m ###

Since the yacht has the lower equivalent annual cost, it is the best choice.

Note that this is a bit of an unusual result since the yacht and jet are both sold for the amount that they are bought for, but the yacht has higher running costs than the jet ($20k vs $12k). Common sense would lead us to conclude that we should buy the thing with the lowest running costs.

But this common-sense approach ignores opportunity costs. The jet costs $2m more than the yacht ($6m vs $4m), and since the interest rate is 10%, that extra $2m means that there is a $200,000 opportunity cost of having that cash tied up in the jet rather than sitting in the bank collecting interest at 10% pa. This is the main reason why the yacht is the more cost-effective choice.

An alternative method to find the equivalent annual cash flow is to use the perpetuity formula to discount the cash flows as if they continue forever. The first step is to find the present value of the cash flows that go forever. Because the cost of the jet (or yacht) is always the same and the sale price at the end of the current jet's life cancels out with the purchase price of the next jet, only the purchase at the very start needs to be included.

###V_\text{0, perpetual} = -C_\text{0, initial cost} - \dfrac{C_\text{1, monthly ongoing costs}}{r_\text{eff monthly}-g_\text{eff monthly}}###

For the jet:

###\begin{aligned} V_\text{0, jet, perpetual} &= -6m - \dfrac{0.012m}{0.00797414-0} \\ &= -7.504864474m \\ \end{aligned}###

The second step is to spread these costs over each year forever, also using the perpetuity formula.

###V_\text{0, jet, perpetual} = \frac{C_\text{EAC jet}}{r_\text{eff annual} - g_\text{eff anual}} ### ###-7.504864474m = \frac{C_\text{EAC jet}}{0.1 - 0} ### ###\begin{aligned} C_\text{EAC jet} &= -7.504864474m \times 0.1 \\ &= -0.7504864474m \\ \end{aligned}###

For the yacht:

###\begin{aligned} V_\text{0, yacht, perpetual} &= -4m - \dfrac{0.02m}{0.00797414-0} \\ &= -6.508107457m \\ \end{aligned}### ###V_\text{0, yacht, perpetual} = \frac{C_\text{EAC yacht}}{r_\text{eff annual} - g_\text{eff anual}} ### ###-6.508107457m = \frac{C_\text{EAC yacht}}{0.1 - 0} ### ###\begin{aligned} C_\text{EAC yacht} &= -6.508107457m \times 0.1 \\ &= -0.6508107457m \\ \end{aligned}###

Both equivalent annual cash flows are the same as before, ignoring the small discrepancy caused by rounding the monthly discount rate.

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.

If a project's NPV is negative, it's IRR will be less than the required return. But the IRR will not necessarily be negative. The IRR could say 4% which is less than the positive required return of say 10%, but more than zero.

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

The NPV of buying any fairly priced asset is zero. Therefore the NPV of buying a fairly priced bond is also zero. Whether the bond is a premium or discount bond is irrelevant, it's unrelated to the NPV of buying it.

The fair price of a bond is the present value (PV) of its expected future cash flows, which is the present value of coupons and face value:

###\begin{aligned} P_\text{0, bond} &= PV(\text{coupons}) + PV(\text{face value}) \\ &= \frac{C_1}{r} \left(1 - \frac{1}{(1+r)^T} \right) + \frac{F_T}{(1+r)^T} \\ \end{aligned}###

The net present value (NPV) of buying an asset is the present value of costs less gains.

###\begin{aligned} NPV &= -PV(\text{costs}) + PV(\text{gains}) \\ \end{aligned}###

The cost of a bond is its price, and the gains from a bond are the coupons and face value. Since the price of a fairly priced bond equals the present value of the coupons and face value, then the net present value of buying a fairly priced bond must be zero.

Mathematically, we can re-arrange the bond price formula to be in the same form as the NPV formula, which shows that the NPV must be zero:

###P_\text{0, bond} = PV(\text{coupons}) + PV(\text{face value}) ### ###\underbrace{0}_{\text{NPV}} = -\underbrace{P_\text{0, bond}}_{PV(\text{costs})} + \underbrace{PV(\text{coupons}) + PV(\text{face value})}_{PV(\text{gains})} ###

Note that premium bonds can also be fairly priced. The NPV of buying a fairly priced premium bond is zero. The term 'premium' does not indicate that the bond's price is above (or below) the fair price, it indicates that the bond's price is above its face value which is usually the $100 or $1,000 that's paid at maturity. Premium bonds have a higher price than their face value because the coupon rate is more than the total required return (the yield). Therefore investors are willing to pay a high price for the bond, higher than the face value, making the bond a premium bond. The highest price investors will pay for the bond will be the price that makes the NPV zero.

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

Projects with a positive NPV will have a Profitability Index greater than one.

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

Since the Profitability Index is less than one, the project should be rejected since the present value of the future cash flows must be less than the initial cost.

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

When the NPV of a project is positive, the project's IRR must be more than its required return.

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

From the bond pricing formula, the required return r is in the denominator of each fraction so any increase in r causes a decrease in the price P and vice versa:

###P_\text{0, bond} = C_\text{1,2,3,...,T} \times \frac{1}{r}\left(1 - \frac{1}{(1+r)^{T}} \right) + \frac{F_\text{T}}{(1+r)^{T}} ###

When the required return rises, the bond price falls.

When the required return falls, the bond price rises.

This is not only true for bonds but for any asset including shares and land.

The required return of a fairly priced bond is also its IRR. Remember that the IRR is the discount rate that makes the NPV zero.

###\begin{aligned} NPV &= C_0 + \frac{C_1}{(1+r)^1} + \frac{C_2}{(1+r)^2} + ... + \frac{C_T}{(1+r)^T} \\ 0 &= C_0 + \frac{C_1}{(1+r_{irr})^1} + \frac{C_2}{(1+r_{irr})^2} + ... + \frac{C_T}{(1+r_{irr})^T} \\ \end{aligned} ###

Re-arranging the bond-pricing equation:

###P_\text{0, bond} = C_\text{1,2,3,...,T} \times \frac{1}{r}\left(1 - \frac{1}{(1+r)^{T}} \right) + \frac{F_\text{T}}{(1+r)^{T}} ### ###\underbrace{0}_{\text{NPV}} = -\underbrace{P_\text{0, bond}}_{PV(\text{cost})} + \underbrace{C_\text{1,2,3,...,T} \times \frac{1}{r_\text{IRR}}\left(1 - \frac{1}{(1+r_\text{IRR})^{T}} \right) + \frac{F_\text{T}}{(1+r_\text{IRR})^{T}}}_{PV(\text{gains})} ###

Because the NPV of buying a fairly priced bond is zero, the bond's yield is equivalent to the IRR of buying it too.

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

Using the dividend discount model (DDM),

###\begin{aligned} P_{0} &= \frac{C_\text{6mth}}{r_\text{eff 6mth} - g_\text{eff 6mth}} \\ &= \frac{C_0(1+g_\text{eff 6mth})^1}{r_\text{eff 6mth} - g_\text{eff 6mth}} \\ \end{aligned} ### ###\begin{aligned} 3 &= \frac{0.14(1-0.01)^1}{r_\text{eff 6mth} - (-0.01)} \\ \end{aligned} ### ###\begin{aligned} r_\text{eff 6mth} &= \frac{0.14(1-0.01)^1}{3} - 0.01 \\ &= 0.0362 \\ \end{aligned} ### ###\begin{aligned} r_\text{eff annual} &= (1+r_\text{eff 6mth})^2-1 \\ &= (1+0.0362 )^2-1 \\ &= 0.07371044 \\ \end{aligned} ###

Since this stock is very low risk, we can guess that it should have a low return close to the risk free rate which is the time value of money. Since this stock returns much more than the risk free rate (7.37% vs 4%), this stock is returning more than what we deserve. It is a good stock that we would like to buy. Its price is too low so it is under-priced, and buying it would have a positive NPV.

Note that this assumes that the stock's high historical rate of return in the past will continue into the future, but this might not be true.

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

The difference in costs between the BYO and bundled plans is 71-50=$21 per month for 24 months. The present value of this difference is the implicit cost of the phone. Since the payments are made in advance (at the start of the month), the ordinary annuity equation needs to be adjusted. This equation is known as the 'annuity due' equation and there are lots of versions of it. Here is one:

###\begin{aligned} V_0 &= C_0(1+r_\text{eff})^1 \times \frac{1}{r_\text{eff}}\left(1 - \frac{1}{(1+r_\text{eff})^{T}} \right) \\ &= (71-50)(1+0.02)^1 \times \frac{1}{0.02}\left(1 - \frac{1}{(1+0.02)^{24}} \right) \\ &= 21(1+0.02) \times 18.9139256 \\ &= 21.42 \times 18.9139256 \\ &= 405.1362864 \\ \end{aligned} ###

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

First convert the interest rate from an APR to an effective rate,

###r_\text{eff monthly} = \frac{r_\text{apr comp monthly}}{12} = \frac{0.12}{12} = 0.01###

The advertised sale price of the product comes with the 2 year interest free loan so it will be paid in 2 years or 24 months. Let that price be ##P_\text{24}##.

If we pay for the product right now we would be silly to pay the full price. We wouldn't pay now unless the price now is equal to or less than the present value of the price in 2 years. Let this discounted price be ##P_0##.

To find out how much less ##P_0## should be compared with ##P_{24}##,

###\begin{aligned} P_0 &= \frac{P_{24}}{(1+r_\text{eff monthly})^{24}} \\ &= \frac{P_{24}}{(1+0.01)^{24}} \\ &= \frac{P_{24}}{1.269734649} \\ \end{aligned} ###

If the advertised sale price ##P_{24}## is $100, then

###\begin{aligned} P_0 &= \frac{100}{1.269734649} \\ &= 78.7566127 \\ \end{aligned} ###

So the highest price that you would be willing to pay now is $78.76 which is 21.24% less than the advertised sale price of $100.

Or for those who prefer an algebraic expression rather than substituting $100,

###\begin{aligned} P_0 &= \frac{P_2}{1.269734649} \\ &= P_2 \times \frac{1}{1.269734649} \\ &= P_2 \times 0.787566127 \\ &= P_2 \times (1-0.212433873) \\ \end{aligned} ###

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

Since the cash flows occur every 6 months, the effective annual rate needs to be converted into an effective 6 month rate:

###r_\text{eff 6mth} = (1+r_\text{eff annual})^{0.5}-1 = (1+0.08)^{0.5}-1 = 0.039230485###

To find the present value, subtract the initial cost now and add the perpetuity of 6 month cash flows which go on forever. The only trick is to remember that the perpetuity formula will give a value one period before the first cash flow. Since the first cash flow will be at t = 4 semi-annual periods (2 years), the perpetuity formula will give a value at t = 3 semi-annual periods (1.5 years).

###\begin{aligned} V_0 &= -C_{0} + \frac{ \left(\dfrac{C_4}{r_\text{eff 6mth}}\right) }{(1+r_\text{eff 6mth})^{3}} \\ &= -100,000 + \frac{ \left(\dfrac{600 \times 100}{0.039230485}\right) }{(1+0.039230485)^{3}} \\ &= -100,000 + \frac{1,529,422.846}{(1+0.039230485)^{3}} \\ &= -100,000 + 1,362,673.923 \\ &= 1,262,673.923 = $1.262674m\\ \end{aligned} ###

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

The IRR is the discount rate that makes the NPV equal to zero.

###\begin{aligned} V_0 &= -C_{0} + \frac{C_2}{(1+r)^{2}} \\ 0 &= -C_{0} + \frac{C_2}{(1+r_\text{IRR})^{2}} \\ 0 &= -400 + \frac{500}{(1+r_\text{IRR})^{2}} \\ r_\text{IRR} &= \left( \frac{500}{400} \right)^{1/2} - 1 \\ &= 0.118033989 \\ \end{aligned} ###

Note that if $400 was the fair price of a 2-year zero-coupon bond, and $500 was the face value, then this IRR would also be the yield to maturity of the fairly priced bond.

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

To find the payback period,

###\begin{aligned} T_\text{payback} &= \left( \begin{array}{c} \text{time of} \\ \text{first positive} \\ \text{cumulative} \\ \text{cash flow} \\ \end{array} \right) - \frac{ \left( \begin{array}{c} \text{first positive} \\ \text{cumulative} \\ \text{cash flow} \\ \end{array} \right) }{ \left( \begin{array}{c} \text{cash flow over} \\ \text{that period} \\ \end{array} \right) } \\ &= 2 - \frac{(-400+500)}{500} \\ &= 2 - \frac{100}{500} \\ &= 1.8 \\ \end{aligned} ###

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

This project is interesting because it has a negative NPV, but since its positive cash flows continue forever, it must eventually pay itself off. So it has a finite payback period. Note that inflation is a red herring since the pay back period doesn't account for inflation. Usually only nominal cash flows are used in pay back period calculations and this is what is given.

Algebraic method: The cumulative cash flow ##C_\text{sum,T}## at time T is:

###\begin{aligned} C_\text{sum,T} &= -50 \times 3 + 10 \times (T - 3) \\ \end{aligned}###

The payback period ##T_\text{payback}## occurs when the cumulative cash flow is zero, so:

###\begin{aligned} C_\text{sum,T} &= -50 \times 3 + 10 \times (T - 3) \\ 0 &= -50 \times 3 + 10 \times (T_\text{payback} - 3) \\ 10 \times (T_\text{payback} - 3) &= 50 \times 3 \\ T_\text{payback} &= 3 + \frac{50 \times 3}{10} \\ &= 3 + 15 \\ &= 18 \\ \end{aligned}###

Table method:

Payback Period Calculation
Time (yrs)	Cash flow ($)	Cumulative cash flow ($)
0	-50	-50
1	-50	-100
2	-50	-150
3	0	-150
4	10	-140
5	10	-130
...	...	...
16	10	-20
17	10	-10
18	10	0

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

### V_{0} = \frac{V_{60}}{(1+r_\text{eff,annual})^{(60-0)/365}} ###

But the discount is 1.5% if paid at ##t=0##, so ###V_{0} = V_{60}(1-0.015)### Substituting into the above, then:

###V_{60}(1-0.015) = \frac{V_{60}}{(1+r_\text{eff,annual})^{(60-0)/365}} ### ###1-0.015 = \frac{1}{(1+r_\text{eff,annual})^{(60-0)/365}} ### ###(1+r_\text{eff,annual})^{(60-0)/365} = \frac{1}{1-0.015} ### ###1+r_\text{eff,annual} = \left(\frac{1}{1-0.015}\right)^{365/(60-0)} ###

###\begin{aligned} r_\text{eff,annual} &= \left(\frac{1}{1-0.015}\right)^{365/60} - 1 \\ &= 0.096300464\\ \end{aligned} ###

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

### V_{0} = \frac{V_{25}}{(1+r_\text{eff,annual})^{365/(25-0)}} ###

But the discount is 1% if paid at ##t=0##, so ###V_{0} = V_{25}(1-0.01)### Substituting into the above, then:

###\begin{aligned} V_{25}(1-0.01) &= \frac{V_{25}}{(1+r_\text{eff,annual})^{(25-0)/365}} \\ 1-0.01 &= \frac{1}{(1+r_\text{eff,annual})^{(25-0)/365}} \\ (1+r_\text{eff,annual})^{(25-0)/365} &= \frac{1}{1-0.01} \\ 1+r_\text{eff,annual} &= \left(\frac{1}{1-0.01}\right)^{365/(25-0)} \\ r_\text{eff,annual} &= \left(\frac{1}{1-0.01}\right)^{365/25} - 1 \\ &= 0.158046928 \\ \end{aligned} ###

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

This is an equivalent annual cost question because the two different types of light bulb last for different amounts of time.

Since the cash flows and discount rate are real, we can apply present value techniques without needing to worry about inflation. If the cash flows and discount rate were nominal, inflation can also be ignored. It's only when the cash flows are real and the discount rate is nominal (or vice versa) that you have to convert the nominal discount rate to a real rate using the Fisher equation. So in this question, there is no need to do this and the inflation figure of 3% is a red herring.

###\begin{aligned} V_\text{0, low energy} &= C_0 + \frac{C_\text{1,2,...T}}{r} \left(1-\frac{1}{(1+r)^{T}} \right) \\ &= 3.50 + \frac{1.60}{0.05} \left(1-\frac{1}{(1+0.05)^{9}} \right) \\ &= 14.87251468\\ \end{aligned} ###

###\begin{aligned} V_\text{0, conventional} &= C_0 + \frac{C_\text{1}}{(1+r)^1} \\ &= 0.50 + \frac{6.60}{(1+0.05)^1} \\ &= 6.785714286 \\ \end{aligned} ###

The conventional light bulb appears cheaper because it has a lower present value of costs, but we have to recognise that it has a shorter life, so of course the present value of costs will be less. We need to use the annuity formula to spread the costs over each light bulb's life so we can get an equivalent annual cost.

For the low energy light bulb that lasts for 9 years:

###V_\text{0, low energy} = \frac{C_\text{EAC low energy}}{r} \left(1-\frac{1}{(1+r)^{T}} \right) ### ###14.87251468 = \frac{C_\text{EAC low energy}}{0.05} \left(1-\frac{1}{(1+0.05)^{9}} \right) ### ##C_\text{EAC low energy} = 2.09241528 ##

For the conventional light bulb that lasts for 1 year:

###V_\text{0, conventional} = \frac{C_\text{EAC conventional}}{r} \left(1-\frac{1}{(1+r)^{T}} \right) ### ###6.785714286 = \frac{C_\text{EAC conventional}}{0.05} \left(1-\frac{1}{(1+0.05)^{1}} \right) ### ###C_\text{EAC conventional} = 7.125 ###

Since the low energy light bulb has the lower equivalent annual cost, it is the best choice.

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

This is an equivalent periodic cash flow (sometimes called equivalent annual cost) question since the chickens can have different life times. The first step is to find the present value of the gain from slaughtering the chickens when they are 6 weeks old and also at 10 weeks old, then in the second step we spread these gains over 6 and 10 weeks using the annuity formula. The best time to slaughter a poor chicken will be at whichever time gives the highest equivalent weekly cash flow.

Note that the increase in chicken meat occurs at the end of each week, but you can't chop off bits of the chicken and sell it each week without killing the poor thing! So the increase in the amount of meat is all received at the end when the chicken is slaughtered.

###\begin{aligned} V_\text{0, 6 weeks} &= -C_\text{0, hatchling} + \frac{-C_\text{0,1,2,...5, feed}}{r} \left(1-\frac{1}{(1+r)^{6}} \right)(1+r)^1 + \frac{6 \times C_\text{6, young meat}}{(1+r)^6} \\ &= -0.50 + \frac{-0.30}{0.005} \left(1-\frac{1}{(1+0.005)^{6}} \right)(1+0.005)^1 + \frac{6 \times 0.70}{(1+0.005)^6} \\ &= 1.798416029 \\ \end{aligned} ###

###\begin{aligned} V_\text{0, 10 weeks} &= -C_\text{0, hatchling} + \frac{-C_\text{0,1,2,...9, feed}}{r} \left(1-\frac{1}{(1+r)^{10}} \right)(1+r)^1 + \\ &\frac{6 \times C_\text{10, young meat} + (10-6) \times C_\text{10, old meat}}{(1+r)^{10}} \\ &= -0.50 + \frac{-0.30}{0.005} \left(1-\frac{1}{(1+0.005)^{10}} \right)(1+0.005)^1 + \frac{6 \times 0.70 + 4 \times 0.40}{(1+0.005)^{10}} \\ &= 2.08409888 \\ \end{aligned} ###

It looks like the chickens should be allowed to live! But remember that the chickens have different lives. It is too early to draw a conclusion, we must find the equivalent weekly cash flow from letting them live for 6 and 10 weeks and then make a decision.

For the 6 week old chickens,

###\begin{aligned} V_\text{0, 6 weeks} &= \frac{C_\text{EWC 6 weeks}}{r} \left(1-\frac{1}{(1+r)^{T}} \right) \\ 1.798416029 &= \frac{C_\text{EWC 6 weeks}}{0.005} \left(1-\frac{1}{(1+0.005)^{6}} \right) \\ C_\text{EWC 6 weeks} &= 0.305003186 \\ \end{aligned} ###

For the 10 week old chickens,

###\begin{aligned} V_\text{0, 10 weeks} &= \frac{C_\text{EWC 10 weeks}}{r} \left(1-\frac{1}{(1+r)^{T}} \right) \\ 2.08409888 &= \frac{C_\text{EWC 10 weeks}}{0.005} \left(1-\frac{1}{(1+0.005)^{10}} \right) \\ C_\text{EWC 10 weeks} &= 0.214184036 \\ \end{aligned} ###

Since the 6 week old chickens have the highest equivalent weekly cash flow, it is best to slaughter the chickens young. This is in fact what happens in commercial chicken meat farms, chickens are killed at 6 weeks^source.

You may think that it doesn't make economic sense to kill them so young because they eat $0.30 of food but put on $0.40 of weight per week, so there is a gain for every extra week that the chickens are alive, which is true. But the opportunity cost of the feeding an old chicken that grows slowly is feeding a new younger chicken that grows quickly. Chicken farmers can make bigger gains per chicken by killing them young. Economically that makes better sense. But of course this ignores some ethical dilemmas raised by this train of thought.