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

We will use the annuity equation and the present value of a single cash flow equation. Keep in mind that the annuity equation gives a value that is one period before the first cash flow at t=3, so the value of the annuity will be at t=2 and needs discounting by 2 periods to get to t=0.

###\begin{aligned} V_{0} &= \dfrac{C_{3} \times \dfrac{1}{r_\text{eff annual}} \left(1 - \dfrac{1}{(1+r_\text{eff annual})^{10}} \right)}{(1+r_\text{eff annual})^2} + \dfrac{C_{5.5}}{(1+r_\text{eff annual})^{5.5}} \\ &= \dfrac{60 \times \dfrac{1}{0.1} \left(1 - \dfrac{1}{(1+0.1)^{10}} \right)}{(1+0.1)^2} + \dfrac{400}{(1+0.1)^{5.5}} \\ &= \dfrac{60 \times 6.144567106}{(1+0.1)^2} + \dfrac{400}{(1+0.1)^{5.5}} \\ &= 304.689278 + 236.810101 \\ &= 541.4993789 \\ \end{aligned} ###

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

Use the annuity formula to find the present value of the 10 payments of $80 starting in 3 years (t=3). Keep in mind that the annuity formula gives a value that is one period before the first cash flow at t=3, so the value of the annuity will be at t=2 and needs discounting by 2 periods to get to t=0.

###\begin{aligned} V_\text{0, annuity} &= C_{1} \times \dfrac{1}{r} \left(1 - \dfrac{1}{(1+r)^{10}} \right) \\ V_\text{2, annuity} &= C_{\mathbf{3}} \times \dfrac{1}{r} \left(1 - \dfrac{1}{(1+r)^{10}} \right) \\ &= 80 \times \dfrac{1}{0.1} \left(1 - \dfrac{1}{(1+0.1)^{10}} \right) \\ &= 80 \times 6.144567106 \\ &= 491.56536848 \\ \end{aligned} ###

Now discount this value at year 2 (t=2) to the present (t=0) using the 'present value of the single cash flow' formula: ###\begin{aligned} V_\text{0, annuity} &= \dfrac{V_\text{2, annuity}}{(1+r)^2} \\ &= \dfrac{491.56536848 }{(1+0.1)^{2}} \\ &= 406.252370645 \\ \end{aligned} ###

To discount the $600 payment in 5 years and 6 months (t=5.5) to a value now (a present value), again use the 'present value of the single cash flow' formula:

###\begin{aligned} V_\text{0, single cash flow} &= \dfrac{C_\text{5.5}}{(1+r)^{5.5}} \\ &= \dfrac{600}{(1+0.1)^{5.5}} \\ &= 355.2151514 \\ \end{aligned} ###

Now we just add the present values of the annuity and the single payment in 5.5 years together:

###\begin{aligned} V_\text{0, all} &= V_\text{0, annuity} + V_\text{0, single cash flow} \\ &= 406.252370645 + 355.2151514 \\ &= 761.4675221 \\ \end{aligned} ###

Here's all the steps in one big formula:

###\begin{aligned} V_{0} &= \dfrac{C_{3} \times \dfrac{1}{r_\text{eff annual}} \left(1 - \dfrac{1}{(1+r_\text{eff annual})^{10}} \right)}{(1+r_\text{eff annual})^2} + \dfrac{C_{5.5}}{(1+r_\text{eff annual})^{5.5}} \\ &= \dfrac{80 \times \dfrac{1}{0.1} \left(1 - \dfrac{1}{(1+0.1)^{10}} \right)}{(1+0.1)^2} + \dfrac{600}{(1+0.1)^{5.5}} \\ &= \dfrac{80 \times 6.144567106}{(1+0.1)^2} + \dfrac{600}{(1+0.1)^{5.5}} \\ &= 406.2523706 + 355.2151514 \\ &= 761.4675221 \\ \end{aligned} ###

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

The annuity formula can be applied to find the present value of the 6 equal payments, but care must be taken since the present value of an annuity is one period before the first cash flow.

One method is to make an annuity of the cash flows from periods 1 to 5 (five cash flows) and add the cash flow at time zero separately.

###\begin{aligned} V_0 &= -C_1.\dfrac{1}{r}\left( 1-\dfrac{1}{(1+r)^5} \right) -C_0 + \dfrac{C_7}{(1+r)^7} \\ &= -1,000 \times \dfrac{1}{0.1}\left( 1-\dfrac{1}{(1+0.1)^5} \right) -1,000 + \dfrac{10,000}{(1+0.1)^7} \\ &= 340.7944 \\ \end{aligned}###

Another method is to make an annuity of the cash flows from periods 0 to 5 (six cash flows) which gives an annuity one period in the past (t=-1), and then grow the annuity forward by one period to get a present value (at t=0).

###\begin{aligned} V_0 &= -C_0.\dfrac{1}{r}\left( 1-\dfrac{1}{(1+r)^6} \right).(1+r)^1 + \dfrac{C_7}{(1+r)^7} \\ &= -1,000 \times \dfrac{1}{0.1}\left( 1-\dfrac{1}{(1+0.1)^6} \right).(1+0.1)^1 + \dfrac{10,000}{(1+0.1)^7} \\ &= 340.7944 \\ \end{aligned}###

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

The annuity formula can be applied to find the present value of the 6 equal payments from t=2 to 7. But care must be taken since the present value of an annuity is one period before the first cash flow (at t=2), so the whole annuity value will be at t=1 so it needs to be discounted back one extra period to get a present value.

###\begin{aligned} V_0 &= C_0 - \dfrac{ C_2.\dfrac{1}{r}\left( 1-\dfrac{1}{(1+r)^6} \right) }{(1+r)^1} \\ &= 5,000 - \dfrac{ 1,000 \times \dfrac{1}{0.1}\left( 1-\dfrac{1}{(1+0.1)^6} \right) }{(1+0.1)^1} \\ &= 5,000 - 3,959.3279 \\ &= 1,040.6721 \\ \end{aligned}###

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.

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.

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 cars have different lives, we need to compare their equivalent annual cash flows (EAC) or equivalent weekly cash flows to make an informed decision about which is the best value for money.

Since cash flows are weekly, the discount rate needs to be converted from an effective annual rate to an effective weekly rate:

###r_\text{eff wkly} = (1+r_\text{eff annual})^{1/52}-1= (1+0.1)^{1/52}-1 = 0.001834569###

Find the NPV of each car's cash flows:

###\begin{aligned} PV(\text{cash flows}) =& -(\text{buy price now}) \\ &- PV(\text{fuel at end of each week for T weeks}) \\ &+ PV(\text{sale price in T weeks}) \\ \end{aligned}### ###\begin{aligned} V_\text{0, old car} &= -C_0 - C_1.\dfrac{1}{r_\text{eff wkly}} \left( 1- \dfrac{1}{(1+r_\text{eff wkly})^{T_\text{wks}}} \right) + \dfrac{C_T}{(1+r_\text{eff wkly})^{T_\text{wks}}} \\ &= -5,000 - 90.\dfrac{1}{0.001834569} \left( 1- \dfrac{1}{(1+0.001834569)^{3 \times 52}} \right) + \dfrac{500}{(1+0.001834569)^{3 \times 52}} \\ &= -16,824.3034 \\ \end{aligned}### ###\begin{aligned} V_\text{0, new car} &= -14,000 - 50.\dfrac{1}{0.001834569} \left( 1- \dfrac{1}{(1+0.001834569)^{4 \times 52}} \right) + \dfrac{1,000}{(1+0.001834569)^{4 \times 52}} \\ &= -21,956.25202 \\ \end{aligned}###

The new car looks more expensive when comparing present values of cash flows for each car's life, but the new car lasts longer so it should cost more. To find the equivalent weekly cash flows (EWC) and take the time value of money into account properly, use the annuity formula to spread the present value of cash flows across each week of each car's life:

###\begin{aligned} V_\text{0, old car} &= \frac{C_\text{EWC old car}}{r_\text{eff wkly}} \left(1-\frac{1}{(1+r_\text{eff wkly})^{T_\text{wks}}} \right) \\ -16,824.3034 &= \frac{C_\text{EWC old car}}{0.001834569} \left(1-\frac{1}{(1+0.001834569)^{3 \times 52}} \right) \\ C_\text{EWC old car} &= -124.1141124 \\ \end{aligned} ###

###\begin{aligned} V_\text{0, new car} &= \frac{C_\text{EWC new car}}{r_\text{eff wkly}} \left(1-\frac{1}{(1+r_\text{eff wkly})^{T_\text{wks}}} \right) \\ -21,956.25202 &= \frac{C_\text{EWC new car}}{0.001834569} \left(1-\frac{1}{(1+0.001834569)^{4 \times 52}} \right) \\ C_\text{EWC new car} &= -127.0724466 \\ \end{aligned} ###

Since the old car's equivalent weekly cash flow is higher (less negative) and therefore less costly, buy the old car.

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

Since the cash flows are daily, find the effective daily rate:

###r_\text{eff daily} = (1+r_\text{eff annual})^{1/365}-1= (1+0.1)^{1/365}-1 = 0.000261157876###

Find the NPV of eating high sugar treats for 5 years, and low-sugar treats for 15 years:

###\begin{aligned} V_\text{0, high sugar} &= -C_1.\dfrac{1}{r_\text{eff daily}} \left( 1- \dfrac{1}{(1+r_\text{eff daily})^{T_\text{days}}} \right) - \dfrac{C_\text{5 years}}{(1+r_\text{eff annual})^{T_\text{years}}} \\ &= \dfrac{-2}{0.000261157876} \left( 1- \dfrac{1}{(1+0.000261157876)^{5 \times 365}} \right) - \dfrac{2,000}{(1+0.1)^{5}} \\ &= -4,144.904064 \\ \end{aligned}### ###\begin{aligned} V_\text{0, low sugar} &= -C_1.\dfrac{1}{r_\text{eff daily}} \left( 1- \dfrac{1}{(1+r_\text{eff daily})^{T_\text{days}}} \right) - \dfrac{C_\text{15 years}}{(1+r_\text{eff annual})^{T_\text{years}}} \\ &= \dfrac{-4}{0.000261157876} \left( 1- \dfrac{1}{(1+0.000261157876)^{15 \times 365}} \right) - \dfrac{2,000}{(1+0.1)^{15}} \\ &= -12,128.5641 \\ \end{aligned}###

The low sugar treats look more expensive (cash flows are more negative) when comparing present values of cash flows, but the time periods are not equal. To find the equivalent annual cash flow (EAC) and take the time value of money into account properly, use the annuity formula to spread the present value of cash flows across each year of the high and low sugar strategies' lives:

###\begin{aligned} V_\text{0, high sugar} &= \frac{C_\text{EAC, high sugar}}{r_\text{eff yrly}} \left(1-\frac{1}{(1+r_\text{eff yrly})^{T_\text{yrs}}} \right) \\ -4,144.904064 &= \frac{C_\text{EAC high sugar}}{0.1} \left(1-\frac{1}{(1+0.1)^{5}} \right) \\ C_\text{EAC high sugar} &= -1,093.41525 \\ \end{aligned}### ###\begin{aligned} V_\text{0, low sugar} &= \frac{C_\text{EAC, low sugar}}{r_\text{eff yrly}} \left(1-\frac{1}{(1+r_\text{eff yrly})^{T_\text{yrs}}} \right) \\ -12,128.5641 &= \frac{C_\text{EAC low sugar}}{0.1} \left(1-\frac{1}{(1+0.1)^{15}} \right) \\ C_\text{EAC low sugar} &= -1,594.588131 \\ \end{aligned}###

Since the high sugar treats are cheaper per year (the cash flows are more positive or less negative), buy them instead of the low-sugar treats.

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

By letting your brother use the suit, it will wear out 1 year earlier. The one year of additional cost should be found using the annuity formula, called the 'equivalent annual cost' (EAC) method:

###\begin{aligned} V_0 &= C_1\times \frac{1}{r}\left(1-\frac{1}{(1+r)^T}\right) \\ 600 &= EAC_1\times \frac{1}{0.1}\left(1-\frac{1}{(1+0.1)^6}\right) \\ EAC_1 &= 600 \div \left( \frac{1}{0.1}\left(1-\frac{1}{(1+0.1)^6}\right) \right)\\ &= 600 \div 4.355260699 \\ &= 137.7644282 \\ \end{aligned} ###

Note that this is not equal to just dividing the cost of the suit ($600) by how long it lasts (6yrs) which is $100. This is because $600 now is a higher cost in present value terms than $100 received at the end of each year for 6 years. Dividing the cost by the total time doesn't take the time value of money into account.

In 4 yrs you will have to buy another suit at a cost of $600 (at t=4). The one year of additional cost is the equivalent annual cost just calculated. But we have to be careful since the annuity equation used to find the EAC actually gives a figure one year ahead (##EAC_1##) of the price now (##V_0##). So even though the new suit will need to be bought at t=4, the EAC is actually one period ahead at t=5. So the present value of this additional cost is:

###\begin{aligned} V_\text{0, additional cost} &= \frac{EAC_5}{(1+r)^5} \\ &= \frac{137.7644282}{(1+0.1)^5} \\ &= 85.54087104 \\ \end{aligned} ###

This is the present value of the additional cost of letting your brother use the suit.

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

By letting your sister use the dress, it will wear out 3 years earlier. Each year of additional cost should be found using the annuity formula, called the 'equivalent annual cost' (EAC) method:

###\begin{aligned} V_0 &= C_1\times \frac{1}{r}\left(1-\frac{1}{(1+r)^T}\right) \\ 600 &= EAC_1\times \frac{1}{0.1}\left(1-\frac{1}{(1+0.1)^6}\right) \\ EAC_1 &= 600 \div \left( \frac{1}{0.1}\left(1-\frac{1}{(1+0.1)^6}\right) \right)\\ &= 600 \div 4.355260699 \\ &= 137.7644282 \\ \end{aligned} ###

Note that this is not equal to just dividing the cost of the dress ($600) by how long it lasts (6yrs) which is $100. This is because $600 now is a higher cost in present value terms than $100 received at the end of each year for 6 years. Dividing the cost by the total time doesn't take the time value of money into account.

In 3 yrs you will have to buy another dress at a cost of $600 (at t=3), 3 years sooner than you budgeted. The three years of additional costs can be calculated using the equivalent annual cost just found. But we have to be careful since the annuity equation used to find the EAC actually gives a figure one year ahead (##EAC_1##) of the price now (##V_0##). So even though the new dress will need to be bought at t=3, the EAC's will occur one year later at t=4, then 5 and 6. So the present value of these additional costs is:

###\begin{aligned} V_\text{0, additional costs} &= \frac{EAC_4}{(1+r)^4} + \frac{EAC_5}{(1+r)^5} + \frac{EAC_6}{(1+r)^6} \\ &= \frac{137.7644282}{(1+0.1)^4} + \frac{137.7644282}{(1+0.1)^5} + \frac{137.7644282}{(1+0.1)^6} \\ &= 257.4002574 \\ \end{aligned} ###

This is the present value of the additional cost of letting your sister use the dress. Notice that even though the dress lasts half as long, the additional cost is not half the value of the dress ($300) or even the present value if we assume the $300 is paid in 3 years ($225.39) since neither of these methods take the time value of money into account properly.

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.