There are many ways to write the ordinary annuity formula.
Which of the following is NOT equal to the ordinary annuity formula?
All answers are mathematically equivalent except (e).
The following cash flows are expected:
- 10 yearly payments of $60, with the first payment in 3 years from now (first payment at t=3 and last at t=12).
- 1 payment of $400 in 5 years and 6 months (t=5.5) from now.
What is the NPV of the cash flows if the discount rate is 10% given as an effective annual rate?
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} ###Question 58 NPV, inflation, real and nominal returns and cash flows, Annuity
A project to build a toll bridge will take two years to complete, costing three payments of $100 million at the start of each year for the next three years, that is at t=0, 1 and 2.
After completion, the toll bridge will yield a constant $50 million at the end of each year for the next 10 years. So the first payment will be at t=3 and the last at t=12. After the last payment at t=12, the bridge will be given to the government.
The required return of the project is 21% pa given as an effective annual nominal rate.
All cash flows are real and the expected inflation rate is 10% pa given as an effective annual rate. Ignore taxes.
The Net Present Value is:
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} ###
The first payment of a constant perpetual annual cash flow is received at time 5. Let this cash flow be ##C_5## and the required return be ##r##.
So there will be equal annual cash flows at time 5, 6, 7 and so on forever, and all of the cash flows will be equal so ##C_5 = C_6 = C_7 = ...##
When the perpetuity formula is used to value this stream of cash flows, it will give a value (V) at time:
The perpetuity equation gives a value one period before the first cash flow.
###V_{t-1} = \dfrac{C_t}{r} ###In this question the first cash flow is at time 5, and there's a one year period between each cash flow. So the value of the perpetuity will be at time 4, one year before time 5.
###V_4 = \dfrac{C_5}{r} ###A stock just paid its annual dividend of $9. The share price is $60. The required return of the stock is 10% pa as an effective annual rate.
What is the implied growth rate of the dividend per year?
The $9 dollar dividend just paid will not be included in the price since buying the stock now will not give the owner the right to receive that dividend since it's already paid, it's too late. The next dividend will be in one year, and it will have grown by g, so ## c_1 = 9 \times (1+g)^1## .
###p_0=\dfrac{c_1}{r-g}### ###60 = \dfrac{9 \times (1+g)^1}{0.1-g}### ###60 \times (0.1-g) = 9 \times (1+g)### ###6 - 60g = 9 + 9g### ###69g = -3### ###g = -3/69 = -0.043478261 ###Two years ago Fred bought a house for $300,000.
Now it's worth $500,000, based on recent similar sales in the area.
Fred's residential property has an expected total return of 8% pa.
He rents his house out for $2,000 per month, paid in advance. Every 12 months he plans to increase the rental payments.
The present value of 12 months of rental payments is $23,173.86.
The future value of 12 months of rental payments one year ahead is $25,027.77.
What is the expected annual growth rate of the rental payments? In other words, by what percentage increase will Fred have to raise the monthly rent by each year to sustain the expected annual total return of 8%?
The perpetuity with growth equation is suitable for valuing real estate since rental payments from land last forever. Re-arranging the perpetuity with growth equation into the 'total return' equation gives:
###P_0 = \frac{C_1}{r-g} ### ###r = \frac{C_1}{P_0} + g ###The growth rate in the rental payments g must equal the capital return which is the growth rate in the house price. See question 3 for an explanation of why.
Values can now be substituted into the equation to find g which is the growth rate in income cash flows (and the capital return). The price is supposed to be the market value now ($500,000) not the historical cost book value ($300,000) years ago. The income yield is the future value of income cash flows divided by the current market price:
###r = \frac{C_1}{P_0} + g ### ###0.08 = \frac{25,027.77}{500,000} + g ### ###\begin{aligned} g &= 0.08 - \frac{25,027.77}{500,000} \\ &= 0.08 - 0.050055546 \\ &= 0.029944454 \\ \end{aligned}###The following is the Dividend Discount Model (DDM) used to price stocks:
### P_0 = \frac{d_1}{r-g} ###Assume that the assumptions of the DDM hold and that the time period is measured in years.
Which of the following is equal to the expected dividend in 3 years, ## d_3 ##?
Re-arranging the dividend discount model we can break up total return into its dividend yield and capital yield components:
###P_0 = \frac{d_1}{r-g} ### ###r-g = \frac{d_1}{P_0} ### ###r = \frac{d_1}{P_0} + g ### ###r_\text{total} = r_\text{dividend} + r_\text{capital} ###So the following expressions are all equal to the dividend yield:
###r_\text{dividend} = \frac{d_1}{P_0} = r_\text{total} - r_\text{capital} = r - g###
Therefore, starting from answer (e),
###\begin{aligned} &P_0(1+g)^2(r-g) \\ &= P_2 \times (r-g) \\ &= P_2 \times \frac{d_1}{P_0} \\ &= P_2 \times \frac{d_1 \times (1+g)^2}{P_0 \times (1+g)^2} \\ &= P_2 \times \frac{d_3}{P_2} \\ &= d_3 \\ \end{aligned} ###
Note that all of the other answers give ## d_4 ##, the dividend in year 4.
The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation.
###p_0=\frac{d_1}{r_\text{eff}-g_\text{eff}}###
Which expression is NOT equal to the expected capital return?
Answer (d) is the dividend yield minus one (which is not very useful!). All of the other expressions will be equal to the firm's capital yield which is the same as the growth rate of the stock price and also the growth rate of the dividend, provided that the assumptions of the DDM hold.
Question 498 NPV, Annuity, perpetuity with growth, multi stage growth model
A business project is expected to cost $100 now (t=0), then pay $10 at the end of the third (t=3), fourth, fifth and sixth years, and then grow by 5% pa every year forever. So the cash flow will be $10.5 at the end of the seventh year (t=7), then $11.025 at the end of the eighth year (t=8) and so on perpetually. The total required return is 10℅ pa.
Which of the following formulas will NOT give the correct net present value of the project?
Answers a, b, c and d are all equivalent and give the same result: 44.73676041. Answer e is not the present value of the payments, it gives a result of 42.8683517. Answer e is very similar to answer a, but answer e discounts the annuity of the first three $10 payments by one too many years.
Answers a and b deal with the growing perpetuity in the same way by making the first perpetual cash flow the $10 at time 6. Because the perpetuity formula gives a value one year before the first cash flow, the value of the perpetuity is discounted by another 5 years.
Answers a and b differ in how they discount the three $10 payments at time 3, 4 and 5. Answer a uses the annuity formula to discount the three $10 payments which gives a value at time 2, one period before the first $10 cash flow at time 3. Then the value of the annuity is discounted by another 2 years. Answer b sums each cash flow's individual present value which gives the same result.
Answers c and d deal with the growing perpetuity in the same way by making the first perpetual cash flow the $10.50 at time 7. Because the perpetuity formula gives a value one year before the first cash flow, the value of the perpetuity is discounted by another 6 years.
Answers c and d differ in how they discount the four $10 payments at time 3, 4, 5 and 6. Answer c uses the annuity formula to discount the four $10 payments which gives a value at time 2, one period before the first $10 cash flow at time 3. Then the value of the annuity is discounted by another 2 years. Answer b sums each cash flow's individual present value which gives the same result.
Estimate the Chinese bank ICBC's share price using a backward-looking price earnings (PE) multiples approach with the following assumptions and figures only. Note that the renminbi (RMB) is the Chinese currency, also known as the yuan (CNY).
- The 4 major Chinese banks ICBC, China Construction Bank (CCB), Bank of China (BOC) and Agricultural Bank of China (ABC) are comparable companies;
- ICBC 's historical earnings per share (EPS) is RMB 0.74;
- CCB's backward-looking PE ratio is 4.59;
- BOC 's backward-looking PE ratio is 4.78;
- ABC's backward-looking PE ratio is also 4.78;
Note: Figures sourced from Google Finance on 25 March 2014. Share prices are from the Shanghai stock exchange.
ICBC's earnings per share (EPS) multiplied by the average of the other Chinese banks' backward-looking PE ratios gives a backward-looking PE multiple valuation of ICBC's stock price:
###\begin{aligned} P_\text{0,ICBC} &= \dfrac{ \left( \dfrac{P_\text{0,CCB}}{EPS_\text{0,CCB}} + \dfrac{P_\text{0,BOC}}{EPS_\text{0,BOC}} + \dfrac{P_\text{0,ABC}}{EPS_\text{0,ABC}} \right) }{3}.EPS_\text{0,ICBC} \\ &= \dfrac{ \left( \text{PE}_\text{0,CCB} + \text{PE}_\text{0,BOC} + \text{PE}_\text{0,ABC} \right) }{3}.EPS_\text{0,ICBC} \\ &= \dfrac{ \left( 4.59 + 4.78 + 4.78 \right) }{3} \times 0.74\\ &= 4.716666667 \times 0.74 \\ &= 3.490333333 \\ \end{aligned}###ICBC's share price actually closed at RMB3.35 on 25 March 2014 so the PE ratio valuation approach gives a number that's pretty close to the market's valuation.
Since the actual market traded price RMB3.35 is lower than the estimated price of RMB4.49 based on similar firms, ICBC stock might be under-priced and therefore should be bought. Or, perhaps ICBC has lower expected future growth potential or higher systematic risk compared to its peers so it's fairly priced or even over-priced after taking these factors into account.
Private equity firms are known to buy medium sized private companies operating in the same industry, merge them together into a larger company, and then sell it off in a public float (initial public offering, IPO).
If medium-sized private companies trade at PE ratios of 5 and larger listed companies trade at PE ratios of 15, what return can be achieved from this strategy?
Assume that:
- The medium-sized companies can be bought, merged and sold in an IPO instantaneously.
- There are no costs of finding, valuing, merging and restructuring the medium sized companies. Also, there is no competition to buy the medium-sized companies from other private equity firms.
- The large merged firm's earnings are the sum of the medium firms' earnings.
- The only reason for the difference in medium and large firm's PE ratios is due to the illiquidity of the medium firms' shares.
- Return is defined as: ##r_{0→1} = (p_1-p_0+c_1)/p_0## , where time zero is just before the merger and time one is just after.
Say a medium-sized firm makes $1m of earnings (also called profit or net income), then it will trade at a price of $5m since its price-to-earnings ratio is 5.
If lots of these firms are bought, merged and sold off as a large listed company then for each $1m of earnings, the larger firm will trade at a price of $15m since its price-to-earnings ratio is 15.
Therefore each medium-sized firm bought for $5m can be sold sold for $15m once it's merged, making a return of 200%.
###\begin{aligned} r_{0→1} &= \dfrac{p_1-p_0+c_1}{p_0} \\ &= \dfrac{15-5+0}{5} \\ &= 2 = 200\% \\ \end{aligned}###When using the dividend discount model, care must be taken to avoid using a nominal dividend growth rate that exceeds the country's nominal GDP growth rate. Otherwise the firm is forecast to take over the country since it grows faster than the average business forever.
Suppose a firm's nominal dividend grows at 10% pa forever, and nominal GDP growth is 5% pa forever. The firm's total dividends are currently $1 billion (t=0). The country's GDP is currently $1,000 billion (t=0).
In approximately how many years will the company's total dividends be as large as the country's GDP?
A low-quality second-hand car can be bought now for $1,000 and will last for 1 year before it will be scrapped for nothing.
A high-quality second-hand car can be bought now for $4,900 and it will last for 5 years before it will be scrapped for nothing.
What is the equivalent annual cost of each car? Assume a discount rate of 10% pa, given as an effective annual rate.
The answer choices are given as the equivalent annual cost of the low-quality car and then the high quality car.
This is an equivalent annual cost question because the two different types of cars last for different amounts of time.
###V_\text{0, low quality} = 1,000 ### ###V_\text{0, high quality} = 4,900 ###The low quality car appears cheaper because it has a lower present value of costs, but we have to recognise that it has a shorter life of just 1 year rather than 5, so of course the present value of costs will be less.
The next mistake that's commonly made is to divide the total costs by the number of years. This makes the high quality car look better since it's annual cost would be $980 per year compared to the low quality car's $1,000 per year. But this per year cost ignores the time value of money since the present value of $980 per year for 5 years is clearly less than the reality of paying $4,900 now.
To take the time value of money into account we need to use the annuity formula to spread the costs over each car's life so we can get an equivalent annual cost.
For the low quality car that lasts for 1 year,
###V_\text{0, low quality} = C_\text{1, EAC low quality} \times \frac{1}{r} \left(1-\frac{1}{(1+r)^{T}} \right) ### ###1,000 = C_\text{1, EAC low quality} \times \frac{1}{0.1} \left(1-\frac{1}{(1+0.1)^{1}} \right) ### ###1,000 = C_\text{1, EAC low quality} \times 0.90909090909 ### ###C_\text{1, EAC low quality} = 1,100 ###Note that it's a little silly to find the low quality car's EAC using the annuity formula over just one year when we could just grow the original $1,000 cost by the 10% cost of capital.
###C_\text{1, EAC low quality} = V_\text{0, low quality}(1+r)^1 = 1,000 \times (1+0.1)^1 = 1,100###For the high quality car that lasts for 5 years, the annuity formula is perfect.
###V_\text{0, high quality} = C_\text{1, EAC high quality} \times \frac{1}{r} \left(1-\frac{1}{(1+r)^{T}} \right) ### ###4,900 = C_\text{1, EAC high quality} \times \frac{1}{0.1} \left(1-\frac{1}{(1+0.1)^{5}} \right) ### ###4,900 = C_\text{1, EAC high quality} \times 3.79078676941 ### ###C_\text{1, EAC high quality} = 1,292.607656 ###Since the low quality car has the lower equivalent annual cost, it's the best choice.
You're advising your superstar client 40-cent who is weighing up buying a private jet or a luxury yacht. 40-cent is just as happy with either, but he wants to go with the more cost-effective option. These are the cash flows of the two options:
- The private jet can be bought for $6m now, which will cost $12,000 per month in fuel, piloting and airport costs, payable at the end of each month. The jet will last for 12 years.
- Or the luxury yacht can be bought for $4m now, which will cost $20,000 per month in fuel, crew and berthing costs, payable at the end of each month. The yacht will last for 20 years.
What's unusual about 40-cent is that he is so famous that he will actually be able to sell his jet or yacht for the same price as it was bought since the next generation of superstar musicians will buy it from him as a status symbol.
Bank interest rates are 10% pa, given as an effective annual rate. You can assume that 40-cent will live for another 60 years and that when the jet or yacht's life is at an end, he will buy a new one with the same details as above.
Would you advise 40-cent to buy the or the ✓?
Note that the effective monthly rate is ##r_\text{eff monthly}=(1+0.1)^{1/12}-1=0.00797414##
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.
You just bought a nice dress which you plan to wear once per month on nights out. You bought it a moment ago for $600 (at t=0). In your experience, dresses used once per month last for 6 years.
Your younger sister is a student with no money and wants to borrow your dress once a month when she hits the town. With the increased use, your dress will only last for another 3 years rather than 6.
What is the present value of the cost of letting your sister use your current dress for the next 3 years?
Assume: that bank interest rates are 10% pa, given as an effective annual rate; you will buy a new dress when your current one wears out; your sister will only use the current dress, not the next one that you will buy; and the price of a new dress never changes.
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.
Question 548 equivalent annual cash flow, time calculation, no explanation
An Apple iPhone 6 smart phone can be bought now for $999. An Android Kogan Agora 4G+ smart phone can be bought now for $240.
If the Kogan phone lasts for one year, approximately how long must the Apple phone last for to have the same equivalent annual cost?
Assume that both phones have equivalent features besides their lifetimes, that both are worthless once they've outlasted their life, the discount rate is 10% pa given as an effective annual rate, and there are no extra costs or benefits from either phone.
No explanation provided.