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I am solving the following problem:

The owner has premises that he decided to rent. Rental costs for the first year are 1000 and each year costs increase by 5% for 5 years.

How much rent should be agreed with the customer to cover all costs?

My solution:

I calculated the price of the rent during following 5 years:

  1. year: 1000

  2. year:1050

  3. year:1102,5
  4. year:1157,625
  5. year:1215,51

Then I added the numbers and then discounted it for five years:

$$\frac{5525,635}{\left ( 1+0,05 \right )^{5}}$$

Is that correct?

  • I am not sure I understand the discount. The numerator looks to be correct. $$1000\sum_{n=0}^4 (1,05)^n = 1000 \dfrac{1,05^5-1}{0,05} = 5525,63$$ which is what you calculated, as well. That total amount is correct, and I am not understanding why you would then divide by $(1+0,05)^5$ – SlipEternal Nov 18 '19 at 19:57
  • Why do you divide by $(1+ 0.05)^5$ rather than by $5$? $\frac {5525,635}{(1+0.05)^5}=4329.48$. That's pretty high rent considering the highest the costs will ever get are $1215.51$. If $5525.635$ is the total costs for $5$ years, then the rent per year to cover it should be $1105.127$ – fleablood Nov 18 '19 at 19:59
  • @InterstellarProbe but what should I do? I thought that I should discount it, because I want the price for rent today –  Nov 18 '19 at 19:59
  • If the customer will be paying the same amount every year, then shouldn't that be $5525.635/5 = 1015.127$ yearly rent? Then the total paid for five equal payments is exactly the total rental costs you computed. – MPW Nov 18 '19 at 19:59
  • Discounting only makes sense if you expect something to grow with time. This is not the case for the rent, assuming it will be the same every year. – MPW Nov 18 '19 at 20:00
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    What do you mean "discount"? I can't think of any meaning of the word that is relevant here. – fleablood Nov 18 '19 at 20:01
  • @MPW But 5525.635 is the future amount, is it not? and I want todays amount –  Nov 18 '19 at 20:01
  • @fleablood I thought that I should calculate todays value –  Nov 18 '19 at 20:02
  • I see what you've done Daniel, but is that a correct interpretation of the question? – Phil H Nov 18 '19 at 20:04
  • Your original question suggests that costs are growing (and you computed them) but rent is constant (if this is not the case, please clarify the question). You want to find $r$ so that $$\underbrace{r+r+r+r+r}{\textrm{5 yrs of rent}} = \underbrace{c_1+c_2+c_3+c_4+c_5}{\textrm{5 yrs of costs}}$$ $$5r = 5525.635$$ $$r=1015.127$$ – MPW Nov 18 '19 at 20:05
  • Todays value is $1000$. I have no idea what you are talking about. I really don't. And I have no idea what adding up the costs of the rent will be for $5$ years and then dividing but how much more the final rent cost compared to today is supposed to signify. – fleablood Nov 18 '19 at 20:05
  • @fleablood, okay, thanks, it is possible that I misunderstood the question –  Nov 18 '19 at 20:07
  • @MPW thanks, probably I just misunderstood the question –  Nov 18 '19 at 20:07

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