<p>Hard to speculate on the inner workings of your programs since you do not share their details; but by thinking about the marginal benefit of each type of asset (painting, money) you can determine that the correct time to sell is <span class="math-container">$T = 10$</span> or <span class="math-container">$11$</span> (in the discrete time model).</p>
<p>If you keep the painting, you receive a benefit of \$10 + \$1, appreciation and psychological benefit. If you sell it, you get 10% marginal benefit in the form of interest. The question is, when will the 10% exceed the constant \$11. Once it does, it will exceed it in every following year because the painting appreciates linearly. The question could be trickier if the interest rate was changing, but in this simple situation this simple analysis does work.</p>
<hr />
<p>Numbers edited after <a href="https://economics.stackexchange.com/questions/58329/present-value-calculation/58330?noredirect=1#comment96861_58330">this comment</a> pointed out a mistake I made.</p>

CC BY-SA 4.0
2024-05-07T16:39:05.857Z
why the appreciation is $10?
PiVoyager
CC BY-SA 4.0
2024-05-07T17:28:58.300Z
My bad, the appreciation is only 1 dollar per year, price is $100 + T$ in year $T$.
Giskard
CC BY-SA 4.0
2024-05-07T16:08:26.897Z
2024-05-07T17:30:13.350Z
2024-05-07T17:30:13.350Z
Giskard
<p>Vincent is deciding whether to buy a painting. The current sale price of the painting is 100 dollars. The resale price of the painting after <span class="math-container">$T$</span> years is <span class="math-container">$100+T$</span> dollars, where <span class="math-container">$T$</span> is a positive integer. Ownership of the painting generates a psychological benefit of 10 dollars per year, where the benefit is received at the end of each year. Assume that the discount rate is <span class="math-container">$r=0.10$</span>.</p>
<p>Suppose that Vincent buys the painting. What is the earliest year after which he should sell the painting? [Hint: maximize the net present value with respect to the year after which he sells the painting.]</p>
<p><span class="math-container">$$
\begin{aligned}
&amp; N P V(T)=-100+\frac{10}{1+10 \%}+\frac{10}{(1+10 \%)^2}+\frac{10}{(1+10 \%)^3}+ \cdots + \frac{10}{(1+10)^T} \\
&amp;+\frac{100+T}{(1+10 \%)^T}
\end{aligned}
$$</span></p>
<p>Let <span class="math-container">$N P V(T)$</span> first derivative equal to zero. Using a spreadsheet software <span class="math-container">$T \approx 5.79$</span></p>
<p>The above is the solution, but I doubt it.</p>
<p>I get a different answer <span class="math-container">$T \approx 10.4921$</span> by using the finite sum formula of geometric series and use Wolfram|Alpha.</p>

<microeconomics>
Present value calculation
PiVoyager
# Present value calculation

(Asked by: username_0 on 2024-05-07)

Vincent is deciding whether to buy a painting. The current sale price of the painting is 100 dollars. The resale price of the painting after $T$ years is $100+T$ dollars, where $T$ is a positive integer. Ownership of the painting generates a psychological benefit of 10 dollars per year, where the benefit is received at the end of each year. Assume that the discount rate is $r=0.10$.

Suppose that Vincent buys the painting. What is the earliest year after which he should sell the painting? [Hint: maximize the net present value with respect to the year after which he sells the painting.]

$$
\begin{aligned}
& N P V(T)=-100+\frac{10}{1+10 \%}+\frac{10}{(1+10 \%)^2}+\frac{10}{(1+10 \%)^3}+ \cdots + \frac{10}{(1+10)^T} \\
&+\frac{100+T}{(1+10 \%)^T}
\end{aligned}
$$

Let $N P V(T)$ first derivative equal to zero. Using a spreadsheet software $T \approx 5.79$

The above is the solution, but I doubt it.

I get a different answer $T \approx 10.4921$ by using the finite sum formula of geometric series and use Wolfram|Alpha.

**Tags:** microeconomics


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## 1 answers

### Answer by username_1 on 2024-05-07. Score: 2 | Accepted answer

Hard to speculate on the inner workings of your programs since you do not share their details; but by thinking about the marginal benefit of each type of asset (painting, money) you can determine that the correct time to sell is $T = 10$ or $11$ (in the discrete time model).

If you keep the painting, you receive a benefit of \$10 + \$1, appreciation and psychological benefit. If you sell it, you get 10% marginal benefit in the form of interest. The question is, when will the 10% exceed the constant \$11. Once it does, it will exceed it in every following year because the painting appreciates linearly. The question could be trickier if the interest rate was changing, but in this simple situation this simple analysis does work.

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Numbers edited after [this comment](https://economics.stackexchange.com/questions/58329/present-value-calculation/58330?noredirect=1#comment96861_58330) pointed out a mistake I made.

> **Comments:**
> **username_0**: why the appreciation is $10?  
> **username_1**: My bad, the appreciation is only 1 dollar per year, price is $100 + T$ in year $T$.  

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