Understanding a specific notation letter.

Question

I have data which looks like:

   AGE   AGE_CAT decile house_price
1   22 bin_18_24      1    4726.947
2   22 bin_18_24      2    8161.651
3   23 bin_18_24      3   11699.392
4   24 bin_18_24      4   14941.351
5   21 bin_18_24      5   17676.896
6   20 bin_18_24      6   21592.982
7   20 bin_18_24      7   29277.973
8   20 bin_18_24      8   37110.146
9   23 bin_18_24      9   47122.003
10  19 bin_18_24     10  112750.115
11  25 bin_25_29      1    7584.590
12  25 bin_25_29      2   17151.103
13  27 bin_25_29      3   25990.247
14  25 bin_25_29      4   32154.051
15  27 bin_25_29      5   38576.356
16  28 bin_25_29      6   47229.818
17  26 bin_25_29      7   54688.024
18  29 bin_25_29      8   64382.058
19  29 bin_25_29      9   80473.481
20  26 bin_25_29     10  187721.742

Where;

AGE is the persons age,

AGE_CAT is the category of peoples ages (between 18 and 24),

decil is a ranking of 1 to 10 of the persons house Price (i.e. decile 10 = most expensive).

I am trying to compute the NPV based on this formula;

Where $(a)$ is the AGE, $b_i(a) = [decile1(a), decile10(a)]$.

I am having a hard time trying to figure out what "a_tilde" is. I have been told that its the index for age and that by putting in "tilde" we allow age to vary independently of the age of the bin (AGE_CAT).

In R I am using this code:

npvs <- df %>%
  mutate(a_tilde = 1, # I don`t know what should be here
         discount = 1 / (1 + rate) ^ (a_tilde - AGE),
         CF_disc = house_price * discount)

Answer 1

The comments of Mauro Allegranza go in the right direction. But it seems that you haven´t understood fully how to interpret the formula. We have

$$NVP(a, b_i(a))=\sum\limits_{\tilde a>a} \frac{f(\tilde a, b_i(\tilde a))}{(1+r)^{(\tilde a-a)}}$$

$\tilde a$ is an index which are integers only. For instance, if $a=25$ then $\tilde a>a$ means that $\tilde a \in \{26,27,28,29\}$. In that case

$$NVP(25, b_i(25))=\sum\limits_{\tilde a>25} \frac{f(\tilde a, b_i(\tilde a))}{(1+r)^{(\tilde a-25)}}=\sum\limits_{\tilde a=26}^{29} \frac{f(\tilde a, b_i(\tilde a))}{(1+r)^{(\tilde a-25)}}$$

$$=\frac{f(26, b_i(26))}{(1+r)^{(26-25)}}+\frac{f(27, b_i(27))}{(1+r)^{(27-25)}}+\frac{f(28, b_i(28))}{(1+r)^{(28-25)}}+\frac{f(29, b_i(29))}{(1+r)^{(29-25)}}$$

$$=\frac{f(26, b_i(26))}{(1+r)^{1}}+\frac{f(27, b_i(27))}{(1+r)^{2}}+\frac{f(28, b_i(28))}{(1+r)^{3}}+\frac{f(29, b_i(29))}{(1+r)^{4}}$$

Answer 2

From the definition you posted, $\tilde{a}$ is simply the summation index when you vary $a$. The particular values are below the capital sigma.

Also, $a$ and $\tilde{a}$ have similar purposes in the calculation, because $f(\tilde{a}, b_i(\tilde{a})$ and $NPV(a, b_i(a))$ have similar forms, as well as the direct subtraction in the denominator of the summed term.