# calculate indices with base year and relative percentage change

Another way would be to use cumprod() after converting the values to percentages:

library(dplyr)

start_tbl %>%
group_by(id, grp) %>%
mutate(idx_value = cumprod(c(100, (100 + value[-1]) / 100)))

# A tibble: 14 x 5
# Groups:   id, grp [4]
id   grp  year value idx_value
<int> <int> <int> <dbl>     <dbl>
1     1     1     7   2       100
2     1     1     8  -7        93
3     1     1     9  -2.3      90.9
4     1     1    10   1.1      91.9
5     1     2     7  -1       100
6     1     2     8 -12        88
7     1     2     9  -4        84.5
8     1     2    10   2        86.2
9     2     1     7   1       100
10     2     1     8  -3        97
11     2     1     9   2        98.9
12     2     2     7  -1       100
13     2     2     8  -4        96
14     2     2     9  -2        94.1


Based on the new dataset

library(purrr)
library(dplyr)
start_tbl2 %>%
group_by(id, grp) %>%
mutate(idx_vlue = accumulate(value[-1], ~ .x * (1 + .y/100), .init = 100 ))
# A tibble: 8 x 5
# Groups:   id, grp [2]
#     id   grp  year value idx_vlue
#  <int> <int> <int> <dbl>    <dbl>
#1     1     1     7   2      100
#2     1     1     8 -12       88
#3     1     1     9 -18.3     71.9
#4     1     1    10 100      144.
#5     1     2     7  15      100
#6     1     2     8  30      130
#7     1     2     9  40      182
#8     1     2    10 -50       91


and using 'start_tbl

start_tbl %>%
group_by(id, grp) %>%
mutate(idx_vlue = accumulate(value[-1], ~ .x * (1 + .y/100), .init = 100 ))
# A tibble: 14 x 5
# Groups:   id, grp [4]
#      id   grp  year value idx_vlue
#   <int> <int> <int> <dbl>    <dbl>
# 1     1     1     7   2      100
# 2     1     1     8  -7       93
# 3     1     1     9  -2.3     90.9
# 4     1     1    10   1.1     91.9
# 5     1     2     7  -1      100
# 6     1     2     8 -12       88
# 7     1     2     9  -4       84.5
# 8     1     2    10   2       86.2
# 9     2     1     7   1      100
#10     2     1     8  -3       97
#11     2     1     9   2       98.9
#12     2     2     7  -1      100
#13     2     2     8  -4       96
#14     2     2     9  -2       94.1