| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 843.1 | | CLT::GILBERT | | Sun Mar 20 1988 16:19 | 25 |
| 1. Suppose two employees start with the same salary. One employee
gets a raise of X% every 12 months, and the other gets a raise
of Y% every 15 months. After 5 years, they again earn the same
salary. Find Y in terms of X.
(1 + X/100) ** 5 = (1 + Y/100) ** 4
Y = 100 (1 + X/100) ** (5/4) - 100
(ex: X = 10%, Y = 12.6525%)
2. Suppose two employees start with the same salary. One employee
gets a raise of X% every 12 months, and the other gets a raise
of Y% every 15 months. After 5 years, they have earned the same
amount of money. Find Y in terms of X.
12 (1 + (1+X/100) + (1+X/100)^2 + (1+X/100)^3 + (1+X/100)^4)
= 15 (1 + (1+Y/100) + (1+Y/100)^2 + (1+Y/100)^3)
12 ( 5 + 10 (X/100) + 10 (X/100)^2 + 5 (X/100)^3 + (X/100)^4)
= 15 ( 4 + 6 (Y/100) + 4 (Y/100)^2 + (Y/100)^3 )
4 ( 1 + (1+X/100)**5 )/ X = 5 ( 1 + (1+Y/100)**4 )/ Y
(ex: X = 10%, Y = 12.4427%)
|
| 843.2 | | LABC::FRIEDMAN | | Mon Mar 21 1988 11:07 | 4 |
| In the problem, only the first period is 15 months long; all succeeding
periods are 12 months long.
|
| 843.3 | | CLT::GILBERT | | Mon Mar 21 1988 12:14 | 11 |
| Yes, I realize that the original problem had successive raises spaced
every 12 months. However, in order to compare a 15-month raise with
a 12-month raise, I compared four 15-month raises with five 12-month
raises, giving 60 months in either case. I then used two reasonable
metrics for comparing the raises -- one compared salaries and the other
compared the total received salary. These yielded two slightly
different answers.
In what sense do you want to "equate" the two? The are, after all,
different, and the only way to make all the effects identical is
to always give a 0% raise.
|
| 843.4 | A = B | VINO::JMUNZER | | Mon Mar 21 1988 12:47 | 51 |
| A gets paid
S0 for the first 12 months
S1 = (1+x) * S0 for the next 12 months
S2 = for the next 12 months
S3 = for the next 12 months
etc.
B gets paid
T0 = S0 for the first 15 months
T1 = (1+y) * T0 for the next 12 months
T2 = for the next 12 months
T3 = for the next 12 months
etc.
Suppose that the ratios T2:S2, T3:S3, T4:S4, ... are the same as the
ratio T1:S1.
Suppose that future money is discounted at interest rate i, so that money
M, t years from now, is worth
v^t * M, where v = 1 / (1+i),
today.
Think about A's and B's present values (today's values of their total
salaries):
APV = S0 + v^1 * S1 + v^2 * S2 + v^3 * S3 + ...
BPV = T0 + v^1.25 * T1 + v^2.25 * T2 + v^3.25 * T3 + ...
Rewrite APV = S0 + (v * S1 + v^2 * S2 + v^3 * S3 + ...)
BPV = S0 + v^.25 * (v * T1 + v^2 * T2 + v^3 * T3 + ...)
If ST = T1 / S1 = T2 / S2 = T3 / S3 = ...,
then BPV = S0 + v^.25 * ST * (v * S1 + v^2 * S2 + v^3 * S3 + ...)
Want A's present value to equal B's.
Rearrange (APV - S0) = (BPV - S0) * v^.25 * ST
or 1 = v^.25 * ST
Given ST = (1+y) / (1+x),
then (1+y) = (1+x) * v^(-.25)
E.g. i = 5%
x = 10%
y = 11.35%
John
|
| 843.5 | re .4 | VINO::JMUNZER | | Tue Mar 22 1988 11:23 | 4 |
| Hey, you forgot that B wants to be paid in the 13th, 14th, and 15th
months! Not as easy as you thought.
John
|