Question

Question

Answer 1

So... Build it. What's preventing you from drawing a map?

Answer 2

You INVENT the specific formula foe each problem. First off, the words "minimum of 20 years" is just silly. Why don't we do EXACTLY 20 years?! There is not a unique solution to "minimum of 20".

Answer 3

Second, it's a little silly to make a deposit and a withdrawal on the same day, 65th birthday, but I guess we can go with that.

Answer 4

Draw a map

65 200000
66 200000*1.03
67 200000*1.03^2
68 etc...
69 ...
70
71
72
73
74
75 200000*1.03^10 + 75000
76 200000*1.03^11
77 etc...
78
79
80
81
82
83
84 200000*1.03^19

Make sense?

Answer 5

It's a map. It's not a formula.
It shows the payments increasing 3% each year. Do you believe?

Answer 6

After we believe we have the right payments, then we can add the interest discounting.

We have i = 0.05
This gives an annual discount rate of v = 1/(1+i) = 1/1.05

Are you comfortable with those definitions?

Answer 7

65 200000
66 200000*1.03*v
67 200000*1.03^2*v^2
68 etc...

75 200000*1.03^10*v^10 + 75000*v^10
76 200000*1.03^11*v^11
77 etc...

84 200000*1.03^19*v^19

How's that. Did I get all the pieces?

Answer 8

We can simplify the whole mess if we make another definition.

r = 1.03*v

65 200000
66 200000*r
67 200000*r^2
68 etc...

75 200000*r^10 + 75000*v^10
76 200000*r^11
77 etc...

84 200000*r^19

Answer 9

@tkhunny is deriving them for you...

Answer 10

Algebra time.

200000(1 + r + r^2 + ... + r^19) + 75000v^10 = \(200000\dfrac{1-r^{20}}{1-r} + 75000v^{10}\)

We just used a present value formula, which is simply a sum of a geometric series.

Answer 11

Evaluate that and you have your answer for Part A.

Answer 12

How do you know? What results are you seeing?

In case you lost track...
j = 0.03 -- The inflation rate.
i = 0.05 -- The annual interest rate.
v = 1/(1.05) -- The annual discount rate
r = (1+j)*v -- A surrogate rate that combines interest and inflation.

It is not easier to put it into an equation. It is unlikely you can develop the necessary confidence that you have done it correctly.

It is the Present Value of an Annuity Immediate with the rate "r-1", increased by the 75000 discounted at 5% for 10 years.

Answer 13

Good catch, that should be $50,000.

Answer 14

$50,000 gift on your 75th birthday

Answer 15

That's not paying attention in a problem where there is too much going on. That's why we draw a map. It's $50,000.

Answer 16

The error stuck out more this way than if you'd plugged it into a formula, I think.

Answer 17

Hogwash. The formula is for the annuity, not the lump.

Answer 18

And if you're serious about doing finance, you need to have an intuitive feel for how these formulas come together.

Hogwash? I'm saying the error was easier to spot!

Answer 19

Absolutely not. You have not been following at all. You cannot solve this problem in the next 6 minutes if you do not pay very close attention. 3% is ONLY the inflation factor. That does not include the interest AT ALL. Look at my 'r'. The correct rate is "r-1".

@whpalmer4 Sorry for the "hogwash". Past my bedtime.

Answer 20

Well, you should try the Present Value of the Geometrically Growing Annuity - the last one. Then, you will need the Future Value of a NonGrowing Annuity in order to answer Part B.

Answer 21

I would argue that credit to the Supreme Court. Learning the actual guts and how to solve ALL SUCH PROBLEMS is somehow inferior to memorizing a few formulas to solve a few ideal problems? It is just silly. I blame your teacher, not you.

Answer 22

36? I thought it was a 20 year horizon.

Answer 23

There is no increasing annuity for Part B. The "Increasing" formulas are not appropriate.

Answer 24

I can no longer tell what you are doing. We're all over the place and the discussion has become unclear.

Answer 25

First, we have to come to terms with that formula, a little bit.

"r-g" is a linear approximation to the exact answer. That is very unfortunate as the exact answer is easily calculated. What, really, is the point of the approximation?

Anyway,

C = 200000
g = 0.03
r = 0.05
n = 20

...seems to be what it wants.

$3.2 Million

Answer 26

Some formatting troubles in there, but yes, that seems okay.

Answer 27

Let's fix those formatting problems: \(2000\cdot\dfrac{1}{0.05-0.03}\cdot\left(1-\left(\dfrac{1.03}{1.05}\right)^{20}\right)\)

Answer 28

Of course, I meant 200000, not 2000. Too late. To much rushing.

Answer 29

Not sure what to tell you. Are you writing down intermediate values and re-entering them? That could lead to rather severe rounding errors.

Answer 30

1/0.02 = 50 That should be fine.

1.03/1.05 = 0.980952380952381 -- That could certainly be chopped inappropriately.

(1.03/1.05)^20 = 0.680704329245671

200000*50*(1-0.680704329245671) = 3,192,957

Answer 31

For Part B, It is not an increasing annuity. Use a level one. It is also an accumulation, not a present value.

Answer 32

Do you have accumulation formulas, or just present value?

Answer 33

Awesome.

Use "Present Value of an Annuity" on the 31st birthday, NOT the 30th birthday. This may give you one less year than you imagined.

Multiply your result by 1.05^34 That will do.

Answer 34

I have to go. That was for Part B, the accumulation phase.

For part A, just match my numbers. It should expose your calculation error.