Course 2 Question-How 2 Pass

[Browser]

Given that (S 2n) - Accumulated Due value with 2n payments = 300 and (An) = 10 (Simple PV formula) find i.

The hint is to change the accumulated value to a present vlaue with 2n payments. I seem to be getting stuck on good ole Algebra. any help would be great

[MathGuy]

Clarification Request:

You state that the first annuity is an annuity due, accumulated 2n periods, so the accumulated value is:

S_2n = (1 + v + ... + v^2n-1) * (1 + i)²ⁿ = 300

(This is correct right?) Anyway, the clarification is whether the second annuity you mention (A_n = 10) is due or immediate. Thanks.

[Browser]

The first is due, the second is immediate.

[Bama Gambler]

Mathguy,

How did you use subscripts in your post?

Thanks,
Bama Gambler

[MathGuy]

Assuming it's what I think it is:

A_n = 1 + v + ... + v^n-1 = 10
vA_n = v + v² + ... + vⁿ = 10v

A_n - vA_n = 1 - vⁿ = 10 - 10v

1 - vⁿ = 10(1 - v) [Formula I]

We'll save this for later.

Next,

S_2n = (1 + v + ... + v^2n-1) * (1 + i)²ⁿ = 300

If we "de-accumulate", which means we just discount back 2n periods using the discount factor v, we get:

1 + v + ... + v^2n-1 = 300v²ⁿ
1 + v + ... + v^n-1 + vⁿ + ... + v^2n-1 = 300v²ⁿ

but, the bolded terms are the same as A_n = 10 from above:

10 + vⁿ + ... + v^2n-1 = 300v²ⁿ
10 + vⁿ(1 + v + ... + v^n-1) = 300v²ⁿ

And there he is again!

10 + vⁿ(10) = 300v²ⁿ
0 = 300v²ⁿ - 10vⁿ - 10
0 = 300(vⁿ)² - 10vⁿ - 10

And use the handy dandy quadratic formula, with vⁿ = x:

vⁿ = (10 + sqrt(100 - 4(300)(-10)))/(2(300)) = (10 + 110)/600 = 120/600 = .20.

Now, we take Formula I from way back at the beginnning and plug in vⁿ = .20:

1 - vⁿ = 10(1 - v)
1 - .20 = 10(1 - v)
.8 = 10(1 - v)
.08 = 1 - v
v = .92
(1 + i)^-1 = .92
(1 + i) = .92^-1
1 + i = 1.087
i = .087

[MathGuy]

Subscripts: use the html tags <sub> and </sub> around what ever you'd like to have as a subscript. (Same for sup).

If the second is immediate I'll have to think a little more.

[Browser]

With my inproved notational abilites the problem reads:

..
S_2n = 300 (Due)

And

a_n=10 (Immediate)

find i

[MathGuy]

This solution will be similar to my previous one:

a_n = v + ... + vⁿ = 10
va_n = v² + ... + vⁿ⁺¹ = 10v

v - vⁿ⁺¹ = 10 - 10v FORMULA I

The "de-accumulation" of S_2n works the same as above, and we get to:

1 + (v + v² + ... + vⁿ) + vⁿ⁺¹ + ... + v^2n-1 = 300v²ⁿ
1 + (10) + vⁿ⁺¹ + ... + v^2n-1 = 300v²ⁿ
11 +vⁿ⁺¹ + ... + v^2n-1 = 300v²ⁿ
11 + vⁿ(v + ... + v^n-1) = 300v²ⁿ
11 + vⁿ(10 - vⁿ) = 300v²ⁿ
11 - 10vⁿ - 2v²ⁿ = 300v²ⁿ
0 = 302v²ⁿ - 10vⁿ - 11

Again, using the quadratic formula, we get vⁿ = .2081

And, using Formula I from above:

v - vⁿ⁺¹ = 10 - 10v
v(1 - vⁿ) = 10 - 10v
v(1 - .2081) = 10 - 10v
.7919v = 10 - 10v
10.7919v = 10
v = 10/10.7919
1/(1+i) = .9266
1 + i = 1.079
i = .079

[Browser]

Thats the right answer, thanks for your help Math Guy.

[c3 taker]

And, using Formula I from above:

v - vⁿ⁺¹ = 10 - 10v
v(1 - vⁿ) = 10 - 10v
v(1 - .2081) = 10 - 10v
.7919v = 10 - 10v
10.7919v = 10
v = 10/10.7919
1/(1+i) = .9266
1 + i = 1.079
i = .079

Rather than use formula 1 you could also just say that since a_n = 10 = (1 - vⁿ) / i
then i = (1 - .2081) / 10 = .079

same premise, I find it's easier to remember this way (if you know the formula for an annuity-immediate, which I assume you won't do very well on interest theory without) :D

[RiSK kid]

Subscripts: use the html tags _and around what ever you'd like to have as a subscript. (Same for sup).


Amazing, I totally forgot that we could use HTML. This is gonna make the questions a lot more readable.
Thanks

[MathGuy]

C2 taker: Formulas are for babies. I prefer the all-out guerilla-math style: Everything on first principles!!! For Interest Theory, this is particularly useful because the entire topic is light and fluffy, but "the man" weighs it down with all sorts of formulas and stuff. If you can write down 1 + v + ... + vⁿ, and you truly understand what v means, you can answer any question about annuities that comes up. Sure it makes for a crazy four hours come exam time, but it means that there is no studiying required.

[c3 taker]

C2 taker: Formulas are for babies. I prefer the all-out guerilla-math style: Everything on first principles!!! For Interest Theory, this is particularly useful because the entire topic is light and fluffy, but "the man" weighs it down with all sorts of formulas and stuff. If you can write down 1 + v + ... + vⁿ, and you truly understand what v means, you can answer any question about annuities that comes up. Sure it makes for a crazy four hours come exam time, but it means that there is no studiying required.

I understand the concept of 1 + v + ... + vⁿ, I just think it's easier to remember a few formulas, but maybe I'm just frightened by your "all-out guerilla-math style" :D