Actuarial Outpost Present value problem, easier solution?
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 Financial Mathematics Old FM Forum

#1
08-02-2004, 08:04 AM
 Chevy Chef Guest Posts: n/a
Present value problem, easier solution?

Here's a problem that I'm looking for an easier way to solve:

At an effective annual interest rate of i, i >0, each of the following two sets of paymtents has a present value of K:

i) a payment of 121 immediately and another payment of 121 at the end of one year.
ii) A payment of 144 at the end of two years and another payment of 144 at the end of three years.

-----------
So, I set up 121 + 121 / (1+i) = 144 / (1 + i)^2 + 144 / (1 + i)^3

Is there an easy way to solve for i?

I applied Newton's method and got the answer, but I'm looking for an easier way.
#2
08-02-2004, 09:44 AM
 Actuwhat? Member Join Date: Jun 2004 Location: Atlanta Posts: 191
solution

---> 121 ----> 121--------------->
(it accumulates to t=1) then move the lump forward 2 years
|----------|----------|----------|
0 1 2 3

---> 141 ---> 141 (right side of equation)

I will denote s angle n as s(n)

121s(2)*(1+i)^2 = 141s(2)

now cancel the s(2) on each side AND divide the 121:

(1+i)^2 = 141/121

i = sqrt(141/121)-1
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#3
08-02-2004, 09:45 AM
 Actuwhat? Member Join Date: Jun 2004 Location: Atlanta Posts: 191

It messed up my time diagram...sorry
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#4
08-02-2004, 10:13 AM
 Chevy Chef Guest Posts: n/a

Ah. I get the idea. Thanks actuwhat? !
#5
08-02-2004, 10:41 AM
 RG Member Join Date: May 2004 Posts: 268

Just in case -

You DO know that your calculator will do this for you? Mine spits out 7.948564428. This is a BA II-plus.

Also - obviously with a MC exam, you just try a few of the answer choices, though that's easiest once you've simplified like actuwhat
#6
08-02-2004, 10:58 AM
 Chevy Chef Guest Posts: n/a

Quote:
 Originally Posted by RG Just in case - You DO know that your calculator will do this for you? Mine spits out 7.948564428. This is a BA II-plus. Also - obviously with a MC exam, you just try a few of the answer choices, though that's easiest once you've simplified like actuwhat
I couldn't see a way to use the cash flow or time value of money functionality of the BA-II Plus to solve it. How do you do it? The interest rate should work out to 9.09%, I believe.
#7
08-02-2004, 11:20 AM
 Wannabe Actuary Member SOA Join Date: Nov 2003 Location: in front of a computer screen Favorite beer: KCCO Gold Lager Posts: 32,317

Of course if can be done similar to what Actuwhat did as well.

You can use the annuity due function (a dot n - hopefully that's understandable, damn math formulas are impossible to type) and look at everything in that mind set and evaluate at t=0.

So you have 121 (a dot 2) = [144 (a dot 2)] / (1 + i)^2

This leads you to the same (1 + i)^2 = 144/121 that Actuwhat shows, but I think this way was more intuitive, given the first stream of payments. It's all a matter of what you're more comfortable with.

And this gives you i = 9.09 or so, but once you solve for that answer just store it as i in your calculator and use the annuity due function (very easy on BA-35 solar), with 2 payments of 121 along with the i you just stored and have it compute PV. Spits back just under 232.

Calculators are great, aren't they!
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Quote:
 Originally Posted by rawl316 I like playing with myself
#8
08-02-2004, 11:53 AM
 RG Member Join Date: May 2004 Posts: 268

On the BA II-Plus, it's the 2nd line of buttons: CF stands for cash flow, NPV and IRR are what you think they are. Look at the manual if this is confusing, but here's this problem:

cf displays cf0=, which we set to
121, enter, down arrow, displays cf1=, which we set to
121, enter, down arrow, which displays f01=1, which is fine,
down arrow, displays cf2=0; set it to -144 and f02 to 2.

Then hit IRR and compute to get 9.090909091

Then to find NPV of either set of payments, delete the other:

I set cf2=0; hit npv and set i=9.0909091, down arrow, npv= compute to get 231.9166667
#9
08-02-2004, 12:02 PM
 Chevy Chef Guest Posts: n/a

Thanks RG.
#10
05-22-2018, 07:07 AM
 eruben1007 SOA Join Date: May 2018 Studying for IFM Posts: 7

Quote:
 Originally Posted by RG Woops, I had read 141, not 144. On the BA II-Plus, it's the 2nd line of buttons: CF stands for cash flow, NPV and IRR are what you think they are. Look at the manual if this is confusing, but here's this problem: cf displays cf0=, which we set to 121, enter, down arrow, displays cf1=, which we set to 121, enter, down arrow, which displays f01=1, which is fine, down arrow, displays cf2=0; set it to -144 and f02 to 2. Then hit IRR and compute to get 9.090909091 Then to find NPV of either set of payments, delete the other: I set cf2=0; hit npv and set i=9.0909091, down arrow, npv= compute to get 231.9166667

Why set 144 to be negative?