1.) The current value of the $1000 payment at time t=2 is:
1000 (1+i*t) or 1000 (1+.05(2)) = 1100
We only need to bring the $1000 payment at t=2 ahead by 2 years.
The current value of the $400 payment at t=7 is:
400v*v*v or (400 v-cubed)
(I'm going to use { for the integral symbol)
(1+i) to the t-th power = exp[{.2/(1+.2t)] from t to 4.
(1+i) to the third power = exp[ln(1+.2t)] from 7 to 4
OR exp[ln(2.4) - ln(1.8)]
1/(1+i)cubed = .75 400*v-cubed = 400(.75) = 300
The current value = 1400
2.) The present value at time t=0 of X at time t=10 has to be broken up into
two pieces.
X*v(to the tenth)
X(1/exp[ln(3)-ln(1.8)])(1/(1+.05(4))) = X(.6)(1/1.2)=.5X
3.) .5X=1400
X=2800
I've included a lot of the steps that I used in case. Hopefully if I screwed
up somewhere, someone can let me know.
Most of the above match Mandy's answer except the current value of the $1000
payment at time t=2.
------------------( Forwarded letter 1 follows )--------------------
Date: Thu Jul 16 14:06:48 1998
To: studygroup4a@lists.casact.org
From: mandy.m.y.seto@ArthurAndersen.com
Sender: studygroup4a-return@casact.org
Subject: 4A interest theory, Chapt 2
Carolyn,
I will try to answer your question, but I'm quite new to 4A right now (my
first attempt in Fall 98).
Years [0,4) uses simple interest 5% or a(t) = 1 + 0.05*t
As for years 4 and beyond, the "dell" assumed is actually a simple
interest of 20% in disguise. Look at Example 1.12 page 19 of Parmenter
text. Therefore, years 4+ has a(t) = 1+ 0.2*t, t>=4.
i) CV at t = 4:
CV = X * exp {-[integral s=0 to s=10 of dell(s) ds]}
CV = X *exp{-[integral s=0 to s=4 of (0.05/(1+0.05s))ds]}*exp{-[integral
s=4 to s=10 of (0.20/(1+0.20s))ds]}
CV = X*(1/(1.2))*(1.8/3.0)
CV = X/2
ii) PV at t = 0:
PV = 1000*{exp[integral s=2 to s=4 of (0.05/(1+0.05s))ds]} +
400{exp{-[integral s=4 to s=7 of (0.20/(1+0.20s))ds]}}
PV = 1000(1.2/1.1) + 400(1.8/2.4)
PV = 1390.91 (rounded)
Note there is not {exp{-[integral ... } in the 1000 integral because it is
an accumulation factor rather than a discount factor.
i) = ii) and so CV = PV
Therefore X = 2*1390.91 = 2781.82
Carolyn, let me know if my answer is even close!
___________________________________________________________________________
_________________________________________________________________
To: studygroup4a @ lists.casact.org @ internet
cc: (bcc: Mandy M. Y. Seto)
From: Carolyn.McElroy @ ey.com ("Carolyn J. McElroy") @ internet
Date: 07/15/98 11:24 AM
Subject: 4A interest theory, Chapt 2
___________________________________________________________________________
_________________________________________________________________
Can someone help out with the following question for exam 4A?
Durning the first 4 years, interest is credited using a simple interest
rate of 5% per year. After 4 years, interest is credited at a force of
interest:
dell=0.2/(1+0.2t), t >=4.
The following are numerically equal:
i) the current value at time t=4 of payments of 1000 at time t=2 and
400 at time t=7; and
ii) the present value at time t=0 of a payment of X at time t=10.
Calculate X.