Question 1
The firm is considering buying office building which still has 25 useful life, and it will guarantee a rental income of $ 150000 for the first 5 years and thereafter the rental income will increase by 10% every year. The total expenses will be $ 45000 for the first year, and it will increase by $3000every year thereafter. The end of 25 years the building will be razed and sold as a lot in which it will realize $50000.
In order to be able to find the maximum amount the firm should be willing to pay now, we need to find the present value of all the future cash flow.
P = PMT x ((1 – (1 / (1 + r) ^ n)) / r)
Where by;
P = the present value of an annuity stream
PMT = the amount of each annuity payment
r = discount rate (interest rate)
n = the number of periods in which payments will be made
year 1; cashflow=$150000-45000
=$105000*0.8929=$93754.5
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Year 2 to year 5; cash flow will be=$150000*(3.6055-0.8931)= $421710
Year 6-10: cash inflow= $165000*(5.650-3.6055)=$337342.50
Year 11-15; cash inflow= $181500*(6.8115-5.650)=$ 210812.5
Year 16-20; cash inflow= $199650*(7.469-6.8115)= $131269.88
Year 21-25; cash inflow= $219615*(7.843-7.469)= $82136.01
Net realized value of the after 25 years= 500000/17.000=2941.17
Total cash in flow for year2-25 =421710+337342.50+210812.5+131269.88+82136.01+2941.17
=$1186212.06
Cash outflow =$50000*(7.843-0.893)= $347500
Continuous expense increase= $3000(7.843-0.893)=$20000
Total outflow= 347500+20000=$ 368350
Net in flow =93754.5+1186212.06-368350=$911616.56
Therefore, the maximum amount he should be willing to pay now is $911616.56
Question 2
There are two deposits for the whole period. The first deposit is now of $25000 while the other one, $30000 is at the end of year 6. There is also the annual withdrawal of C for the first 6 years and (C +$1000) per year for the next other 6 years.
Therefore, in this question, it seems that the total future value of deposits would be equal to the total withdrawal for 12 years.
FV = I * ((1 + R) ^ T)
Where FV is future value
R is interest rate
T is time
Therefore the future value of deposit is
FV=$25000(1+0.10)^6+ $30000
= $ 44289.025+$30000
=$74289.025
At the end of the year 6, the future value will be $74289.025, but he has been making a withdrawal. So we need to find the total withdrawal;
For the first 6 years the withdrawals will be =C*6=6C and for the next 6 other years the withdrawal will be=(C +$1000)*6= C+$6000
Therefore total withdrawal for the whole period will be= C+6C+$6000
To find C, we will equate total deposit to total withdrawals;
Total deposit=total withdrawals
=$74289.025=C +6C+$6000
=$74289.025-$6000=7C
=68289.025/7=7C/7
C=$9755.575
Therefore the value of Cis $9755.575
Question 3
The nominal annual interest rate is the rate which is quoted on an annual period.
Therefore, nominal interest rate=monthly interest rate * 12 months in a year
=1.8%* 12 months
=21.6%
Therefore, nominal annual interest rate is 21.6%
Effective interest rate is the actual interest which is earned or paid in a given year.
i= (1 + r/M)^M -1
where;
r is the nominal interest rate per year
I refers to the effective annual interest rate
M is the number of interest periods per year=12 months in a year
I= (1+(0.216/12)^12)-1
= 23.87%
The number of years it will take the investment to triple if the interest is compounded continuously.
The bank offers loan at 1.8% per month. Hence interest rate per period is 1.8%= 0.018
the nominal is 1.8%*12=21.6%=0.216
Continuous compounding is a formula which assumes compounding is at very minimal intervals.
The formula for this will be given by
A = Pe^(rt )
where:
A = Future Value of the present Investment.
P = Present value of the Amount to be invested.
e = 2.718281828… is a scientific constant
r = nominal interest rate.
t = time
in this question, we assume that P = 1(initial investment)
Therefore, A=1*3=3
e = 2.718281828 since it’s a constant
r = 21.6% = .216
t = x
Putting the figures in the formula we will have;
3 = 1*e^ (0.216*x)
then insert log in both sides of the equation in order to find x
=log (3) = log (e^ (0.216*x) =0.216*x*log (e)
Then we divide each sides of the equation by 0.216*log (e), we will get
log (3)/ (.216*log) e) = x using scientific calculator we will get,
x = 5.086168003
Therefore for the money to triple, it will take 5.086 years.
Question 4
In this Question, Adam Smith made deposits $15000 in a savings account and this will pay6% interest compounded monthly. After 3 years he deposits another $14000. In the next 2 years after the $14000 deposit, he deposits another $12500. Then 4 years after the $12500 deposit, half of the accumulated funds is transferred to a fund that pays 8% interest compounded quarterly. So we need to find the amount of money in each account 6 years after the transfer.
We need to first find the future value of all the deposits up to the time he transfer half accumulated to the fund that pays 8%
FV = I * ((1 + R) ^ T
Where Fv is future value
R is interest rate
T is time
I is initial investment
1st year he deposits $15000
3rd year he deposits $14000
6th year he deposits $12500
All the deposits stayed in an account earning cumulative interest until half of it was withdrawn in year 10.
Therefore, as at year 10 the future value of deposits is;
FV=$15000(1+.06)^10+14000(1+0.06)^6+12500(1+0.06)^4
=$26862.72+$19859.27+$15780.96
=$62502.95
Therefore after 10 years, the total value of all the deposits is $62502.95, and at this point, Adam Smith put half of this to a fund giving a yearly return of 8% while the other half remained in the saving account earning 6% for the next 6 other years.
=$62502.95/2=$31251.48
Hence amount left in 6% saving account is 31251.48 while in the 8% saving fund is 31251.48. So we need to find the future value for the money in each account for the next 6 years.
In the 6% saving account the amount will be;
=$31251.48(1+0.06)^6
=$44330.82
While in the 8% saving fund will be;
=$31251.48(1+0.08)^6S
=$49592.17
Question 5
The uniform gradient factor finds the present worth of a uniformly increasing cash flow. The cash flow starts in year 2, not in year 1.
n = 10:
A = [800 + 20(A/G, 6%, 7)] x (P/A, 6%, 7)(A/P, 6%, 10) + [300(F/A, 6%, 3) – 500](A/F, 6%, 10
P= ($800)(P/A, 6%, 10) + ($300)(P/G, 6%, 10)
= $800((1+0.06)^9)-1)/(0.06)(1+0.06)^9)+$500((1+0.06)^9-1)/0.06^2(1+0.06)^9)-($300(9/(0.06)(1+0.06)^9)
= $800(6.8017) +$300(24.5768)
=$12814.4
