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# FIN100 – Principles of finance Chapter 9: P6, P9, P10, P11, P12, P13, P15

P6

6. Determine the present value (PV) if $5,000 is received in the future (i.e., at the end of each indicated time period) in each of the following situations: (same problem as in assignment week4)

a. 5 percent for ten years

PV= $5000/ (1+0.05) ^10 = $5000/1.6288 =

**PV=$3,069.7445**

b. 7 percent for seven years

PV= $5000/ (1+0.07) ^7 = $5000/1.6057

**PV= $3,113.7487**

c. 9 percent for four years

PV = $5000/ (1+0.09) ^4 = $5000/1.4115

**PV = $3,542.1260**

P9

**9.** Assume you are planning to invest $5,000 each year for six years and will earn 10 percent per year. Determine the future value of this **annuity** if your first $5,000 is invested __at the end of the first year. __

**FVa = PMT {(1+r)^n -1/r} **

Year#1 = **$5,000 **for the first year there would not be a big difference but every year for the next six years it would increase.

FV= $5000{(1+0.10)^1 -1 /0.10} = $5000{1.10-1/.10}

FV = $5000 (.10/.10) **FV= $5,000.00**

Second year results;

$5000{(1+0.10)^2 -1 /0.10} = $5000{(1.10)^2 -1 /0.10}

= $5000{1.21-1 /0.10}

= $5000{.21/.10}= $5000(2.10)

**FV= $10,500.00**

Third year results;

$5000{(1+0.10)^3 -1 /0.10} = $5000{(1.10)^3 -1 /0.10}

= $5000{1.331-1 /0.10}

= $5000{.331 /0.10} = $5000{3.31}

**FV = $16,550.00**

Fourth year results;

$5000{(1+0.10)^4 -1 /0.10} = $5000{(1.10)^4 -1 /0.10}

= $5000{1.4641-1/0.10}

= $5000{.4641 /0.10}

= $5000{4.641}

**FV= $23,205.00**

Fifth year results;

$5000{(1+0.10)^5 -1 /0.10}

= $5000{(1.10)^5 -1 /0.10}

= $5000{1.6105-1 /0.10}

= $5000{(.6105 /0.10}

= $5000{6.1051}

**FV = $30,525.50**

After six years of investing results;

$5000{(1+0.10)^6 -1/0.10}

= $5000{1.7715-1/.10}

= $5000{0.7715/0.10}

= $5000(7.7156)

**FV= $38,578.05**

**P10**

**10.** Determine the present value now of an investment of $3,000 made one year from now and an additional $3,000 made two years from now if the annual **discount rate** is 4 percent.

Using formula; PV = FV/(1+r)^n

__First year__; $3,000/(1+0.04)^1

= 3000/1.04

= **FV= $2,884.61**

__Second year;__ $3,000/(1+0.04)^2

= 3000/1.0816

= **FV= $2,773.66**

**Total after the second year= $5,658.2786**

**P11**

**11.** What is the present value of a loan that calls for the payment of $500 per year for six years if the **discount rate** is 10 percent and the first payment will be made one year from now? How would your answer change if the $500 per year occurred for ten years?

**Using formula PV = FV/(1+r)^n**

Year#1

$500/(1+10%)^1

= $500/(1+0.10)^1

= $500/1.10

**First year total $454.5454**

Year#2

=$454.5454 + $500/(1+10%)^2

= $454.5454 + $500/(1.10) ^2

= $454.5454 + $500/1.21

= $454.5454 + $413.2231

**Second year total $867.7685**

Year#3

=$867.7685 + $500/ (1+10%)^3

= $867.7685 + 500/ (1.10) ^3

= $867.7685 + $500/1.331

= $867.7685 + $375.6574

**Third year total $1,243.4259**

Year#4

= $1,243.4259 + $500/(1+10%)^4

= $1243.4259 + $500/(1.10) ^4

= $1243.4259 + $500/1.4641

= $1243.4259 + $341.5067

**Fourth year total** $**1584.9326**

Year#5

= $1584.9326 + $500/ (1+10%)^5

= $1584.9326 + $500/ (1.10) ^5

= $1584.9326 + $500/1.6105

` = $1584.9326 + $310.4625

**Fifth year total $1895.3951**

Year#6

= $1895.3951 + $500/ (1+10%)^6

= $1895.3951 + $500/ (1+0.10) ^6

= $1895.3951 + $500/ (1.10) ^6

=$1895.3951 + $282.2466

**Six year total $2177.6417**

__How would it be for 10 years__? 500/(1+10%)^10 = 500/(1.10)^10 =500/2.5937 **=**192.7748

**By the tenth year would be a total of $3,072.50**

**P12**

**12.** Determine the annual payment on a $500,000, 12 percent business loan from a commercial bank that is to be **amortized** over a five-year period.

A= P{r(1+r)^n/(1+r)^n-1}

P= $500,000 initial principal loan,

A= payment amount per period,

r = 12% interest rate per period,

n = 5 years (it is an annual payments)

=500,000{.12(1+.12)^5/(1+.12)^5-1}

= 500,000{.12(1.12)^5/(1.12)^5-1}

= 500,000{.12(1.7623)/1.7623-1}

= 500,000{.2114/.7623}

= 500,000(.2774)

**A = $138,709.1696**

**P13**

**P13.** Determine the annual payment on a $15,000 loan that is to be amortized over a four-year period and carries a 10 percent interest rate. Also prepared a loan **amortization** schedule for this loan

P= principal loan $15,000, r=interest rate 10%, n=number of payments (which payments are annual), A=payment amounts per period

=15,000{0.10(1+0.10)^4/(1+.10)^4-1}

= 15,000{.10(1.10)^4/(1.10)^4-1}

= 15,000{.10(1.4641)/1.4641-1}

= 15,000{.14641/.4641}

= 15,000{.3154}

**A= $4732.06**

Basic loan info;

Loan Amount

$15,000

Annual Interest rate

10%

Term of Loan in Years

4

First Payment Date

9/1/2015

Payment Frequency

Annually

Compound Period

Annually

Payment Type

End of Period

Summary of loan;

Rate per period

7%

Number of payments

4

Total money paid

$18,928.24

Total in Interest

$3,928.24

Loan Schedule

Every year

Principal Vs Interest

Loan Balance

First Payment 9/1/2015

$4,732.06

$3,750 on principal ($982.06 on interest)

$11,250 in principal loan

Second Payment 9/1/2016

$4,732.06

$3,750 on principal ($982.06 on interest)

$7,500 in principal loan

Third Payment 9/1/2017

$4,732.06

$3,750 on principal ($982.06 on interest)

$3,750 in principal

Fourth Payment 9/1/2018

$4,732.06

$3,750 on principal ($982.06 on interest)

$0.00

**P15**

15. Assume a bank loan requires an interest payment of $85 per year and a principal payment of $1,000 at the end of the loan’s eight-year life.

a. At what amount could this loan be sold for to another bank if loans of similar quality carried an 8.5 percent interest rate? That is, what be the present value (PV of this loan?

Principal payment per year= $1,000/8= $125.00

$125+$85= $210

=$2101 8.5%

Total PV of **$2,470.5882**

b. Now if the interest rate in the other similar quality loans is 10 percent, what would be the PV of this loan?

$210/10% **Total PV of $2,100.00**

c. What would be the PV of the loan if the interest rate is 8 percent on similar quality loans?

$210/8% **Total PV of $2,625.00**