# HOMEWORK SET #2

HOMEWORKSET #2

InstitutionAffiliated:

Dateof submission:

QUESTION1:

Netpresent value is a discounting capital budgeting technique thatcalculates cash flows taking into account time value of money throughthe discounting rate. The negative value for the first year indicatesthe initial outlay of the project.

 YEAR Cash flows (\$) Discounting factor (10%) Present values Year 0 (50) 1 (50) Year 1 100 0.9091 90.91 Year 2 75 0.8264 61.98 Year 3 50 0.7513 37.57 Net Present Value 140.46

Discountingrate of 10% e formula = (1 + r) ^ -n.

Presentvalues are obtained through the multiplication of discounting factorby cash flow.

QUESTION2:

Ata rate of 20%, time duration to double the sales would be?

FutureValue (FV) = Present Value (PV) (1+R) ^ n (the Number of years)

Assumedoubling of sales of \$ 1

\$2 = \$ 1 (1+.02) ^ n

Log2 = n Log 1.2

Log2= n

Log1.2

N= 3.8 years.

QUESTION3:

Assetsthat pay interest frequently semi-annually, quarterly or on a dailyperiod will have higher future values as their interests are gainedoften and frequently after compounding. The more an investment isfrequently compounded, the greater the future value obtained[ CITATION Eug09 l 1033 ].Effective Annual Rate increases as an investment if frequentlycompounded resulting to a resulting to increment in the investment’sfuture value.

Exampleto show the interest differences on an annual and quarterlyperspective

Presentvalue = \$ 1,000

Interestrate = 10%

Formula= Future Value (FV) = Present value (PV) (1 + r/period (p)) ^ (year *period)

Annualperspective:

FV= \$ 1,000 (1 + 0.1/1) ^ 1

FV= \$1,100

Interestgain = \$ 1,100 – \$ 1,000

Interestgain= \$ 100

Quarterlyperspective:

FV= \$ 1,000 (1 + 0.1/4) ^ 4

FV= \$1,103.81289

Interestgain = \$ 1,103.80 – \$ 1,000

Interestgain= \$ 103.80

Asproven in the earlier statement

\$103.80 for the quarterly period &gt \$ 100 for annual perspective

QUESTION4:

Effectiveannual rate at a nominal rate of 12%

Formula:

EffectiveAnnual Rate (EAR) = (1+ r/n) ^n – 1

1. Semi-annual basis Compounding

EAR= (1+ 0.12/2) ^2 – 1

EAR= (1.06) ^2 – 1

EAR= 1.1236 – 1

EAR= 0.1236

EAR= 12.36%

1. Quarterly basis Compounding

EAR= (1+ 0.12/4) ^4 – 1

EAR= (1.03) ^4 – 1

EAR= 1.125509 – 1

EAR= 0.125509

EAR= 12.5509%

1. Monthly basis Compounding

EAR= (1+ 0.12/12) ^12 – 1

EAR= (1.01) ^12 – 1

EAR= 1.126825 – 1

EAR= 0.126825

EAR= 12.6825%

1. Daily basis Compounding

EAR= (1+ 0.12/365) ^365 – 1

EAR= (1.01) ^365 – 1

EAR= 1.127475 – 1

EAR= 0.127475

EAR= 12.7475%

QUESTION5:

Depositvalue = \$ 100

Interestrate = 11.33463%

Tobe compounded daily for a period of 9 months.

Convertthe interest rate to a daily rate through the division of the annualrate by 365.

=11.33463%/365

=0.03105378%.

Convertthe duration into days.

=9/12 * 365

=273.75 which is equivalent to 274 days

Compounding= Deposit value (1+ Rate) ^ period

=100 (1 + 1.0003105378) ^ 274

=108.8798

Asat October 1st,the deposit would have grown to \$108.88

QUESTION6 &amp 7:

Bonddetails:

Period= 10 years.

Parvalue of the bond = \$ 1,000.

Annualcoupon rate = 10%.

Requiredrate of return = 10%.

QUESTION6:

Inflationrose by 3% resulting in a rate of return of 13%.

BondPrice = C * (1 – (1/ (1 + R) ^n))/R + P/ (1 + R) ^n

Where

C= Coupon rate value of the bond

=10% * 1,000

=\$ 100.

R= required rate of return

n= Period of bond maturity.

P= Par value of the bond.

=\$ 100 * ((1-(1/ (1.13) ^10)) + \$ 1,000/ (1.13) ^10

=\$ 100 * 0.70541165/0.13 + 294.588348

=542.624346 + 294.588348

=837.2127

Itis considered to be a Discount Bond. The coupon rate remains constantat 10%, but the market rate goes up due to investor’s expectationsto 13% resulting in the fall in the bond price below the par valuestated. A par value is expected to be fixed, but it fell to \$837.2127 due to the increase in expected rate of return that is abovethe coupon rate making it a discount bond.

QUESTION7:

Similarformula will be used to determine the bond price and the type.

BondPrice = C * (1 – (1/ (1 + R) ^n))/R + P/ (1 + R) ^n

=\$ 100 * ((1-(1/ (1.07) ^10)) + \$ 1,000/ (1.07) ^10

=\$ 100 * 0.49165071/0.07 + 508.349291

=702.358157 + 508.349291

=1,210.70745

Itis considered to be a Premium Bond. It is because the bond price hasrisen from \$ 1,000 to \$ 1,210.70745 due to a decline in the marketrate expected by the investors. A par value is expected to be fixed,but it rose to \$ 1,210.70745 due to the decrease in expected rate ofreturn that is below the coupon rate making it a premium bond.

QUESTION8:

Part1

Presentvalue of bond = \$ 887

Futurevalue of bond = \$ 1,000

Interestper year = \$ 90 (9% * \$ 1,000)

Periodof bond = 10 years

Bondprice (B) = C/2 * ((1-(1+YTM/2) ^ -2*N)) / (YTM/2))) + ((P/ (1+YTM/2)^ 2*N))

887= 90/2 * ((1-(1+YTM/2) ^ -2*10)) / (YTM/2))) + ((1,000/ (1+YTM/2) ^2*10))

YTMrate using a YTM calculator will be 10.88%

Part2

Presentvalue of bond = \$ 1,134.20

Futurevalue of bond = \$ 1,000

Interestper year = \$ 90 (9% * \$ 1,000)

Periodof bond = 10 years

Bondprice (B) = C/2 * ((1-(1+YTM/2) ^ -2*N)) / (YTM/2))) + ((P/ (1+YTM/2)^ 2*N))

1,134.20= 90/2 * ((1-(1+YTM/2) ^ -2*10)) / (YTM/2))) + ((1,000/ (1+YTM/2) ^2*10))

YTMrate using a YTM calculator will be 7.10%

Therelationship between required rate of return and bond’s coupon rateis that for a discount bond, the rate of return will be higher thanthe bond’s coupon rate resulting in a decline in the price of thebond. In the case of a premium bond, the rate of return will be lowerthan the bond’s coupon rate resulting in an increase in the priceof the bond[ CITATION Eug09 l 1033 ].Such a relationship indicates if the rate of return were equal to thecoupon rate, the bond price would be equal to the par value of \$1,000.

QUESTION9:

CalculatingTotal Return, current yield and the capital gains yield for the twotypes of bonds discount and premium bonds. Since there is no timestipulated for the consideration of values, I shall calculate using 2years and the maturity period of 10 years.

Part1: Discount Bond

Period= 10 years

Parvalue = \$ 1,000

Annualcoupon rate = 9%

Requiredrate of return = 10.88%

Bondprice was \$ 887

Usingthe par value level in 2ndyear

Calculatingthe capital gain that arose in the second year will resultcalculating the bond price using n (the number of years) as 9 yearsto maturity.

BondPrice = C * (1 – (1/ (1 + R) ^n))/R + P/ (1 + R) ^n

=\$ 90 * (1 – (1/ (1 + 10.88%) ^9))/ 10.88% + \$ 1,000/ (1 + 10.88%)^9

==\$ 90 * ((1-(1/ (1.1088) ^9)) + \$ 1,000/ (1.1088) ^9

=\$ 90 * 0.60525099/0.1088 + 394.749006

=500.667269 + 394.749006

=895.416275

Thecapital gain between the first and second year will be

=\$ 895.4 – \$ 887.0

=\$ 8.40

Currentyield

=Annual coupon rate interest/ Discount bond price in the first year

=\$ 90/ \$ 887

=\$ 0.1015

=10.15%

Capitalgain yield

=Capital gain in between the years/ Discount bond price in the firstyear

=\$ 8.40/ \$ 887

=0.0095

=0.95%

Totalrate of return

=Current yield + Capital gains yield

TR= 10.15%+0.95%

=11.10%

Usingthe par value level in 10thyear

Calculatingthe capital gain that arose in the tenth year will result calculatingthe bond price using n (the number of years) as 0 years to maturity.

BondPrice = C * (1 – (1/ (1 + R) ^n))/R + P/ (1 + R) ^n

=\$ 90 * (1 – (1/ (1 + 10.88%) ^0))/ 10.88% + \$ 1,000/ (1 + 10.88%)^0

==\$ 90 * ((1-(1/ (1.1088) ^0)) + \$ 1,000/ (1.1088) ^0

=\$ 90 * 0 + \$ 1,000

=0 + \$ 1,000

=\$ 1,000

Thecapital gain at maturity

=\$ 1,000 – \$ 887.0

=\$ 113.0

Currentyield

=Annual coupon rate interest/ Discount bond price in the first year

=\$ 90/ \$ 887

=\$ 0.1015

=10.15%

Capitalgain yield

=Capital gain in between the years/ Discount bond price in the firstyear

=\$ 113/ \$ 887

=0.1274

=12.74%

Totalrate of return

=Current yield + Capital gains yield

TR= 10.15%+12.74%

=22.89%

Period= 10 years

Parvalue = \$ 1,000

Annualcoupon rate = 9%

Requiredrate of return = 7.10%

Bondprice was \$ 1,134.20

Usingthe par value level in 2ndyear

Calculatingthe capital gain that arose in the second year will resultcalculating the bond price using n (the number of years) as 9 yearsto maturity.

BondPrice = C * (1 – (1/ (1 + R) ^n))/R + P/ (1 + R) ^n

=\$ 90 * (1 – (1/ (1 + 7.10%) ^9))/ 7.10% + \$ 1,000/ (1 + 7.10%) ^9

==\$ 90 * ((1-(1/ (1.071) ^9)) + \$ 1,000/ (1. 071) ^9

=\$ 90 * 0.46062009/0.071 + 539.379906

=583.884621 + 539.379906

=\$ 1,123.26453

Thecapital gain between the first and second year will be

=\$ 1,123.3–\$ 1,134.20

=- \$ 10.9

Currentyield

=Annual coupon rate interest/ Discount bond price in the first year

=\$ 90/ \$ 1,134.2

=\$ 0.07935108

=7.94%

Capitalgain yield

=Capital gain in between the years/ Discount bond price in the firstyear

=- \$ 10.9/ \$ 1,134.20

=- 0.009610298

=- 0.96%

Totalrate of return

=Current yield + Capital gains yield

TR= 7.94%-0.96%

=6.98%

Usingthe par value level in 10thyear

Calculatingthe capital gain that arose in the tenth year will result calculatingthe bond price using n (the number of years) as 0 years to maturity.

BondPrice = C * (1 – (1/ (1 + R) ^n))/R + P/ (1 + R) ^n

=\$ 90 * (1 – (1/ (1 + 7.10%) ^0))/ 7.10% + \$ 1,000/ (1 + 7.10%) ^0

==\$ 90 * ((1-(1/ (1.071) ^0)) + \$ 1,000/ (1.071) ^0

=\$ 90 * 0 + \$ 1,000

=0 + \$ 1,000

=\$ 1,000

Thecapital gain at maturity

=\$ 1,000 – \$ 1,134.20

=- \$ 134.20

Currentyield

=Annual coupon rate interest/ Discount bond price in the first year

=\$ 90/ \$ 1,134.20

=\$ 0.07935

=7.94%

Capitalgain yield

=Capital gain in between the years/ Discount bond price in the firstyear

=- \$ 134.20/ \$ 1,134.20

=- 0.1183

=- 11.83%

Totalrate of return

=Current yield + Capital gains yield

TR= 7.94%-11.83%

=- 3.89%

Reference

Eugene Brigham, J. H. (2009). Fundamentals of Financial Management. Mason, South Western: Cengage Learning.