Mathematics



Financial Mathematics

Question 1

Zero-coupon yield curve for years 1 and 2:

Zero coupon bond: A title that does not pay interest during his lifetime, but does so entirely in the moment that pays off, i.e. when the amount of the bond is returned. In return, its price is lower than its nominal value.

To calculate the price of some products, such as interest rate swaps, it is necessary to determine a particular rate curve called zero-coupon curve. This yield curve is used to represent the function followed by interest rates for different maturities

To construct this curve, it is assumed that rates have the same periodicity of payment of interest (coupon detachment) and use the same convention calculation.

i) Zero coupon bonds are those bonds in which company does not pay any coupon payment.

[(F/PV)^n] - 1

The yield formula of zero coupon bonds is widely used to calculate the periodic return for a zero coupon bond, or sometimes referred to as a discount bond.

Yield formula is (1+i) ^n

(1+0.08)^1 =1.1664

(1+0.076)^2=1.1577

Furthermore:

100= 80/ (1+0.08) ^1+1080/(1+t)^2

= 5.45 is for two years

ii)Forward rate from the zero-coupon yield curve

f (1) = (DF (1)/DF (2)) - 1

= (0.08*1/0.076*2)-1

=1.105293

Return from Investment in property

Calculate the return from investment in property by calculating

i) Monthly payment.

Amount= 1000000

Down payment= 200000

Remaining= (1000000-200000) = 800000

Interest on loan = 800000*0.03= 24000 yearly

Monthly payment = 24000/12 = 2000

He will be paying 2000 monthly for 10 years

Loan balance at end of December 31, 2012

Loan amount = 800000

Internet on loan= 2000 monthly

Balance: Loan amount - monthly interest payment

Bal Jan 2012: 800000-2000= 798000

Bal Feb 2012: 798000-2000 =796000

Bal March 2012: 796000-2000 = 794000

Bal April 2012: 794000-2000= 792000

Bal May 2012: 792000-2000 = 790000

Bal June 2012: 790000-2000 = 788000

Bal July 2012: 788000-2000 = 786000

Bal Aug 2012 786000-2000 = 784000

Bal Sept 2012: 784000-2000 = 782000

Bal Oct 2012: 782000- 2000= 780000

Bal Nov 2012: 780000-2000 = 778000

Bal Dec 2012: 778000- 2000= 776000

Loan balance at end of December 31, 2012 = 776000

iii) Gain on sale of property on December 31, 2012

Loan amount = (800000)

Payment for 1 year = (24000)

Rental income 1 year= 57600

Gain= invested amount + rental income

=1057600

Initial cost= 200000+24000+800000

= 1024000

Gain = 1057600-1024000

= 33600

iv) Cash flow at end of each month during 2012

Bal Jan 2012: 800000-2000+4800= 802,800

Bal Feb 2012: 798000-2000 +4800= 800,800

Bal March 2012: 796000-2000 +4800= 798,800

Bal April 2012: 794000-2000+4800= 796,800

Bal May 2012: 792000-2000 +4800= 794,800

Bal June 2012: 790000-2000 +4800= 792,800

Bal July 2012: 788000-2000 +4800= 790,800

Bal Aug 2012 786000-2000 +4800= 788,800

Bal Sept 2012: 784000-2000 +4800= 786,800

Bal Oct 2012: 782000- 2000+4800= 784,800

Bal Nov 2012: 780000-2000 +4800= 782,800

Bal Dec 2012: 778000- 2000+4800= 780800

v) Return from investing in property

Return= Gain/ initial investment

= 33600/1000000

=0.0336

=3.36%

C. Calculate return from investing in preferred stock by calculating

i) Current required rate of return on preferred stock

Rf = 6%

Rm = 14%

B = 0.75

R= Rf + (Rm - Rf)*B

R= 6% + (14-6)* 0.75

R= 12%

ii) Current price of preferred stock

Pp = Dp/r

Pp0= Do (1+g)/r-g

= (0.52(1+0.08)/ (0.12-0.08)

= 0.5616/0.04

Pp1 = 14.04

iii) Number of preferred stocks to be purchased

Number of stock = amount/current price

= ...