QUESTION1

i. Solve for the equilibrium Interest Rate obtaining a formula in analytical formS( r, Mˢ ) = αs + βsr + βmMˢD( r, W )= αd + βdr + βwW

S( r, Mˢ ) = D( r, W )

αs + βsr + βmMˢ = αd + βdr + βwW

βsr – βdr = -αs + αd + βwW - βmMˢ

r(βs - βd) = -αs +αd + βwW - βmMˢ

r = αd - αs + βwW - βmMˢ (βs - βd)

ii.a) Find the value of excess demand or excess supply at interest rates 5% and

20%

r = 0.05

S( r, Mˢ ) = 20 + 200(0.05) + (0.2X 300)= 90

D(r, W) = 100- 120(0.05) + (0.15X200) = 124

Demand (124) > Supply (90)Hence we have an excess Demand of 34

r = 0.20

S( r, Mˢ ) = 20 + 200(0.2) + (0.2X 300)= 120

D( r, W ) = 100- 120(0.2) + (0.15X200)= 106

Supply (120) > Demand (106)Hence we have an excess supply of 14

b) Find the equilibrium Interest rate using the formula derived in part i r = αd - αs + βwW - βmMˢ

(βs - βd)

=100 – 20 + 0.15(200) – 0.2(300)200-(-120)

= 0.15625= 15.625 %

c) Find the equilibrium loanable funds used by the economyS( r, Mˢ ) = 20 + 200r + 0.2X 300

= 20 + 200(0.15625) + 0.2 X 300 = 111.25

D(r, W) = 100 – 120r + 0.15 X 200 = 100 – 120(0.15625) + 0.15X 200 = 111.25

Answer = 111.25 (at equilibrium Interest rate of 15.63 %)

d) Total wealth falls from 200 to 150. What will be the impact of this event on equilibrium interest rate and equilibrium amount of loanable funds?

r = αd - αs + βwW - βmMˢ (βs - βd)

=100 – 20 + 0.15(150) – 0.2(300)200-(-120)

= 0.1328= 13.28 %

S( r, Mˢ ) = 20 + 200r + 0.2X 300 = 20 + 200(0.1328) + 0.2 X 300 = 106.56

D(r, W) = 100 – 120r + 0.15 X 200 = 100 – 120(0.1328) + 0.15X 200 = 106.56

∆ Equilibrium interest rate = 15.625% – 13.28 % = 2.345%

The equilibrium interest rate decreases by 2.345 %, when the total wealth in the economy falls from 200 to 150.

∆ Equilibrium loanable funds = 111.25 – 106.56 = 4.69

The equilibrium amount of loanable funds decreases by 4.69, when the total wealth in the economy falls from 200 to 150.

QUESTION2

i. Find the value of X

Present Value=Future value(1+r )n

Consider A, $2 million in 3.5 years

r = 0.025 F.V = 2 000 000 n=7

x01=2,000,000(1+0.025)7

=$ 1,682,530.47

Consider B, $3 million in 5 yearsr = 0.025 F.V= 3 000 000 n=10

x02=3,000,000

(1+0.025)10

= 2,343,595.21

X=1,682,530.47+2,343,595.21 = $4, 026125.68

ii. Showing that X can satisfy the amount needed

Method 1:

Step 1: pay $2 million in 3.5 years

n = 7 i=0.025 p.v = 4026125.68 (calculated value of X)

F.V = P.V (1+r) n = 4026125.68 * (1.025)7

= $4785798.238

This is the amount we have in hand after 3.5 years to pay $ 2 million and reinvest the remaining for another 1.5 years to cover the $ 3 million left to pay

Step 2: reinvest the remaining amount after payment

Amount after payment = 4785798.238 - 2000000 = $2785798.24

p.v: 2785798.24 n = 3 i = 0.025

F.V = P.V (1+r) n = 2785798.24 * (1.025)3

This show that the value of X calculated is able to satisfy both payments we have to make

= 3000000.006 (rounding up to 3 million)QUESTION 3

(a) Find the spot rate for maturities 2, 3, 4 & 5 years. Then plot the term structure of interest curve for the maturities 2, 3, 4 & 5 years.

(1+0Rn) n = (1+ 0Rm) m * (1+mFn) n-m

Consider maturity 2 years, (0R2)

(1+0R2)2 = (1+0R1)1 * (1+1F2)2-1

(1+0R2)2 = 1.05 *1.045

(1+0R2)2 = 1.0973

(1+0R2) = √ 1.0475

0R2 = 0.04752 = 4.75%

Consider maturity 3 years, (0R3)

(1+0R3)3 = (1+0R1)1 * (1+1F3)3-1

(1+0R3)3 = 1.05 * 1.0422

(1+0R3)3 = 1.140

(1+0R3) = 3√1.140 = 1.0447

0R3 = 0.0447 = 4.47%

Consider maturity 4 years, (0R4)

(1+0R4)4 = (1+0R3)3 * (1+3F4)4-3

(1+0R4)4 = 1.044663 * 1.04

(1+0R4)4 = 1.1857

(1+0R4) = 4√1.1857 = 1.04350

0R4 = 0.04350 = 4.35 %

General equation

Consider maturity 5 years, (0R5)

(1+0R5)5 = (1+0R2)2 * (1+2F5)5-2

(1+0R5)5 = 1.04752 * 1.053

(1+0R5)5 = 1.270

(1+0R5) = 5√1.270 = 1.04896

0R5 = 0.04896 = 4.90 %

Maturity Spot Rates1 5.00%2 4.75%3 4.47%4 4.35%5 4.90%

0 2 4 6 8 10 120.00%

200.00%

400.00%

600.00%

800.00%

1000.00%

1200.00%

Yield Curve

Maturity year

Spot

Rat

e

(b) State which of the 3 term structure theories are consistent with graph plotted. Briefly explain your answer.

It is consistent with the Expectation hypothesis. The latter explains two scenarios which relates to our graph, namely a downward sloping and an upward sloping yield curve.

The graph plotted is download sloping until maturity year 4 and then on till year 5(our data stops here), there is a rise in the spot rate resulting in an upward sloping curve.

The expectation hypothesis suggests that expected spot rate values in the future are equal to the corresponding forward rates known today.

The downward sloping part (maturity 1-4)

It suggests lower spot rates are to be expected in the future

The upward sloping part (maturity 4-5)

There is a sharp rise, it is more a straight line, meaning that for maturity year 5, we need to expect higher spot rates in the future.

(c) Based on the expectation hypothesis, find the expected value of the 1 year spot rate in 4 years.

(1+0Rn) n = (1+ 0Rm) m * (1+mFn) n-m

(1+0R5)5 = (1+0R4)4 * (1+4F5)1

1.0495 = 1.04354 * (1+4F5)

(1+4F5) = 1.0495 /1.04354

= 1.07129

4F5 = 0.0713 = 7.13 %

Answer = 7.13 %

QUESTION 4

(a) Find the price of the bond issued under strategy I, where the firm issues 4.5 % 8 year plain vanilla bonds. How many bonds does 3D printing need to issue to raise $10 million?

Price of bond = ci ∗(1−

1(1+i )n )+ F

(1+i )n

Cr = 0.045 i/2 = 0.0206 n=16 F= 10 000

C = (C r∗F)

2 = (0.045 * 10000)/2 = 225

Price = 225

0.0206∗(1− 1(1.0206 )16 )+ 10000

(1.0206 )16

= $ 10256.75

$10256.75 = 1 bond

$ 10 000 000 = 1/10256.75 * 10 000 000

= 974.97 bonds (rounding up to 975 bonds)

(b) Find the conversion ratio for the convertible bond issued under strategy ii

Convertion ratio= Face valueConversion Price

= 10 000/250

= 50 (1 bond = 50 shares)

Find the value of the conversion option associated with the convertible bonds

Price of convertible bond = price of plain vanilla bond + value of conversion ratio

11500 = 10256.75 + value of conversion ratio

Value of conversion ratio = 11500 – 10256.75 = $1243.25

What will be the minimum value of the convertible bond if the stock price suddenly jumps to $300 per share?

1 share = $ 300

Conversion ratio = 50

Price of convertible bond is equivalent to 50 shares

50 shares = $ 15000

Minimum price = 15000 + 1243.25

= $ 16243.25 (assuming that value of conversion option is constant)

What will be the minimum value of the convertible bond if the stock price suddenly drops to $50 per share?

1 share = $ 50

Conversion ratio = 50

Price of convertible bond is equivalent to 50 shares

50 shares = $ 2500

Minimum price = 2500 + 1243.25

= $ 3743.25 (assuming that value of conversion option is constant)

Find the price of the put option associated with the putable bond

Putable bond price = Plain vanilla bond price + Put Option Value

11000 = 10256.75 + Put Option Value

Put Option Value = 11000 – 10256.75

= $ 743.25