Midterm Solution

FIN500 (Spring 2012) Instructor: Matthew Marcinkowski

Midterm Exam Solutions

PROBLEM 1 (10 points):

Consider a 2-asset portfolio consisting of Asset 1 and Asset 2, whose rates of return are denoted by r1 and r2. The assets have the following risk/return characteristics: 0 < μ1 = E[r1] < E[r2] = μ2 < ∞, 0 < σ1 = Std.Dev [r1] < Std.Dev [r2] = σ2 < ∞, and Corr[r1,r2] = ρ1,2 = -1. Show that the minimum variance portfolio (MVP) constructed out of these two assets has σ = 0.

Let’s denote the weight on Asset 1 by w1 and on Asset 2 by w2 = 1 – w1.

The portfolio’s variance can be written as:

σp2 = w12σ12 + w22σ22 + 2 w1 w2σ1σ2ρ1,2 =

= w12σ12 + w22σ22 – 2 w1 w2σ1σ2 =

= (w1σ1 – w2σ2)2

Take both sides to the power of ?.

σp = w1σ1 – w2σ2 = w1σ1 – (1 – w1) σ2

Set to 0 and solve for w1.

w1σ1 – (1 – w1) σ2 = 0

w1σ1 – σ2 + w1σ2 = 0

w1 (σ1 + σ2) = σ2

w1 = σ2/(σ1 + σ2)

What we have shown is that if ρ1,2 = -1, there exists a portfolio with σ = 0. Since σ≥ 0, this portfolio must also minimize the value of σ, which makes it a MVP. In turn, it implies that the MVP has σ = 0.

PROBLEM 2 (5 points) :

Consider a portfolio consisting of a risky asset A (whose rate of return is r A) and a risk-free asset (whose rate of return is r f). One can express the expected rate of return of this portfolio as

E[r P] = x A E[r A] + (1-x A)r f

and its standard deviation as

σP= x AσA

where x A is the relative value (weight) of asset A, σA is the standard deviation of the risky asset’s rate of return, and σP the standard deviation of the portfolio’s rate of return. Based on the

equations given above, show that the portfolio’s expected rate of return (E[r P]) can be expressed as a linear function of its total risk (i.e. its standard deviation σP).

Solve eq. [2] for x A.

x A = (σP / σA)

Substitute into eq. [1].

E[r P] = (σP / σA) E[r A] + (1- (σP / σA))r f

Expand.

E[r P] = (σP / σA) E[r A] + r f - (σP / σA)r f

Collect terms.

E[r P] = r f + (σP / σA) (E[r A] - r f ) =

= {r f} +{(E[r A] - r f )/ σA} σP

= {intercept} + {slope} σP

PROBLEM 3:

Consider a bond with annual coupon payments. You purchased the bond when it was originally issued. Immediately afterwards, the YTM changed and remained at this new level indefinitely. Today, at the end of year 5 (immediately after the 5th coupon payment), your bond investment has the following characteristics:

Total Interest (Coupons) = $12,466.53

Interest-on-Interest (I2) = $4,782.28

Total Income = $7,406.13

Realized Yield (annual) = 4.6055%

Please, calculate the following:

a. (1 pt) Annual coupon

$12,466.53/5 = $2,493.306

b.(1 pts) Today’s YTM

N = 5

PV = 0

PMT = -2,493.306

FV = $12,466.53 + $4,782.28 = $17,248.81

CPT I/Y = 16.3%

c.(5 pts) Face value

[($7,406.13+ P0)/P0]1/5 – 1 = 0.046055

[($7,406.13+ P0)/P0]1/5 = 1. 0.046055

($7,406.13+ P0)/P0 = 1. 0.0460555

($7,406.13+ P0)/P0 = 1.252485188

$7,406.13+ P0 = 1.252485188x P0

0. 252485188 x P0 = $7,406.13

P0 = $29,333

d.(2 pts) Today’s market value

Cap G/L = Total Income - Total Interest - Interest-on-Interest =

= $7,406.13 - $12,466.53 - $4,782.28 = -$9,842.68

P5 = $29,333 - $9,842.68 = $19,490.32

e.(6 pts) Time to maturity at issue (round to the nearest year)

Time to maturity at the end of year 5:

I/Y = 16.3

PV = -19,490.32

PMT = 2,493.306

FV = 29,333

CPT N = 8

Time to maturity at issue = 8 + 5 = 13 years

PROBLEM 4:

A $20,000 face value 10% coupon corporate bond matures on July 24, 2031. You purchase the bond on November 16, 2011 at quoted price of 101.25.

a. (1 pt)What is the settlement date? Disregard weekends/holidays.

November 16, 2011 + 3 days = November 19, 2011

b. (3 pts)Compute the number of days since the last coupon payment as of the settlement date.

This is a CORPORATE bond. You were told numerous times that the vast majority of corporate bonds in the U.S. pay semi-annual coupon. You were also told to assume any corporate bond was in fact semi-annual unless you were explicitly informed it had a different coupon frequency.

The coupon dates are on January 24 and July 24. Last coupon was paid on July 24, 2011.

Days since last coupon = 4 x 30 – 5 = 115

c. (2 pts)What is the accrued interest (in dollars and cents) on the settlement date?

Accrued interest = (10%/2 of $20,000) x (115/180) = $1000 x (115/180) = $638.89

d. (2 pts)What is the invoice price (in dollars and cents)?

IP(M4) = (101.25% of $20,000) + $638.89= $20,250 + $638.89 = $20,888.89

e. (2 pts)What is the YTM as of the settlement date?

SDT = 11.1911

CPN = 2000

RDT = 7.2431

RV = 20000

360

2/Y

PRI = 20250

YLD CPT = 9.8518

f. (5 pts)What was the dirty price (in dollars and cents) on the purchase date?

P(M1) = IP(M4) – 3 days of AI = $20,888.89 – 3 x ($1000/180) = $20,872.22

PROBLEM 5:

A 6% corporate bond was issued in July of 1996. At issue, the bond had 30 years to maturity and

15 years of call protection.

a.(5 points) In July of 2010, the bond trades to yield 5.5%. If the call price (as of the first call) is 105%, compute the yield to first call.

Assume some face value – say $10,000

Market value as of July 2010:

N = (30 – (2010-1996)) x 2 = 32

I/Y = 5.5/2 = 2.75

PMT = (6%/2 of $10,000) = 300

FV = 10000

CPT PV = 10,527.51

Yield to first call (assume the bond will be called in July of 2011) as of July 2010.

N = 2 (1 year remains until call protection expires)

PV = -10,527.51 (actual market value of the bond in July of 2010).

PMT = 300

FV = 10500 (call price)

CPT I/Y = 2.72 x 2 = 5.44 (don’t forget to annualize)

b.(10 points) If, at the expiration of the call protection, similar non-callable bonds yield 4.5%, will it be profitable for the issuer to call this bond? Assuming a $10,000 par value, what will be the gain or loss per bond (expressed in PV terms) if the bonds are called.

Call protection expires in July of 2011. If you call, you will be replacing $10,000 of debt at 6% with $10,500 of debt at 4.5% (the call price is 105%).

Semi-annual savings = $10,000 x (6%/2) - $10,500 x (4.5%/2) = $300.00 - $236.25 = $63.75 PV of semi-annual savings:

N = 15 x 2 = 30

I/Y = 4.5/2 = 2.25

PMT = 63.75

FV = 0

CPT PV = 1,379.89

PV of additional cost at maturity:

500/1.0225 30 = 256.49

Net savings = 1,379.89 - 256.49 = 1,123.40 > 0 CALL

PROBLEM 6 (7 points):

A 10% convertible corporate bond with a face value of $1,000 and 10 years remaining to maturity trades to yield 2.0%. Its conversion ratio is 40. Compute the upper bound (the maximum possible value) of the price of one share of the underlying stock.

Market price:

N = 10 x 2 = 20

I/Y = 2/2 = 1

PMT = (10%/2) x $1000 = 50

FV = 1000

CPT PV = 1,721.82

The underlying stock cannot be cheaper than $1,721.82/40 = $43.05 per share.

PROBLEM 7 (8 points):

You used second order approximation (with convexity correction) to establish the following: YTM increases by 1% →Bond A’s price decreases 4.9207%

YTM decreases by 1% →Bond A’s price increases 5.4330%

Use 2nd order approximation to calculate Bond A’s volatility for a 2% decrease in YTM. State your answer as a percentage with four digits after the decimal point.

- D + C = -4.9207

D + C = 5.4330

Solve for D and C.

C =

D - 4.9207

D + D - 4.9207 = 5.4330

2D = 10.3537

D = 5.17685

C = 0.25615

When ?r doubles, D would double as well, but C would quadruple.

Volatility for 2% decrease in r = 2 x D + 4 x C = 2 x 5.17685 + 4 x 0.25615 = 11.3783% PROBLEM 8 (7 points):

Springfield mogul Montgomery Burns, age 80, wants to retire at age 100 so he can steal candy from babies full time. Once Mr. Burns retires, he wants to withdraw $500 million at the beginning of each year for 10 years from a special off-shore account that will pay 18% annually. In order to fund his retirement, Mr. Burns will make 20 equal end-of-the-year deposits in this same special account that will pay 18% annually. How large of an annual deposit must be made to fund Mr. Burns retirement plans?

Calculate the required balanced of his retirement account when he turns 100 (use BGN mode): BGN mode

N = 10

I/Y = 18

PMT = 500

FV = 0

CPT PV = 2,651.51 million

Calculate the end year deposit require to reach that goal (use END mode).

END mode

N = 20

I/Y = 18

PV = 0

FV = 2,651.51

CPT PMT = 18.08 million annual deposit

PROBLEM 9 (8 points):

Your parents ask your advice on financing a new car purchase. Nissan has been running a national sales promotion that gives buyers of a new Maxima the choice of a $2,000 rebate or 0.9% APR financing for 60 months. In addition, the local Nissan dealer is offering 5.9% APR financing for 60 months on all car purchases through a local bank, which could be used if your parents decide to take the rebate and use it as an additional down payment on their Maxima. Your parents found a fully loaded Maxima 3.5 SL for a price of $31,000 and have $3,000 as a down payment. Should your parents take the $2,000 rebate with 5.9% APR financing or the

0.9% APR financing and no rebate? Show your work to justify your decision.

Objective: minimize monthly payments With the rebate:

N = 60

I/Y = 5.9/12 = 0.491667

PV = 31,000 – 3,000 – 2,000 = 26,000

FV = 0

CPT PMT = 501.44

With subsidized interest rate but no rebate: N = 60

I/Y = 0.9/12 = 0.075

PV = 31,000 – 3,000 = 28,000

FV = 0

CPT PMT = 477.42

Choose 0.9% APR over the rebate.

TRUE OR FALSE QESTIONS (Answer TRUE or FALSE in the spaces provided)

(this section is worth 10 points – 1/2 points per question)

1.F__ The PV of an ordinary annuity is larger than the PV of an otherwise identical (same

payments, same number of payments, same discount rate) annuity due.

2.T__ At IRR (Internal Rate of Return) the NPV (Net Present Value) is equal to zero.

3.F__ Consider a series of negative cash flows followed by a series of positive cash flows

(i.e. a normal project with an initial outlay). The NPV of such stream of cash flows would

decrease as the discount rate (cost of capital) goes down.

4.T__ In relative terms, prices of bonds with longer time to maturity are more sensitive to

changing interest rates than those with shorter time to maturity.

5.F__ In relative terms, prices of bonds with higher coupon rates are more sensitive to

changing interest rates than those with lower coupon rates.

6.F__ Modified duration measures the absolute sensitivity of bond prices to changing

interest rates.

7.T__ At duration, the realized yield of a bond is approximately equal to the YTM that was

in effect when the bond was purchased.

8.F__ First order approximation produces an asymmetric price response to changing

interest rates.

9.T__Except for the initial YTM, first order approximation always understates the true

value of a bond.

10.F__ Consider a portfolio consisting of two risky assets A (E[r A]=10%, σA=8%) and B

(E[r B]=5%, σA=3%). A trivial portfolio consisting solely of asset A must be inefficient.

11.T__ Under our standard assumptions, all investors would choose a combination of the

risk free asset and the market portfolio.

12.F__ Beta measures the amount of unsystematic risk associated with an investment.

13.F__ A stock whose beta < 1 would earn a higher expected return higher than the market

portfolio.

14.F__ If a risk free asset and the market portfolio are available to be included in an

investment portfolio, the investment portfolio’s efficient frontier is represented by a

curve.

15.F__ Stock A has σA=8% and Stock B has βB=20%. Based on this information, we can

conclude that the total risk of Stock A is lower than that of Stock B.

16.T__ Stock A’s expected return is 10% and the s.d. 8%. Stock B’s expected return is 20%

and the s.d. 10%. If the na?ve (50-50) portfolio has a s.d. of 9%, the rates of return of the two stocks must be perfectly positively correlated.

17.F__ A portfolio consisting of a risk-free asset and shares of some stock has only

systematic risk.

18.T__ By increasing the number of stocks in one’s portfolio, all systematic risk can be

essentially eliminated.

19.F__ Convertible bonds trade at higher yields than comparable non-convertible bonds.

20.F__ An asset that lies above the SML is overpriced.