Suppose you have an equally$-$weighted portfolio consisting of two securities$.$ One is a corporate bond and the other is a share of stock in the Delta Corporation. As of the beginning of the year, the bond has a coupon rate of $6\%,$ matures $5$ years from today and has a yield to maturity $($YTM$)$ of $4\mathrm{.}73$ percent. The Delta stock pays a constant dividend of $95$ cents per share which is expected to remain the same into the foreseeable future and is currently selling for $$41\mathrm{.}30.$ Economists forecast there is a $40$ percent chance the economy will go into a recession and a $60$ percent chance it will remain normal. In the event the economy.

Compute the current price of both securities$.$

Construct a table showing the expected price of both securities under a recession and normal state of the economy.

Reconstruct the table in part b to show the expected return of both securities under a recession and normal state of the economy.

Compute the expected return you would earn on your portfolio over the year.

Will you earn the expected return you compute in part d above? If not, then what return will you actually earn?

There are $4$ steps to solve this one.

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Step $1$

To compute the current price of both securities$,$ we need to use the present

value formulas for the bond and the stock.

$1.$ Corporate Bond:

The coupon rate is $6\%,$ and it matures in $5$ years. The yield to maturity $($YTM$)$

is $4\mathrm{.}73\%.$ We can use the formula for the present value of a bond to calculate

its price:

PV $=($C x $(1-(1+$ r$)^(-$n$)))/$ r $+($F $/(1+$ r$)^$n$)$

Explanation:

Where:

PV $=$ Present value or price of the bond

C $=$ Coupon payment per period

r $=$ Yield to maturity $($expressed as a decimal$)$

n $=$ Number of periods $($in this case, the number of years until maturity$)$

F $=$ Face value or par value of the bond

Let's calculate the price of the bond:

C $=0\mathrm{.}06\backslash$times F $($Coupon rate of $6\%)$

r $=0\mathrm{.}0473($Yield to maturity$)$

n $=5($Maturity in $5$ years$)$

F $=100($Assuming a face value of $$100)$

PV $=(0\mathrm{.}06\backslash$times $100\backslash$times $(1-(1+0\mathrm{.}0473)^(-5)))/0\mathrm{.}0473+(100/(1+0\mathrm{.}0473)^5)$

Explanation:

The current price of the corporate bond is approximately $$102\mathrm{.}32.$

Step $2$

Delta Stock:

The current price of the Delta stock is $$41\mathrm{.}30,$ and it pays a constant dividend of $95$ cents per share. To calculate the price of a stock with a constant dividend,

we can use the Gordon Growth Model:

P$0=$ D $/($r $-$ g$)$

Explanation:

Where:

P$0=$ Price of the stock

D $=$ Dividend per share

r $=$ Required return on the stock

g $=$ Dividend growth rate $($assumed to be zero in this case$)$

Let's calculate the price of the Delta stock:

D $=$ $$0\mathrm{.}95($Constant dividend of $95$ cents per share$)$

r $=$ Required return on the stock

In the event of a recession: r will increase by $20\%.$

In the event of an improvement in the economy: r will decrease by $10\%.$

Recession:

P$0\_$recession $=0\mathrm{.}95/(1+0\mathrm{.}2)$ $$0\mathrm{.}7917$

Normal State of the Economy:

P$0\_$normal $=0\mathrm{.}95/(1-0\mathrm{.}1)$ $$1\mathrm{.}0556$

The current price of the Delta stock under a recession is approximately $$0\mathrm{.}7917,$ and under a normal state of the economy, it is approximately $$1\mathrm{.}0556.$

Step $3$

Now, let's construct a table showing the expected price of both securities under a recession and normal state of the economy:

Recession

Normal State

BOND

$$102\mathrm{.}32$

$$102\mathrm{.}32$

DELTA STOCK

$$0\mathrm{.}79$

$$1\mathrm{.}06$

To reconstruct the table and show the expected return of both securities$,$ we need to consider the dividend yield and capital gains$/$losses$.$

Recession:

Explanation:

Bond: No change in yield to maturity $($YTM$),$ so no capital gains$/$losses$.$

Expected return is the coupon rate: $6\%.$

Delta Stock: Dividend yield is $0\mathrm{.}95/0\mathrm{.}79171\mathrm{.}20\%.$ Capital losses due to increased required return: $20\%.$ Expected return is $-20\%+1\mathrm{.}20\%=-18\mathrm{.}80\%.$

Normal State of the Economy: Bond: No change in yield to maturity $($YTM$),$ so no capital gains$/$losses$.$ Expected return is the coupon rate: $6\%.$ Delta Stock: Dividend yield is $0\mathrm{.}95/1\mathrm{.}05560\mathrm{.}90\%.$ Capital gains due to decreased required return: $-10\%.$ Expected return is $-10\%+0\mathrm{.}90\%=-9\mathrm{.}10\%.$

Step $4$

The reconstructed table showing the expected return of both securities under a recession and normal state of the economy is:

Recession

Normal State

Bond

$6\%$

$6\%$

Delta Stock

$-18\mathrm{.}80\%$

$-9\mathrm{.}10\%$

To compute the expected return you would earn on your portfolio over the year, we need to consider the portfolio weights and expected returns of the securities$.$

Given that the portfolio is equally$-$weighted, each security has a weight of $50\%.$

Explanation:

Expected portfolio return under a recession:

$(0\mathrm{.}5\backslash$times $6\%)+(0\mathrm{.}5\backslash$times $(-18\mathrm{.}80\%))=-6\mathrm{.}40\%$

Expected portfolio return under a normal state of the economy:

$(0\mathrm{.}5\backslash$times $6\%)+(0\mathrm{.}5\backslash$times $(-9\mathrm{.}10\%))=-1\mathrm{.}55\%$

So$,$ the expected return you would