Risk And Return Question Answers

1. Is this morning’s CNN forecast of tomorrow’s temperature a random variable? Is tomorrow’s temperature a random variable?

Solution – CNN forecast of tomorrow‘s temperature is not a random variable rather it is an expected variable . They are making an assumption of the temperature. But the tomorrow‘s temperature is a random variable .As it is random assumption.

2. Does a higher reward (expected rate of return) always come with more risk?

Solution - No, the higher reward not always comes with the more risk. Sometimes the higher reward also comes with the better investment plan or with the better profit made by the company in which the investment is made. There are different factors that affects the expected rate of return other than risk.

3. Would a single individual be effectively more, equally, or less risk averse than a pool of such investors?

Solution – As in the case of single individual, he/she will be effectively more risk averse as compare to the pool of such investors because in the pool of investor the risk is shared amount the different investor as in the case of individual the single person has to bear all the risk for it alone.

Therefore, we can say that the effective risk aversion of the group will be lower than that of any of its members. And this is exactly how financial markets work: Their aggregate risk absorption capabilities are considerably higher than those of their individual investors. In effect, the financial markets are less risk averse than individual investors.

4. A bond will pay off $100 with probability 99% and will pay off nothing with probability 1%. The equivalent risk-free rate of return is 5%. What is an appropriate promised yield on this bond?

Solution – Payoff

Payoff  Rate of return   Frequency  
$100         +0%               99% of the time
$ 0          -100%             1% of the time

Expected Payoff = Prob * Cash flow + Prob * Cash flow
                = 99%* $100 + 1%*$0
                = $99

Expected return $99 but we promised to pay rate of return @5%
Promised R = $105 - $100/$100 = 5%
Our expected rate of return - $99 - $100 / $100 = -1%

For an expected rate of return to be 5% the cash flow would be 

= 99% * X + 1% * $0 + 4% * $0 = $105
= 99x = $105
= X = $106.06

The cash flow for an expected rate of return @5% is $ 106.60
The promised rate of return at a cash flow will be =
Promised (r) = $106.06 - $100 / $100 = 6.06%
The promised rate will provide us the expected rate of return to be = 
Expected Interest Rate – Prob * Rate of Return (1)  +  Prob * Rate of return (2)  + 
= 99% * (+6.06%) + 1% * (-100%) 
= .059994 + ( - 0.01) = .05 ~ 5%

5. An L.A. Lakers bond promises an investment rate of return of 9%. Time-equivalent Treasuries offer 6%. Is this necessarily a good investment? Explain.

Solution – Yes, it is a good investment. L.A. Laker bond promises an investment rate of return offer 9%. Time equivalent treasuries offer 6%. Therefore, it promises a rate of return of 9%, but more than treasuries offer 6%. If the Risk is more than that of treasuries, the bond is not suitable otherwise it is suitable. The investor may also evaluate the coefficient of variance to determine the suitability of bond. So, it is a good investment.

6. A financial instrument will pay off as follows:

Prob      50%     25%    12.5%    6.25%    3.125%    3.125%
Payoff    $100   $110   $130       $170     $250      $500

Assume that the risk-free interest rate is 0.

1. What price today would make this a fair bet?
2. What is the maximum price that a risk-averse investor would be willing to pay?

Solution – Expected today price –

 Prob*payoff (1) + Prob * payoff (2) + Prob*payoff (3) + Prob * payoff (4) +  Prob*payoff (5) + Prob * payoff (6) 
50% * $100 + 25% *$110 + 12.5% * $130 + 6.25% * $170 + 3.125% * $250 + 3.125% * $500
= $127.8125
So the price today would make a fair bet is $127.8125.

2. In the case of Risk – Averse, with the lowest probability there will be the highest risk. So in the case of maximum price, we will consider the lowest probability price i.e $500.

So the maximum price that a risk-averse investor would be willing to pay is $500.

7. Go to the Vanguard website. Look at funds by asset class, and answer this question for bond funds with different durations.

a. What is the current yield-to-maturity of a taxable Vanguard bond fund invested in Treasuries?

b. What is the current yield-to-maturity of a taxable Vanguard bond fund invested in investment-grade bonds?

c. What is the current yield-to-maturity of a taxable Vanguard bond fund invested in high-yield bonds?

Solution

a. - The current yield-to-maturity of a taxable Vanguard bond fund invested in Treasuries is 1.19%

b. The current yield-to-maturity of a taxable Vanguard bond fund invested in investment-grade bonds is 1.94%

c. The current yield-to-maturity of a taxable Vanguard bond fund invested in high-yield bonds is 1.33%

8. A Disney bond promises an investment rate of return of 7%. Time-equivalent Treasuries offer 7%. Is the Disney bond necessarily a bad investment? Explain.

Solution – No it is not a bad investment. A Disney bond promises an investment rate of return of 7%. If the risk in Disney bond is more than that of treasuries than the bond is not suitable otherwise suitable. So, it is not a bad investment. The investor may also evaluate the coefficient of variance to determine the suitability of bond.

9. The probability of receiving full payment of $210 in one year is only 95%, the probability of receiving $100 is 4%, and the probability of receiving absolutely no payment is 1%. If the bond quotes a rate of return of 12% what is the time premium, the default premium, and the risk premium?

Solution

Let the quoted rate of return = 12%
At the promised interest rate of 12% the expected rate of return will be – 
Expected payoff – Prob * Cash flow (1)  +  Prob * Cash flow (2) +   Prob * Cash flow (3)
= 95% * $210 + 4% * $100 + 1% * $0
= $199.5 + $4 +$0
= $203.5

Expected return of $203.5 is less than the $224 that the govt promises. 
We promise to pay a rate of return of 12%
Promised (r) = $224 - $200/$200 = 12%
But XYZ ltd expected rate of return is =
Expected rate of return = $203.5 - $200/$200 = 1.75%
For an expected rate of return to be 12% the cash flow would be 
=   95% * X + 4% * $100 + 1% * $0= $224
=   95x = $224 - $4
=   X =   $220/95
=   X =   $231.5

The cash flow for an expected rate of return @12% is $ 231.5
The promised rate of return at a cash flow will be =
Promised (r) = $231.5 - $200 / $200 = 15.75%
The promised rate will provide us the expected rate of return to be = 

Expected Interest Rate – Prob * Rate of Return (1) + Prob * Rate of return (2) +  Prob * Rate of Return (3)  
= 95% * (+15.75%) +4% * (-50%) + 1% * (-100%)
= .15 + (- .2) + (-.1)
= .12
~ 12%

10. Using information from a current newspaper or the WWW, what is the annualized yield on corporate bonds (high-quality, medium-quality, high-yield) today?

Solution - High-yield corporate bonds - 5.92USD
High-quality corporate bonds – 1.08
Medium Quality corporate bonds – 1.07

11. How is a credit swap like an insurance contract? Who is the insurer in a credit swap? Why would anyone want to buy such insurance?

Solution - Certain risk characteristics are required before protection, in its classical sense, can be considered true “insurance.” Each characteristic of insurance compared to the characteristics of credit default swaps produces a gap leading to the conclusion that these derivative products are not insurance in its most basic form.

Pure vs. Speculative Risk. Insurance was designed to insure against pure risk not a speculative risk.

The seller of a credit default swap (rarely an insurance company, usually another financial institution) agrees to protect the buyer against poor speculation and bad investment decisions. That is not the purpose of insurance.

Credit default swaps do not qualify as insurance in the classical sense because:

1) They protect against speculative losses;
2) No insurable interest is required for their purchase. Essentially, they are an investment instrument used by all counterparties to protect against the loss of and to generate income.

A credit default swap (CDS) is the insurer in a credit swap. We want to buy such insurance because it allows different funds to hold different premium of a bond.

12. A bond promises to pay $12,000 and costs $10,000. The promised discount on equivalent bonds is 25% per annum. Is this bond a good deal?

Solution

Promised to pay = $12000
After discount = $12000*25%
= $12000 - $3000
= $9000
Cost is $12000 and we are getting $9000.
So it is not a good deal.

13. A project costs $19,000 and promises the following cash flows:

                Y1          Y2          Y3
Cash Flows      $12,500     $6,000      $3,000

The appropriate discount rate is 15% per annum. Should you invest in this project?

Solution

Cost of the project – $19000

Year        Cash Flow     Discount rate   Total 
   1        $12500        .869	          $10862.5
   2        $6000         .756            $4536
   3        $3000         .657            $1971
                                     Total = 17369.5

NPV – Cash inflow – cash ouflow 
            $ 17369.5 – $ 19000
            $1630.5

We should not invest in this project.

14. Under risk neutrality, a factory can be worth $500,000 or $1,000,000 in two years, depending on product demand, each with equal probability. The appropriate cost of capital is 6% per year. The factory can be financed with proceeds of $500,000 from loans today. What are the promised and expected cash flows and rates of return for the factory (without a loan), the loan, and the levered factory owner.

Solution – Let us assume that the bad outcome is “Rain” and the good outcome is “Sun”. ( value is in $10000)

Without a loan

Expected future value of Factor = Prob * Value Rain + Prob * Value Sun
                                = 1/2 * $50 + 1/2 *$100
                                = $75

Present value of the Factory =  Future Value/(1+Expected rate )
                             = $75/(1+6% )
                             = $70.75

Promised rate of return of factory :
If it is Sunny 

In Sun = $100 - $70.75/$70.75 ~ +41.34%

If it is Rainy

In Rain = $50 - $70.75 /$70.75 ~ -29.32%
Therefore the expected rate of return is 

= Prob * Rain rate of return + Prob * Sun rate of return
= 1/2 * (-29.32%) + 1/2 (+41.34%)
= 6%

Loan

Event        Prob        Value         Discount Factor
Rain         1/2         $50           1/1.06
Sun          1/2         Promised      1/1.06

Solution - The creditor receives the property worth $50 if it rains, or the full promised amount (to be determined) if the sun shines. 

Lending amount $50 today 
Promised payoff 
Loan value = Prob * Rain Loan PV + Prob * Sun loan PV.
$50   = 1/2 * ($50/(1+6%) ) + 1/2 * ( Promise/(1+6% ))

Promise =            ({(1+6%)*$50 -1/2*$50)/(1/2)

Promise Payoff = $56

In repayment, Paid by the borrower only if the sun shines.

Promised Rate of return r = ( $56-$50/$50 )
                          = + 12%

If it rain than the lender rate of return will be – 
( ($50-$50)/$50 )
= 0%
Therefore, the lender expected rate of return is 

= Prob * Rain rate of return + Prob * Sun rate of return
= 1/2 * (0%) +1/2 (+12%)
= 6%

The stated rate of return is 12% (and it is not an exorbitant rate!), but the expected rate of return is 6%. After all, in our risk-neutral perfect market, anyone investing for one year expects to earn an expected rate of return of 6%

Levered equity owner

If the sun shines – 
Payoff = Value of the Factory – promised value 
       = $100 - $56
       = $ 44
If the rain come –
Payoff = Value of the Factory – promised value 
       = $50 - $56
       = $0
Payoff Table 

Event      Prob       Value    Discount Factor
Rain       1/2        $0       1/1.06
Sun        1/2        $44      1/1.06

Expected future levered building ownership payoff 
= 1/2*$0 + 1/2*$44 
= $22
Present value of levered building ownership 
PV       = 1/2 * ($0/(1+6% ) ) + 1/2 * ( $44/(1+6% ))
         = $20.75

The rate of return – 

If Sun – r =  ($44-$20.75)/$20.75 
         r = +112%
If Rain – r = ($0-$20.75)/$20.75
          R = -100%
Expected rate of return of levered equity ownership =
= Prob * Rain rate of return + Prob * Sun rate of return
= 1/2 * (-100%) + 1/2 (+112%)
= 6%.

15. Debt is usually safer than equity. Does the risk of the rate of return on equity go up if the firm takes on more debt, provided the debt is low enough to remain risk free? Illustrate with an example that you make up.

Solution - An item that qualifies as debt is interest rates while an item that qualifies as equity is the internal rate of return, and together debt and equity refer to how much money the company needs to finance. The cost of debt is usually 4% to 8% while the cost of equity is usually 25% or higher.

Debt is a lot safer than equity because there is a lot to fall back on if the company does not do well. Therefore, in many ways debt is a lot cheaper than equity.

Let us understand with the help of an example of why debt is cheaper than equity:

Say X is a business owner and X need $10,000.00 in order to get the business up and running. X has two options to get this $10,000.00. X first option is to take out a bank loan at 5% interest rate, while X second option is to have someone take out a share at 30% for $10,000.00.

If X business earns $30,000.00 after one year of being opened and X took out the bank loan then the money X would have to pay to the bank would be $500.00 which would leave X with a total of $29,500.00 in profit.

However, if X sold a share of its company to someone at 30% X would have to give that person a total of $9,000.00 which would leave X, the business owner, with a total of $21,000.00.

So had X taken out debt instead of equity X would have received $8,500 more in profit.

In this example we see why debt is a lot safer than equity and why X could potentially receive a lot more money with debt than X can by equity.

But if the firm take more debt, then the rate of return on equity goes down as the company has to pay the return from the equity share capital of the business only.

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