Assignment 8 Answers Valuing Risky Bonds PDF

Preview

Summary

Download Assignment 8 Answers Valuing Risky Bonds PDF

Description

FNCE 30001 – Investments Valuing Risky Bonds Problem Set Due the 12 October 2020 at 10:00am

This is Part A and it is UNMARKED. It will be discussed in your tutorial during the week of 12 October. Part B, the marked Quiz part of this assignment is graded pass/fail purely for the attempt at answering. You must fill in an answer for full credit. Part A Where relevant, assume that you are answering questions for/about a risk-neutral investor, i.e. 𝑬[𝒓] = 𝒓𝒇. Assume the annualized rates are the same for all maturities (a flat yield curve). Feel free to check your work in Excel (in fact, I encourage it), but please do the questions by hand. Why? It forces you to think about the economic intuition and to better understand what you are doing.

The following two tables from Moody’s investor services, may help answers some of the questions below. The first table has an unclear heading. It is the cumulative percentage of firms that default within X years (years are in blue in the row with “Rating”)

1. Suppose a 5-year zero coupon senior unsecured bond with a face value of $1000 has a Moody’s credit rating of Ba. Please assume that agents are risk neutral (or for those of you who remember CAPM, an alternate assumption would be assume that investors are well diversified and the bond’s default is completely uncorrelated with the rest of the economy so that 𝛽 = 0. This assumption is not far from true for many bonds) and that the annual risk-free rate is 4%. Assume that the default rate is similar to the average default rate for Ba-rated bonds. If the bond defaults, assume that the recovery rate is typical for other senior unsecured debt. a. What is the expected return? Because you are told to assume that agents are risk neutral, Annualised 𝐸[𝑟] = 𝑟𝑓 = 4% Or 𝐸[𝑟𝐻𝑃𝑅 ] = 21.665% = 0.21665 = 1.21665 − 1 = 1.045 − 1 b. What are the expected cash flows when this bond matures? Please use the tables above to obtain default and recovery rates. From the table on page 1, 8.5% of Ba rated companies default within 5 years. The recovery rate for Ba-rated bonds that default sometime within 5 years is 41.59%. This is a bit overly precise, but I’ll use this number. 𝐸[𝐶𝐹] =

𝐷𝑒𝑓𝑎𝑢𝑙𝑡

∑

𝑝𝑟(𝑠)𝐶𝐹(𝑠)

𝑠=𝑁𝑜 𝐷𝑒𝑓𝑎𝑢𝑙𝑡

𝐸[𝐶𝐹] = 𝑝𝑟(𝑁𝑜 𝐷𝑒𝑓𝑎𝑢𝑙𝑡)𝐶𝐹(𝑁𝑜 𝐷𝑒𝑓𝑎𝑢𝑙𝑡) + 𝑝𝑟(𝐷𝑒𝑓𝑎𝑢𝑙𝑡)𝐶𝐹(𝐷𝑒𝑓𝑎𝑢𝑙𝑡) 𝐸[𝐶𝐹] = 0.915 × $1000 + 0.085 × (0.4159 × $1000) 𝐸[𝐶𝐹] = $915 + 35.35 𝐸[𝐶𝐹] = $950.35 c. What is the value/price of this bond? ] $950.35 𝐸[𝐶𝐹 = $781.12 𝑃= 𝑇 = ( 1 + 0.04)5 (1 + 𝐸 [𝑅 ]) d. What is the interest rate that would be printed in a newspaper or on a financial website for this bond? Those places like BEY, but since this is a zero-coupon bond, we’ll just calculate the annual rate directly (recall that when annual coupons or zeros BEY=EAR). 𝐹𝑉 𝑃𝑟𝑜𝑚𝑖𝑠𝑒𝑑 𝐶𝐹 = 𝑃= 𝑇 (1 + 𝑌𝑇𝑀 ) (1 + 𝑌𝑇𝑀)𝑇 $1000 781.12 = (1 + 𝑌𝑇𝑀 )5 𝑌𝑇𝑀 = 5.06% e. What is the default premium? Default Premium = Promised YTM − 𝑟𝑓 =5.06%-4%=1.06% f. If investors are risk neutral, how can you explain the fact that the default premium is greater than zero?

s. T

(sometimes more, sometimes less, but on average the risk-free rate). If we had risk-averse investors, the default premium represents the extra return needed to exactly offset the losses in the bad states of the world when the bond defaults, so that on average the investor earns (or whatever asset pricing model you think best explains the world)(sometimes more, sometimes less, but What if I had given you the price ($781.12), cash flows and probabilities and asked you to calculate the expected return. How would your solution change? Answer: The answer 1b would be the same. But, how you calculate expected returns changes. So, how would you calculate the expected return, if you know the pay offs and the probabilities? a. With the payoffs and probabilities, you can still calculate [𝐶𝐹] = $950.35 as in 1b.

b. 𝐸[𝑟𝐻𝑃𝑅 ] =

$950.35 781.15

− 1 = 0.21665

5 5 c. To get the annualised 𝐸[𝑟 ] = √1 + 𝐸 [𝑟𝐻𝑃𝑅 ] − 1 = √ 1 + 0.21665 − 1 = 0.04

The above is the most straightforward way of doing this calculation, but if you are more mathematically inclined then let me offer you one more way of doing this, as well as a technical note. d. You could also calculate the expected return from the payoffs. $415.90 $1000 [𝑟𝐻𝑃𝑅 ] = 0.915 × ( − 1) + 0.085 × ( − 1) = 0.21665 781.15 781.15 Note that you must calculate the 𝐸[𝑟𝐻𝑃𝑅 ] first and only after calculating the 𝐸[𝑟𝐻𝑃𝑅 ], annualise. If you annualise first, your answer will be incorrect. This is because in most circumstances the expectation of a function does not equal the function of an expectation: 𝑔𝑒𝑛𝑒𝑟𝑎𝑙𝑙𝑦 𝐸[𝑔(𝑥)] ≠ 𝑔(𝐸[𝑥]) A specific example of this property that relates to concave functions is “Jensen’s Inequality” e. If you don’t believe me, try it. See if you can annualise first. The answer will be less than the correct answer due to the function being concave. The key thing to remember is to calculate [𝑟𝐻𝑃𝑅 ] before annualising.

2. Credit ratings only capture the probability of default. In truth, recovery rates are noticeably different for firms from different industries. Suppose you have a similar bond to question 1, but because of its industry the recovery rate is a much higher 65%, but the default rate is the same. a. What is the price of this bond with the 65% recovery rate? 𝐸[𝐶𝐹] =

𝐷𝑒𝑓𝑎𝑢𝑙𝑡

∑

𝑝𝑟(𝑠)𝐶𝐹(𝑠)

𝑠=𝑁𝑜 𝐷𝑒𝑓𝑎𝑢𝑙𝑡

𝐸[𝐶𝐹] = 𝑝𝑟(𝑁𝑜 𝐷𝑒𝑓𝑎𝑢𝑙𝑡)𝐶𝐹(𝑁𝑜 𝐷𝑒𝑓𝑎𝑢𝑙𝑡) + 𝑝𝑟(𝐷𝑒𝑓𝑎𝑢𝑙𝑡)𝐶𝐹(𝐷𝑒𝑓𝑎𝑢𝑙𝑡) 𝐸[𝐶𝐹] = 0.915 × $1000 + 0.085 × (0.65 × $1000) 𝐸[𝐶𝐹] = $915 + 55.25 𝐸[𝐶𝐹] = $970.25 𝑃=

] 𝐸[𝐶𝐹

(1 + 𝐸 [𝑅 ])

𝑇

=

$970.25 = $797.47 (1 + 0.04)5

b. What is the promised rate of return (yield to maturity)? How does it compare to your answer above? 𝐹𝑉 𝑃𝑟𝑜𝑚𝑖𝑠𝑒𝑑 𝐶𝐹 = 𝑃= 𝑇 (1 + 𝑌𝑇𝑀 ) (1 + 𝑌𝑇𝑀)𝑇 $1000 797.47 = (1 + 𝑌𝑇𝑀 )5 𝑌𝑇𝑀 = 4.63% It is lower, because the price is higher, as a result of the

c. Sometimes investors will value bonds by using the promised return (yield to maturity) for similarly rated bonds. The bonds in questions 1 and 2 have the same rating. How big of a mistake is it to use the promised return from (1) to price the bond in (2 )? Note the at the correct price is $797.47. But if we “discounted” promised cash flows with the same YTM as in question 6 we would get the same price, because the promised cash flows are the same. 𝑃𝑟𝑜𝑚𝑖𝑠𝑒𝑑 𝐶𝐹 𝑃= (1 + 𝑌𝑇𝑀 )𝑇 $1000 = $781.12 𝐼𝑛𝑐𝑜𝑟𝑟𝑒𝑐𝑡 𝑃 = (1 + 0.0506)5 So, the error is pretty large, $16.35 (=$797.47 - $781.12)

3. Suppose you are a company analyst, whose job, among other things is to assess the risk and return for new bond issues. A mining company called, Acme Mining, intends to issue zero coupon bonds with a $100 face value which will mature in exactly 3 years. The tables on page 1 imply that default and recovery rates are similar across time. In fact, they are not. During recessions the probability of default increases and recovery rates drop. Suppose the default rate for similar mining companies within 3 years is 5% in a strong economy and 15% in a weak economy. In addition, in a strong economy the recovery rate is 60%, but 30% in a week economy. There is a 10% chance of a weak economy. Assume investors are risk neutral and that the risk-free rate is 6%. a. Calculate the cash flows for each state of the world in three years. There are 4 states of the world: 1) Strong Economy and No Default (Strong, No Def) 2) Strong Economy and Default (Strong, Def) 3) Weak Economy and No Default (Weak, No Def) 4) Weak Economy and Default (Weak, Def) Payouts: Strong, No Def Strong, Def Weak, No Def Weak, Def

$100

$100 × .60 = $60

$100

$100 × .30 = $30

b. Calculate the probability of each state of the world in three years. Probability of each state: Strong, No Def Strong, Def Weak, No Def Weak, Def

0.90 × .95 = 0.855

0.90 × .05 = 0.045

0.10 × .85 = 0.085

0.10 × .15 = 0.015

c. What are the expected cash flows? 𝐸[𝐶𝐹] =

∑

𝑝𝑟(𝑠)𝐶𝐹(𝑠)

𝑠=𝐴𝑙𝑙 𝑆𝑡𝑎𝑡𝑒𝑠

𝐸[𝐶𝐹] = 𝑝𝑟(𝑆𝑡𝑟𝑜𝑛𝑔, 𝑁𝑜 𝐷𝑒𝑓)𝐶𝐹(𝑆𝑡𝑟𝑜𝑛𝑔, 𝑁𝑜 𝐷𝑒𝑓) + 𝑝𝑟(𝑆𝑡𝑟𝑜𝑛𝑔, 𝐷𝑒𝑓)𝐶𝐹(𝑆𝑡𝑟𝑜𝑛𝑔, 𝐷𝑒𝑓) + 𝑝𝑟(𝑊𝑒𝑎𝑘, 𝑁𝑜 𝐷𝑒𝑓)𝐶𝐹(𝑊𝑒𝑎𝑘, 𝑁𝑜 𝐷𝑒𝑓) + 𝑝𝑟(𝑊𝑒𝑎𝑘, 𝐷𝑒𝑓)𝐶𝐹(𝑊𝑒𝑎𝑘, 𝐷𝑒𝑓)

𝐸[𝐶𝐹] = 0.855 × $100 + 0.045 × $60 + 0.085 × $100 + 0.015 × $30 𝐸[𝐶𝐹] = $97.15

d. What is the price of this bond? Assuming risk-neutral investors, [𝑟] = 𝑟𝑓 = 6% ] 𝐸[𝐶𝐹 .15 3 = $81.57 𝑃= (1$97 + 0.06) = 𝑇 (1 + 𝐸 [𝑅 ]) e. What is the promised yield to maturity? 𝑃𝑟𝑜𝑚𝑖𝑠𝑒𝑑 𝐶𝐹 𝐹𝑉 𝑃= = 𝑇 (1 + 𝑌𝑇𝑀 ) (1 + 𝑌𝑇𝑀)𝑇 $100 $81.57 = (1 + 𝑌𝑇𝑀 )3 𝑌𝑇𝑀 = 7.03% 4. Part A question: Credit quants infer default probabilities from prices. Suppose two zerocoupon bonds have just been issued and they both have a face value of $100 and a maturity of 5 years. The first bond is risk free and has a price of $70. The second bond is risky and has a recovery rate of 50% and a price of $65. If the spread between the risky bond and the default free bond is 1.61%, what is the default probability implied by the risky bond’s price? First, I need to know what the risk-free rate is for 5-year zeros: 𝑃=

𝐶𝐹𝑇

(1 + 𝑟𝑓 )

$70 =

For the risky bond

100

𝑇

(1 + 𝑟𝑓 )

𝑟𝑓 = 7.39%

𝑃=

5

𝑇] 𝐸[𝐶𝐹 (1 + 𝐸[𝑟])𝑇

Assume that agents are risk neutral, 𝐸[𝑟 ] = 𝑟𝑓 = 7.39% 𝑃=

$65 =

𝑇] 𝐸[𝐶𝐹

(1 + 𝑟𝑓 )

𝑇

(1 − 𝑝𝑟(𝑑𝑒𝑓𝑎𝑢𝑙𝑡))$100 + 𝑝𝑟(𝑑𝑒𝑓𝑎𝑢𝑙𝑡)(0.5 × $100) (1 + .0739)5

$92.85 = $100 − $100𝑝𝑟(𝑑𝑒𝑓𝑎𝑢𝑙𝑡) + $50 𝑝𝑟(𝑑𝑒𝑓𝑎𝑢𝑙𝑡) $50𝑝𝑟(𝑑𝑒𝑓𝑎𝑢𝑙𝑡) = $7.16 𝑝𝑟(𝑑𝑒𝑓𝑎𝑢𝑙𝑡) = 14.3%

Part B 1. Suppose there are 4 states of the world with the following probabilities of occurring and the following payoffs. What is the expected pay off? State Probability Cash Flow Awesome .05 100 Pretty good .40 10 Isn’t bad .50 -2 OMG! .05 -50 𝐸[𝐶𝐹] =

𝐴𝑤𝑒𝑠𝑜𝑚𝑒

∑

𝑝𝑟(𝑠)𝐶𝐹(𝑠)

𝑠=𝑂𝑀𝐺!

= .05 × (−50) + 0.5 × (−2) + 0.4 × 10 + 0.05 × 100 = 5.5 2. Consider a bond with a 10% coupon and a yield to maturity of 8%. If necessary, you can assume that the yield to maturity of 8% is annualized as a bond equivalent yield. If the bond’s yield to maturity remains constant, then in one year, will the bond price be higher, lower or unchanged? Hint 1: what is the relation between price and yields? Hint 2: What is the YTM of a bond priced at par? There are 2 ways to answer this question. Faster is just by intuition: a) If the bond is priced at par, then the coupon rate equals the bond equivalent yield. b) By your answer to hint 1, 8% < 10%  P > FV, so the bond is trading at a premium c) , then as the bond approaches maturity the price must or the Done – this is enough for either the mid-term or final.

The solution below is absolutely, certainly NOT needed for any exam in this subject. I show it to you “for fun.” For those of you who do not trust your intuition or are sceptics and need proof, we can also show this mathematically. Proving intuition can be a little tricky, but below I solve it mathematically. a) The coupon rate = 𝐹𝑉 where C is the coupon payment per period and n is the number of periods per year over which the bond’s coupons are paid (n is usually, 1, 2 or 4). The bond equivalent yield is 𝑟𝐵𝐸𝑌 = 𝑛𝑟𝑝𝑒𝑟𝑖𝑜𝑑𝑖𝑐 𝑌𝑇𝑀 . If the Coupon rate =BEY, then: 𝑛𝐶

𝐶

𝐹𝑉

𝑛𝐶

𝐹𝑉

= 𝑛𝑟𝑝𝑒𝑟𝑖𝑜𝑑𝑖𝑐 𝑌𝑇𝑀

= 𝑟𝑝𝑒𝑟𝑖𝑜𝑑𝑖𝑐 𝑌𝑇𝑀 = 𝑌𝑇𝑀

(a1) (a2)

From the definition of YTM: 𝑃 = ∑ 𝑇𝑖=1

Substitute in (a2)

𝐶𝐹𝑖

(1+𝑌𝑇𝑀 )𝑖

𝑃 =𝐶(

1−

𝑃 = 𝐹𝑉 −

= 𝐶(

𝑇 1 (1+𝑌𝑇𝑀)

1−

1

𝑇 (1+𝐶 ⁄𝐹𝑉 )

𝐶⁄ 𝐹𝑉

𝐹𝑉

𝑌𝑇𝑀

)+

(1+𝐶⁄𝐹𝑉 )

𝑇

𝑃 = 𝐹𝑉

+

)+

𝐹𝑉

𝐹𝑉 (1+𝑌𝑇𝑀 )𝑇

(a3)

𝑇

(1+𝐶⁄𝐹𝑉 ) 𝐹𝑉

(a4)

𝑇

(1+𝐶 ⁄𝐹𝑉 )

(a5) (a6)

Q.E.D. b) We can show that as YTM and prices are inversely related, again by using the definition of YTM: 𝑖 𝑃 = ∑𝑇𝑖=1 (1+𝑌𝑇𝑀 )𝑖

𝜕𝑃

𝜕𝑌𝑇𝑀

= ∑ 𝑇𝑖=1 (

𝐶𝐹

(b1)

−𝑖×𝐶𝐹𝑖

1+𝑌𝑇𝑀 )𝑖−1

(b2)

For every i, the denominator is positive and the numerator is negative, so 𝜕𝑃 0 when the bond is trading at a premium, then the price is declining as the bond approaches maturity (just as we saw in lecture on the first day). The goal below

𝜕𝑃 𝜕𝑇

is to show

𝜕𝑃

𝜕𝑇

> 0 when the bond is trading at a premium.

From (a3) note: 𝑃 =𝐶(

𝑃=

𝐶

𝑌𝑇𝑀

1−

−

𝑃=

1 (1+𝑌𝑇𝑀)𝑇

𝑌𝑇𝑀

For ease of notation let’s call Note that:

𝐹𝑉×𝑌𝑇𝑀−𝐶 𝑌𝑇𝑀

𝐹𝑉

(1+𝑌𝑇𝑀 )𝑇

+

𝐹𝑉

(1+𝑌𝑇𝑀 )𝑇 𝑌𝑇𝑀(1+𝑌𝑇𝑀 )𝑇 𝐶 𝐹𝑉×𝑌𝑇𝑀−𝐶 + 𝑌𝑇𝑀(1+𝑌𝑇𝑀)𝑇 𝑌𝑇𝑀 𝐹𝑉×𝑌𝑇𝑀−𝐶 −𝑇

𝑃 = 𝑌𝑇𝑀 + 𝐶

𝐶

)+

= 𝑚𝑒𝑠𝑠

𝑌𝑇𝑀

(1 + 𝑌𝑇𝑀)

If P=FV then mess = 0 (this follows from (a2)) If P > FV then mess < 0 (this follows from (a2) = just substitute the “=” with “>”)

(a3’) (b1) (b2) (b3)

If P < FV then mess > 0 (this follows from (a2) = just substitute the “=” with “0

(b8)

We know from above that the bond is trading at a premium, so mess 0 and because YTM must be positive, that ln(1 + 𝑌𝑇𝑀) > 0, therefore 𝑑𝑃

𝑑𝑇

So, as the time to maturity decreases, the price must also decrease.

3. If two bonds have the same risk, same coupon payments and same face value, which bond will be have a higher price, a callable bond or a convertible bond? Investors would demand a higher interest rate for a callable bond because a callable bond gives the issuer the right to buy back the bond if interest rates decline (a good thing for the issuer, because they can refinance at lower rates). A higher interest rate implies a lower price. Convertible bonds are more valuable to investors because the bonds can be converted to stock if the stock price increases. More valuable means the bond holder is willing to pay a higher price and accept a lower interest rate.

This text is for questions 4 through 6: An internet company, e-Money, is offering credit card with an A.P.R. of 20%. What is the effective annual interest rate offered by e-Money if the compounding interval is 4. Annual 𝑟𝐸𝐴𝑅 = (1 +

𝑟𝐴𝑃𝑅

5. Monthly 𝑟𝐸𝐴𝑅 = (1 + 6. Daily 𝑟𝐸𝐴𝑅 = (1 +

𝑟𝐴𝑃𝑅

365

1

) − 1 = 20%

1 𝑟𝐴𝑃𝑅

)

12

12 365

)

− 1 = 21.94%

− 1 = 22.13%

This text is for questions 7 through 9: You need a financial calculator or Excel to solve for the periodic yield to maturity of a semiannual coupon-paying bond. In Excel, the function is RATE. A 20year maturity bond with par value $1000 makes semiannual coupon payments at a coupon rate of 8%. Find the bond equivalent and effective annual yield to maturity of the bond if the bond price is:

7. $950 - BEY We need a financial calculator or Excel to solve for the periodic yield to maturity of a semiannual coupon-paying bond. In Excel, the function is RATE. =RATE(40,40,-950,1000,0) → Periodic YTM, 𝑟𝑝𝑒𝑟𝑖𝑜𝑑𝑖𝑐 𝑌𝑇𝑀 = 4.26%

8. $950 - EAR

𝐵𝐸𝑌 = 2 × 4.26% = 8.52%

𝐸𝐴𝑅 = (1.0426)2 − 1 = 8.70%

9. $1000 - BEY YTM= 4% (use RATE(40,40,-1000,1000,0)) 𝐵𝐸𝑌 = 2 × 4% = 8% 10. $1000 - EAR 𝐸𝐴𝑅 = (1.04)2 − 1 = 8.16% 11. $1050 - BEY Using use RATE(40,40,-1050,1000,0)) → Periodic YTM, 𝑟𝑝𝑒𝑟𝑖𝑜𝑑𝑖𝑐 𝑌𝑇𝑀 = 3.76% 𝐵𝐸𝑌 = 2 × 3.76% = 7.52% 12. $1050 - EAR 𝐸𝐴𝑅 = (1.0376)2 − 1 = 7.66%...