Course FM Manual by Dr. Krzysztof Ostaszewski, FSA, CERA, FSAS, CFA, MAAA 2013 Edition Errata As of May 10, 2014, this document is no longer updated Posted November 2, 2013 For Problem 13 in Practice Examination 12, the answer choices should be: A. $65 or more B. Less than $65 but more than $55 C. Less than $55 but more than $45 D. Less than $45 but more than $35 E. Less than $35 and the solution should be: Given that the futures price is $660, spot price is $645, cost of carry is 2% (effectively, a negative dividend), if we write r for the current risk-free force of interest, we must have

660 = F

Forward price = Se−δ t

Ex-dividend spot price ⋅ ert

Accumulated at risk-free rate = 645 ⋅e0.02⋅1 ⋅er⋅1 = 645 ⋅e0.02 ⋅er .

Based on this, 660645

= er+0.02 , and

r = ln 660645

⋅e−0.02⎛⎝⎜

⎞⎠⎟ = ln660 − ln645 − 0.02 ≈ 0.2989518%.

An ounce of gold costs now $645, but whoever owns it will have to pay storage cost, and the price with that storage cost included is $645 ⋅e0.02 . An investor who buys an ounce of gold pays that and at the end of one year will either have one ounce worth $700 with probability pUP or once ounce worth $600 with probability 1− pUP , where probabilities are risk-neutral. Since these probabilities are risk-neutral, we must have

$645e0.02 = pUP ⋅$700er

+ 1− pUP( ) ⋅ $600er

.

Therefore,

pUP =645er+0.02 − 600700 − 600

=645 ⋅ 660

645− 600

100= 0.6,

and 1− pUP = 0.4. One year from now, the futures price will be either

FUP = 700 ⋅er+0.02 = 700 ⋅ 660

645≈ 716.28,

or

FDOWN = 600 ⋅er+0.02 = 600 ⋅ 660645

≈ 613.95.

The European call with the strike price $650 will pay $716.28 - $650 = $66.28 in the up state, and 0 in the down state, resulting in the price of the European call being

pUP ⋅66.28 ⋅e−r = 0.6 ⋅66.28 ⋅ 645

660⋅e0.02 ≈ 39.65.

Answer D. Posted October 27, 2013 In the solution of Problem 19 in Practice Examination 19, the expression or 1+ s0.5( )0.5 = 0.98 and 1+ s0.5 = 0.9604. that follows the fifth displayed equation should be: or 1+ s0.5( )0.5 = 1/ 0.98 and 1+ s0.5 = 1/ 0.9604. Posted October 23, 2013 Problem 20 in Practice Examination 16 is, unfortunately, the same as Problem 31 in Practice Examination 3, whose solution is corrected just below. Posted October 23, 2013 The solution of Problem 31 in Practice Examination 3 should be: The excess of LIBOR over 8% is 2%. Participation rate applies to the case when the rate is below the cap rate, and the only payment here is the excess of 10% over 8%, i.e., 2%, which is 25% of LIBOR. Answer E. Posted October 23, 2013 Discussion of a participating cap on page 65 should be: Similar to a regular cap, a participating cap provides protection from floating rates rising above a specified maximum cap level, but a participating cap requires no upfront fee, and instead the buyer of a cap agrees to forgo a portion of the rate benefit when floating rates decline. For example, in a 50% participating cap at 10%, if floating rate is about 10%, the holder of a cap is paid the excess over 10%, and if the floating rate is below 10%, the holder of a cap has to actually pay 50% of the excess of 10% over the floating rate. A participating floor is structured similarly. Posted October 5, 2013 The solution to Problem 22 in Practice Examination 1 should be: Let us write x for the fraction of the portfolio invested in the ten-year bond, and y for the fraction of the portfolio invested in the twenty-year zero coupon bond. Because the current interest rate is the same as the coupon rate of the ten-year coupon bond, the Macaulay duration of that ten-year bond is

a10 10% ≈ 6.13356711. If the fraction invested in the ten-year bond is x, then since the duration of the remainder of the portfolio is 15.50, and the duration of the entire portfolio is 12, therefore

x ⋅6.13356711+ 1− x( ) ⋅15.5= 12. We solve for x and obtain:

x = 15.5−12

15.5− 6.13356711≈ 37.411417%.

The answer choice is unaffected. Posted October 3, 2013 The answer choices in Problem 17 in Practice Examination 1 should be: A. Less than $0.20 B. $0.20 or more but less than $0.45 C. $0.45 or more but less than $0.75 D. $0.75 or more but less than $0.89 E. $0.89 or more The solution of this problem should be: The non-callable bond consists of a risk-free payment of $2 in one year, and a risky payment of $102 at time 2, which at time 1 will be worth either $102 ⋅1.02998−1 ≈ $99.03 or $102 ⋅1.017192−1 ≈ $100.28. Current price of a payment of $102 in two years made with certainty is $102 ⋅1.019−1 ⋅1.021−1 ≈ $98.04. Thus, in this market, over the first year, the risk-free rate is 1.9%, while there is a risky security (that security is the payment of $102 at time 2), which is now worth $98.04 and in a year it will be worth either $99.03 or $100.20. Based on this information, the risk-neutral probabilities are: • Risk-neutral probability that in one year the one-year rate is 2.9980% equals

p*≈1.019− 99.03

98.04100.2898.04

− 99.0398.04

≈ 0.6996,

• Risk-neutral probability that in one year the one-year rate is 1.7192% equals

1− p*≈

100.2898.04

−1.019

100.2898.04

− 99.0398.04

≈ 0.3004.

At time 1, if one-year rate is 2.9980% and the market price of a payment of $102 at time 2 is $99.03, the bond will not be called as it does not make sense to pay $100 for something that is worth $99.03 in the market. On the other hand, if one-year rate is 1.7192% and the market price of $102 payment at time 2 is $100.28, it will make sense to call it and pay $100 for it. Thus the callable bond will be worth in one year (just after the payment of $2 coupon, as the coupon is paid at the end of the first year):

• $100 with probability p* ≈ 0.6996, • $99.03 with probability 1− p*≈ 0.3004.

Hence, the current price of the callable bond, also including a coupon of $2 paid with certainty at the end of the first year, is approximately

0.6996 ⋅ $1001.019

+ 0.3004 ⋅ $99.031.019

⎛⎝⎜

⎞⎠⎟+ $2

1.019≈ $99.81.

The current price of the non-callable bond is $2 ⋅1.019−1 + $102 ⋅1.019−1 ⋅1.021−1 ≈ $100.00. Since there is no credit risk, the only difference between the two bonds, the callable one and the non-callable one, is the call option, and therefore that option’s value must be $100.00 – $99.81 = $0.19. Answer A. Posted September 11, 2013 Problem 25 in Practice Examination 19, also online Exercise No. 216 was, unfortunately, the same as Problem 13 in Practice Examination 18. Problem 25 in Practice Examination 19, which is also online exercise No. 248 will be now replaced by this one: November 1990 Course 140 Examination, Problem No. 11, and Dr. Ostaszewski’s online exercise 248 posted February 13, 2010 A bond with a par value of 1000 and 6% semiannual coupons is redeemable for 1100. You are given: (i) The bond is purchased at P to yield 8%, convertible semiannually; and (ii) The amount of principal adjustment for the 16-th semiannual period is 5. Calculate P. A. 720 B. 770 C. 790 D. 800 E. 820 Solution. The expression “6% semiannual coupons” refers to 6% as the annual rate with semiannual coupons, so that the coupon amount is 3% of par value of 1000, i.e., Fr = 30. We also have C = 1100, i = 4% per half a year, and n, the number of half-year periods until bond maturity, is unknown. Furthermore, the modified coupon rate is

g = FrC

= 301100

= 3110

≈ 2.727273%.

The price of the bond is

P = C 1+ g − i( )an i( ) = 1100 1+ 3

110− 0.04⎛

⎝⎜⎞⎠⎟1−1.04−n

0.04⎛⎝⎜

⎞⎠⎟=

= 1100 + 30 − 440.04

⋅ 1−1.04−n( ) = 1100 − 350 ⋅ 1−1.04−n( ) = 650 + 350 ⋅1.04−n.

The amount of principal adjustment in the k-th coupon is C g − i( )vn− k−1( ), or, more precisely, the price of the bond is reduced by C g − i( )vn− k−1( ) after the k-th coupon payment. Since in this case we are effectively told the price of the bond increases by 5 in the 16-th payment (which makes sense, since the modified coupon rate is lower than the

market yield, so the bond was purchased at a discount and it appreciates as it approaches maturity), we have C g − i( )vn− k−1( ) = −5, or specifically

1100 30

1100− 0.04

⎛⎝⎜

⎞⎠⎟⋅1.04−n+15 = 30− 44( ) ⋅1.04−n+15 = −14 ⋅1.0415 ⋅1.04−n = −5.

This implies

1.04−n = 5

14⋅1.04−15.

Therefore,

P = 650 + 350 ⋅1.04−n = 650 + 350 ⋅ 514

⋅1.04−15 ≈ 719.41.

Answer A. Posted September 11, 2013 Modify the definition of the bond yield on page 26 in the section 5. Bonds: i = Bond yield, also called the yield to maturity, calculated as the internal rate of return on bond’s cash flows, given its market price, as of the date of purchase, it is the effective interest rate over the coupon payment period. It should not be confused with the current yield, which is unlikely to show up on the Course FM/2 examination. Current yield is defined as the ratio of the coupon Fr to the bond market price P. Posted September 10, 2013 and corrected September 11, 2013 The second to last formula in the solution of Problem 20 in Practice Examination 18 should be

100 = P ⋅a10 4.5%

+ X ⋅1.045−10 =100a

10 0.045

a10 4%

+ 2 ⋅1.04−10 ⋅a20 4%

+ X ⋅1.045−10,

instead of

100 = P + X ⋅1.04−10 =100a

10 0.045

a10 4%

+ 2 ⋅1.04−10 ⋅a20 4%

+ X ⋅1.04−10,

a10 4.5% was missing in the first part of the formula, and the interest rate for

discounting in the first formula should have been 4.5% throughout. Posted September 7, 2013 The syllabus for Course FM/2 for October 2013 is posted here: http://www.beanactuary.org/exams/preliminary/exams/syllabi/2013-10-exam-fm.pdf It differs slightly from that for August 2013. Please take the following items into account when studying for the October 2013 Course FM/2 examination: • Current value: The concept of current value is added to the syllabus. This concept is presented on page 9 of the manual, and it was already covered in the manual. • Portfolio and investment year allocation methods: Portfolio method and investment year allocation method, covered on page 11 of the manual, are no longer explicitly

listed in the syllabus. They are still covered in the manual, as the word “portfolio” is, somewhat strangely, listed among the topics still covered, and I still believe these concepts are worth learning. • Drop payment: While the concept of a balloon payment was discussed in the manual, the term “drop payment” was not, so please use this expanded definition of P at the beginning of Section 4, Loan Amortization:

P = amount of the level payment, paid at the end of each period, assumed to repay the loan in full (sometimes we consider a situation when the last payment is larger than regular payment, such a larger payment is called a balloon payment, and sometimes the last payment is smaller than the regular payment, such a smaller payment is called a drop payment),

• Refinancing: Add this statement just before discussion of the concept of a sinking fund, on page 22: When an outstanding loan is paid off with money obtained from another loan, such process is called refinancing. Calculations related to refinancing are always based on equating the balance of the old loan with the amount of the new loan. • Mandatory convertible bonds – on page 26 of the manual, at the end of the first paragraph, after the definition of a convertible bond, add this: A convertible bond that actually must be converted, either on a specified date, or by that date, is called a mandatory convertible. Mandatory convertible bonds have higher yields to compensate investors for the mandatory conversion. This type of bonds are issued because the issuer prefers to issue bonds instead of stocks, but wants to eventually issue shares (note that stock market generally views issuing new shares as a negative signal, while issuing bonds is perceived as a positive one). • Duration: The new syllabus does not refer to Macaulay duration, modified duration, and effective duration, it only talks about Macaulay duration and modified duration, and the concept of modified duration is used interchangeably with the concept of duration. This is, of course, the standard terminology in finance, already used in the manual. • Ask price, bid price, bid-ask spread: These concepts are removed from the syllabus, but the manual still has a discussion of them, as they provide good background information about functioning of financial markets. • Mark to Market: This concept is now explicitly stated in the syllabus, but of course it was always in the background, as it is vital in the understanding of futures contracts. To further explain it, add this statement just before section entitled “Hedging spot prices with futures” on page 60: We see that one key feature of futures contracts is that the value of the contract is adjusted daily to the market price. Doing this is termed: mark-to-market. In general, mark-to-market refers to the accounting act of recording the price or value of a security, portfolio or account to reflect its current market value rather than some theoretical value, e.g., book value. • No-Arbitrage: This concept is now explicitly stated in the syllabus. Arbitrage is discussed in the manual in several places, starting on page 47. • Risk averse: This concept is added in the syllabus. The concept of risk-neutrality is discussed in the manual. Let us also add this statement at the end of Section 13:

The probability so obtained is called risk-neutral because we can use it to calculate the value of a security (stock, or option, etc.) as the expected present value of its future cash flows, with risk-free interest rate used for discounting, under risk-neutral probability. This means that the value of risky future payments equals to their expected present value. A person who values risky payments equally to their expected value is said to be risk-neutral. For example, if you play a game where you win $100 with probability 0.5, and $0 with probability 0.5, would you pay $50, the expected winnings, to play it? If so, you are risk-neutral. If, on the other hand, you would pay less, you value $50 paid with certainty more than expected payment of $50, and you are risk-averse. If you are willing to pay more than $50 to play this game, you are a risk-lover. • Option Writer: This concept is now explicitly added in the syllabus. It has always been covered in the manual. • Implied repo rate: This concept is now deleted from the syllabus. It is still discussed in the manual, to provide background information. • Synthetic forward: This concept is now deleted from the syllabus. • Paylater Strategy: This concept is now deleted from the syllabus. • Option Spread: This concept is now explicitly added in the syllabus. It has always been covered in the manual. • Vertical Spread: This concept is now explicitly added in the syllabus. Add the following sentence in Section 12 of the manual: A vertical spread is a simultaneous long position and a short position in two options of the same type that have the same expiration dates but different strike prices. • Collar Width: This concept is now explicitly added in the syllabus. It has always been covered in the manual. • Collared Stock: This concept is now explicitly added in the syllabus. Add this statement to the definition of a collar: If a collar is purchased in addition to holding a long position in the underlying stock, such portfolio is called a collared stock. Posted September 6, 2013 On page 10, the sentence Instead, the time-weighted rate of return, which is the return earned by a unit of a fund under consideration. should be Instead, they use the time-weighted rate of return, which is the return earned by a unit of a fund under consideration. Posted August 1, 2013 In the solution of Problem 3 in Practice Examination 4, the calculation of Y uses the interest rate of 3%, instead of 6%, as mistyped. The actual number calculated is correct.

Posted July 23, 2013 In Problem 24 in Practice Examination 2, the answer choices should be: A. 1440 B. 1480 C. 1500 D. 1515 E. 1555 and the solution should be: Solution. We have the general formula BVk 1+ i( ) = Fr + BVk+1. In this case, for k = 2 and k + 1 = 3, we obtain 1479.65 1+ i( ) = 120 + BV3, and for k = 2 and k + 1 = 3, BV3 1+ i( ) = 120 +1439.57. From the first of the above two equations, BV3 = 1479.65 1+ i( )−120, and substituting this into the second of the above two equations, we obtain 1479.65 1+ i( )−120( ) 1+ i( ) = 120 +1439.57, so that 1479.65 1+ i( )−120( ) 1+ i( ) = 120 +1439.57, or 1479.65 1+ i( )2 −120 1+ i( )−1559.57 = 0, resulting in

1479.65 1+ i( )2 −120 1+ i( )−1559.57,

1+ i = 120 ± 1202 + 4 ⋅1479.65 ⋅1559.572 ⋅1479.65

≈ 1.06800187,−0.9869016.

⎧⎨⎪

⎩⎪

Therefore, i ≈ 6.800187%. The purchase price of the bond is its book value at time 0, which is related to the book value at time 2 through two recursive equations BV0 1+ i( ) = Fr + BV1, BV1 1+ i( ) = Fr + BV2. Substituting all quantities known, we obtain BV0 ⋅1.06800187 = 120 + BV1, BV1 ⋅1.06800187 = 120 +1479.65. Based on this,

BV1 =120 +1479.651.06800187

≈1497.797,

and

BV0 =120 + BV11.06800187

= 120 +1497.7971.06800187

≈1514.79.

Answer D. Posted June 11, 2013 The first sentence of the solution of Problem 25 in Practice Examination 17 should be

The sinking fund payment is 1000s10 0.04

.

instead of

The sinking fund payment is 1000s10 0.04

⋅ s4 0.04

.

Posted June 10, 2013 The end part of the solution of Problem 12 in Practice Examination 18 should be: But the proposed price is $1600 per ounce, or $1,600,000 for the contract. This means that the investor is getting a price at contract maturity that is $32,664.69 below the fair market price, and to make up for this the investor must pay that amount in six months, or equivalent amount

$32,664.691+ 0.10%

≈ $32,632.06

at inception. Answer E. Posted May 20, 2013 The second to last formula on page 7 should be:

C τ( ) = C 0( )eδ s ds

0

τ

∫+ γ t( )e

δ s dst

τ

∫

0

τ

∫ dt = C 0( ) ⋅a τ( ) + γ t( ) ⋅ a τ( )a t( )0

τ

∫ ⋅dt.

instead of

C τ( ) = C 0( )eδ s ds

0

τ

∫+ γ t( )e

δ s dst

τ

∫

0

τ

∫ dt = C 0( ) ⋅a t( ) + γ t( ) ⋅ a τ( )a t( )0

τ

∫ ⋅dt.

The accumulated value of the initial deposit C 0( ) ⋅a τ( ) was mistyped as

C 0( ) ⋅a t( ).

Posted December 27, 2012 The solution of Problem 28 in Practice Examination 8 should be modified slightly to become: Five contracts for 5,000 bushels total 25,000 bushels. The current futures price is $2.40 per bushel, for a total futures price for 25,000 bushels for the entire position of 25,000 ⋅$2.40 = $60,000. The investor posts initial margin of five times $2,000, i.e., $10,000. Following that, maintenance margin is five times $1,500, i.e., $7.500. The investor will be able to withdraw exactly $2,000 from the margin account, when the margin account balance exceeds required maintenance margin by exactly $2,000, i.e., when margin account balance is $9,500. The first time such withdrawal can happen is at the end of the first day of trading, and the margin account will have a balance of $9,500 exactly if a loss of $500 occurs during that day (causing the margin account balance to go from $10,000 to $9,500). For that loss to be produced, futures price must decrease by

$50025,000

= $0.02.

Thus the futures price must be $2.38. Note: The solution published by the CAS assumes that $2,000 withdrawn must be a gain, but the problem does not say it must be a gain. Answer A. Posted December 27, 2012 The solution of Problem 29 in Practice Examination 8 should be simplified to say: If we replicate the pure platinum transaction (borrow 1 ounce of platinum, repay 1.025 ounces of platinum) with market transaction involving money and platinum (buy 1 ounce of platinum with borrowed money now, sell 1.025 ounces of platinum in a year, repay the cash loan), the net result will give us the answer we are seeking. Note that the original platinum transaction and its replication involve paying the same storage cost for platinum, so that the direct storage cost will not matter the calculation (but its indirect effect in forward pricing may). Recall that storage cost is treated as a negative dividend in forward pricing. Let us write S0 for the current spot price of one ounce of platinum. The platinum loan has the net effect of the corporate client spending S0 to buy an ounce of platinum today, and then committing to sell 1.025 ounces in a year on the forward. We know that the current forward price is S0e

0.07+0.004 . The effective annual return on money on this transaction (earned by the bank, from which platinum is borrowed, and paid by the corporate client, if borrowing in platinum) is

1.025S0 ⋅e0.07+0.004

S0−1≈10.372698%.

But the corporate client pays 10% to borrow money, thus we see that the platinum loan results in money cost of approximately 10.372698%. The platinum loan rate is too high. By borrowing money at 10%, buying an ounce of platinum now and selling 1.025 ounces of platinum on the forward, the corporate client could earn, per ounce, approximately 1.025S0 ⋅e

0.07+0.004 −1.1S0 ≈ 0.00372698S0, or, as a fraction of the price of an ounce of platinum 0.372698%. We conclude that the platinum loan rate is approximately 37 basis points too high. Answer A.