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Exam FM/2

Exam FM/2. PSU Study Session Fall 2010 Dan Sprik. Outline. 3 formulas for pricing bonds Amortization of Premium / Accumulation of Discount Dealing with callable bonds 7 SoA practice problems (or until we run out of time). Bonds. n = number of periods (term of bond)

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Exam FM/2

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  1. Exam FM/2 PSU Study Session Fall 2010 Dan Sprik

  2. Outline • 3 formulas for pricing bonds • Amortization of Premium / Accumulation of Discount • Dealing with callable bonds • 7 SoA practice problems (or until we run out of time)

  3. Bonds • n = number of periods(term of bond) • j= interest rate • r = coupon rate • F = face value (“par value”) • C = redemption value • C and F assumed equal, unless otherwise stated. When C = F the bond is “redeemed at par” • g = Fr / C • = PV of redemption value

  4. Price Formulas • Basic Formula (“Frank Formula”) • Premium/ Discount Formula P = C + (Fr – Cj) an┐j% OR P = C + C(g – j) an┐j% • Makeham’s Formula

  5. Price in relation to interest rate • Bond prices move in the opposite direction of interest rates • This makes sense: a higher interest rate means a larger present value factor (v) which means that the payments a bond promises to pay in the future are worth less at time 0.

  6. Bonds sold at Premium/Discount • A bond promises to pay the redemption value (C ) at time n in the future. In return for this (and periodic coupon payments) an investor agrees to pay P. • The difference between price and redemption value is P-C. • If P > C: the bond is sold at a premium of P – C • If P < C: the bond is sold at a discount of C - P • P = C + C(g – j) an┐j%P – C = C(g – j) an┐j%so sold at premium if g > jsold at discount if g < j

  7. Amortization of Premium • The term “book value” is used instead of “outstanding balance.” It means the same thing, just in the context of bonds. • Book value iswritten down over time (in the case of a premium) or accumulates over time (in the case of a discount) • Interest earned = • Amount for amortizing premium = • Amortization reduces (or increases) P to C over bond period • BVt= BVt-1- PRt

  8. Price Between Coupon Dates • Theoretical (“dirty price”): BVt+k= BVt(1 + i)k , • Financial Press (“clean price”): BVt+k= BVt(1 + i)k– kFr,

  9. Callable Bonds • Bonds with a call provision allow the corporation that issued the bond to end the bond contract by paying the redemption value at a specified future date before maturity • Premium bonds – coupon payments are “good;” bond buyer would like more coupons if possible, value bond based on earliest redemption date (worst case). • Discount bonds – coupon payments are “bad;” assume call date is latest redemption date (worst case). • If you forget this rule, think about the extreme case of a bond sold at premium or discount that is called immediately. In which situation does the buyer benefit? • For minimum yield, determine lowest price at all redemption dates at this rate.

  10. SOA November 2001 • You have decided to invest in two bonds. Bond X is an n-year bond with semiannual coupons, while bond Y is an accumulation bond redeemable in n/2 years. The desired rate is the same for both bonds. You also have: • Bond X: • Par value is 1000. The ratio of the semi-annual bond rate and the semiannual yield rate, r/i, is 1.03125. The present value of the redemption value is 381.50. • Bond Y • Redemption value is the same as the redemption value of Bond X. The price to yield an effective rate i per half year is 647.80. • What is the price of Bond X? • A) 1019 B) 1029 C) 1050 D) 1055 E) 1072

  11. SOA November 1993 • An n-year 1000 par value bond with 4.2% annual coupons is purchased at a price to yield an annual effective rate of i. You are given: • If annual coupon rate had been 5.25% instead of 4.2%, the price of the bond would have increased by 100. • At the time of purchase, the present value of all the coupon payments is equal to the present value of the bonds redemption value of 1000. • Calculate i. • A) 5% B) 5.5% C) 5.9% D) 6.3% E) 6.5%

  12. SOA May 2003 • A 10,000 par value 10-year bond with 8% annual coupons is bought at a premium to yield an annual effective rate of 6%. Calculate the interest portion of the 7th coupon. • A) 632 B) 642 C) 651 D) 660 E) 667

  13. SOA November 1987 • A 30-year 10,000 bond that pays 3% annual coupons matures at par. It is purchased to yield 5% for the first 15 years and 4% thereafter. Calculate the amount for accumulation of discount for year 8. • A) 78 B) 83 C) 88 D) 93 E) 98

  14. SOA Sample 1983 • A 1000 20-year 8% bond with semiannual coupons is purchased for 1,014. The redemption value is 1000. The coupons are reinvested at a nominal annual rate of 6%, compounded semiannually. Determine the purchaser’s annual effective yield rate over the 20 year period. • A) 6.9% B) 7% C) 7.1% D) 7.2% E) 7.3%

  15. SOA May 1995 • A 1000 par value 5-year bond with semiannual coupons of 60 is purchased to yield 8% convertible semiannually. Two years and two months after purchase, the bond is sold at a price which maintains the same yield for the buyer. Calculate this price. • A) 1089 B) 1099 C) 1105 D) 1113 E) 1119

  16. SOA May 1993 • Matt purchased a 20-year par value bond 8% semiannual coupons at a price of 1,722.25. The bond can be called at par value X on any coupon date starting at the end of year 15. The price guarantees that Matt will receive a nominal semiannual yield of at least 6%. Bert purchases a 20-year par value bond identical to the one purchased by Matt except it is not callable. Assuming a nominal semiannual yield of 6%, the cost of the bond purchased by Bert is P. Calculate P. • A) 1,700 B) 1,725 C) 1,750 D) 1,775 E) 1,800

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