Exam FM Review Part 1

1. Exam FM Review Part 1

2. Introduction Jeffrey Baer 3B Actuarial Science Work terms at Manulife and Towers Watson Waterloo SOS President, May 2009 – Aug 2010 Stat 231 and Act Sci 231 Tutor this term!

3. Outline 1. Time Value of Money 2. Interest/Discount Rates and Force of Interest 3. Level Annuities, Due and Immediate 4. Increasing Annuities, Arithmetic and Geometric 5. Yield Rates 6. Loan Amortization and Sinking Funds 7. Bonds

4. Time Value of Money Accumulation Functions a(t) = (1 + i)t under compound interest a(t) = (1 + it) under simple interest Future Value (FV/AV) = a(t) * Present Value (PV), regardless of interest type Discount Functions v(t) = 1/a(t) v = v(1) = 1/(1 + i) under compound interest Comparing PV or FV of two cash flows is equivalent! A(t) = A0*(1+i)tA(t) = A0*(1+i)t

5. Example 1 Jeffrey puts $100 into Fund X and $100 into Fund Y. Y earns compound interest at the annual rate of j > 0, and X earns simple interest at the annual rate of 1.05j. At the end of 2 years, the amount in Fund Y equals the amount in Fund X. Calculate the amount in Fund Y at the end of 5 years. j=10% 100*1.1^5 = 161.05 j=10% 100*1.1^5 = 161.05

6. Example 2 Barack Obama borrows $1000 from his wife Michelle at an annual effective interest rate i. He agrees to pay back $1000 after six years and $1366.87 after another six years. Three years after his first payment, Barack repays the outstanding balance. What is the amount of Barack’s second payment? y = 1.7716 (reject the negative answer) i = 10% 1026.95y = 1.7716 (reject the negative answer) i = 10% 1026.95

7. i, d, and d Interest Rates Effective: it = [a(t) – a(t-1)]/a(t-1) Nominal (convertible/compounded mthly): 1 + i = (1 + i(m)/m)m We never do calculations directly with nominal rates Discount Rates Effective: dt = [a(t) – a(t-1)]/a(t) d = 1 – v = i/(1+i) i = d/(1-d) Nominal: 1- d = (1 – d(m)/m)m (1 + i)t = v-t = (1 – d)-t 1/d(m) – 1/i(m) = 1/m Effective calculation also works with A(t), A(t-1)Effective calculation also works with A(t), A(t-1)

8. i, d, and d Will d be constant under simple interest? No: = i/(1+it) Will d be constant under simple interest? No: = i/(1+it)

9. Example 3 The force of interest is dt = 0.02t, where t is the number of years from January 1, 2001. If $1 is invested on January 1, 2003, how much is in the fund on January 1, 2008? Show both ways e^0.45 = 1.57Show both ways e^0.45 = 1.57

10. Example 4 The accumulated value of $1 at time t (0<=t<=1) is given by a second degree polynomial in t. You are given that the nominal rate of interest convertible semi-annually for the first half of the year is 5% per annum, and the effective rate of interest for the year is 4% per annum. Calculate d3/4. Three equations: a(0), a(0.5), a(1) C = 1, A = -0.02, B = 0.06 0.03/1.03375 = 2.90%Three equations: a(0), a(0.5), a(1) C = 1, A = -0.02, B = 0.06 0.03/1.03375 = 2.90%

11. Level Annuities Level means payments are constant To get PV/FV of entire annuity, look at PV/FV of each individual cash flow and sum (will form geometric series)Level means payments are constant To get PV/FV of entire annuity, look at PV/FV of each individual cash flow and sum (will form geometric series)

12. Level Annuities Convert interest rates if necessary to correspond to frequency of annuity payments (i.e. monthly effective if payments made monthly)—there are sections in the manuals about off-payments—instead of memorizing fancy formulas, just convert to effective interestConvert interest rates if necessary to correspond to frequency of annuity payments (i.e. monthly effective if payments made monthly)—there are sections in the manuals about off-payments—instead of memorizing fancy formulas, just convert to effective interest

13. Level Annuities Illustrate second relationship with timeline s5¯| Questions about financial calculator: ask me after the sessionIllustrate second relationship with timeline s5¯| Questions about financial calculator: ask me after the session

14. Level Annuities For deferred annuities, we have the choice between treating it as an annuity due or an annuity immediate (neither may be intuitive)—i.e. annuity with payments starting at time 3 Can illustrate block with timelineFor deferred annuities, we have the choice between treating it as an annuity due or an annuity immediate (neither may be intuitive)—i.e. annuity with payments starting at time 3 Can illustrate block with timeline

15. Level Annuities Deferred perpetuity has the same cash flows as a normal perpetuity: PV of perpetuity at year 0 is still the PV of the perpetuity at year n—same stream of payments in the futureDeferred perpetuity has the same cash flows as a normal perpetuity: PV of perpetuity at year 0 is still the PV of the perpetuity at year n—same stream of payments in the future

16. Example 5 At time t =0, Bob deposits P into a fund crediting interest at an annual effective rate of 8%. At the end of each year in years 6 through 20, Bob withdraws an amount sufficient to purchase an annuity due of 100 per month for 10 years at a nominal interest rate of 12% compounded monthly. Immediately after the withdrawal at the end of year 20, the fund value is zero. Calculate P. Compare Present Value: i = 12.68%, X = 7,039.75 P = Xv^5(a angle 15 at 8%) = $41,009.64Compare Present Value: i = 12.68%, X = 7,039.75 P = Xv^5(a angle 15 at 8%) = $41,009.64

17. Example 6 An annuity due has the following present value and accumulated value: än+2¯| = 13.987 n ¯| = 51.632 What is the PV of a perpetuity with annual payments of $5 per year, starting at the end of year 2, at the same interest rate? a angle n+1 = 13.987 – 1 = 12.987 s angle n+1 = 51.632 + 1 = 52.632 Then (1+i)^(n+1) = 52.632/12.987 = 4.053 -> plug into s double dot angle n+1 to arrive at i = 5.80% Therefore PV = (5/0.058)/1.058 = 81.48a angle n+1 = 13.987 – 1 = 12.987 s angle n+1 = 51.632 + 1 = 52.632 Then (1+i)^(n+1) = 52.632/12.987 = 4.053 -> plug into s double dot angle n+1 to arrive at i = 5.80% Therefore PV = (5/0.058)/1.058 = 81.48

18. Level Annuities Remember: interest rate compounded continuously = deltaRemember: interest rate compounded continuously = delta

19. Example 7 Given dt = 2/(10 + t), t>=0, calculate a4¯| . Pmts themselves are discrete! a(t) = [(10+t)/10]^2 PV = v(1) + v(2) + v(3) + v(4) = 2.62Pmts themselves are discrete! a(t) = [(10+t)/10]^2 PV = v(1) + v(2) + v(3) + v(4) = 2.62

20. Example 8 For a fixed force of interest d, you are given: a10¯| = 7.52 d/dd (a10¯| ) = -33.865. Calculate d. Keep in mind that v^10 contains delta! Manipulate to get delta = 6%Keep in mind that v^10 contains delta! Manipulate to get delta = 6%

21. Example 9 Given the accumulation function a(t) = 1 + 0.04t2 for t>=0, calculate the accumulated value at time 5 of deposits of $1 at times 1, 2, 3, 4, and 5. PV, then AV to get 7.34PV, then AV to get 7.34

22. Increasing Annuities Only changes between immediate and due: beginning part and i to d Decreasing annuity: Q is –veOnly changes between immediate and due: beginning part and i to d Decreasing annuity: Q is –ve

23. Example 10 Perpetuity X has annual payments of $1, 2, 3, … at the end of each year. Perpetuity Y has annual payments of $q, q, 2q, 2q, 3q, 3q, … at the end of each year. The present value of X = present value of Y at i = 10%. Calculate q. PV X = 1/(0.1^2/1.1) = 110 PV Y = q(1.1 + 1)/(0.21^2/1.21) = 2.1q*27.43 = 57.62q Therefore q = 110/57.62 = 1.91PV X = 1/(0.1^2/1.1) = 110 PV Y = q(1.1 + 1)/(0.21^2/1.21) = 2.1q*27.43 = 57.62q Therefore q = 110/57.62 = 1.91

24. Increasing Annuities Again, to get AV in the continuous case, multiply by a(n) or (1+i)nAgain, to get AV in the continuous case, multiply by a(n) or (1+i)n

25. Example 11 A perpetuity immediate pays $100 per year. Immediately after the fifth payment, the perpetuity is exchanged for a 25-year annuity immediate paying X at the end of the first year, with each subsequent payment 8% greater than the preceding payment. Immediately after the 10th payment of the 25 year annuity, the annuity is exchanged for a perpetuity immediate paying Y per year, growing at 3% annually. Calculate Y assuming an interest rate of 8%. 100/0.08 = Xvn -> X = 54 Value of remaining annuity = 54*1.08^10*(15v) = 1619.19 1619.19= Yv*(a double dot infinity at j), j = 0.05/1.03 = 4.85% Y = 80.96100/0.08 = Xvn -> X = 54 Value of remaining annuity = 54*1.08^10*(15v) = 1619.19 1619.19= Yv*(a double dot infinity at j), j = 0.05/1.03 = 4.85% Y = 80.96

26. Example 12 Payments are made to an account at a continuous rate of (8k + tk), where 0<=t<=10. Interest is credited at a force of interest dt = 1/(8+t). After 10 years, the account is worth 20,000. Calculate k. v(t) = 8/(8+t) PV = integral 0 to 10 [h(t)v(t)dt] = 80k 80k * 18/8 = 20,000 -> k = 111.11v(t) = 8/(8+t) PV = integral 0 to 10 [h(t)v(t)dt] = 80k 80k * 18/8 = 20,000 -> k = 111.11

27. Yield Rates Take advantage of multiple choice: (i.e. (a angle 24 – a angle 4)/(a angle 4)) Same rules apply for reinvestment Yield Rates 1Take advantage of multiple choice: (i.e. (a angle 24 – a angle 4)/(a angle 4)) Same rules apply for reinvestment Yield Rates 1

28. Example 13 Ritika invests $100 at the beginning of each year for 10 years at 10% annual effective interest. The interest is reinvested at 5%. What is the total AV at the end of 10 years? AV = 1000 + 10*(s double dot angle 10 – 10 )/i = 1641.36AV = 1000 + 10*(s double dot angle 10 – 10 )/i = 1641.36

29. Yield Rates I: What would happen if nothing had been deposited or withdrawn? Assuming over one year Yield Rates 2I: What would happen if nothing had been deposited or withdrawn? Assuming over one year Yield Rates 2

30. Example 14 On January 1, an investment account is worth $100. On May 1, the value has increased to $120 and $D is deposited. On November 1, the value is $100 and $40 is withdrawn. On January 1 of the following year, the account is worth $65. The time weighted rate of interest is 0%. Calculated the dollar weighted rate of interest. D = $10 I = 65 – 100 – 10 + 40 = -5 (straight sum of cashflows, no weighting) DWRR = I/(100 + 20/3 - 40/6) = -5% D = $10 I = 65 – 100 – 10 + 40 = -5 (straight sum of cashflows, no weighting) DWRR = I/(100 + 20/3 - 40/6) = -5%

31. Loan Amortization and SFs Almost always annuity immediate (payment 1 month (i.e. mortgage)/1 year after t0) Prospective does not require knowledge of L Pts form a geometric progressionAlmost always annuity immediate (payment 1 month (i.e. mortgage)/1 year after t0) Prospective does not require knowledge of L Pts form a geometric progression

32. Example 15 A loan is amortized by means of level monthly payments at an annual effective interest rate of 8%. The amount of principal repaid in the 12th payment is $1000, and the amount of principal repaid in the tth payment is $3700. Calculate t. Convert interest rate to effective monthly v = 0.994 X*v^(n-11) = 1000; X*v^(n-t+1) = 3700 -> 3.7 = v^(12-t) -> t = 216Convert interest rate to effective monthly v = 0.994 X*v^(n-11) = 1000; X*v^(n-t+1) = 3700 -> 3.7 = v^(12-t) -> t = 216

33. Example 16 Jeffrey purchases a $100,000 home. Mortgage payments are to be made monthly for 30 years, with the first payment 1 month from now. The annual effective rate of interest is 5%. After 10 years, the amount of each monthly payment is increased by $325.40 to repay the mortgage more quickly. Calculate the amount of interest paid over the duration of the loan. X = 530.06 B 120 = 81,068.47 With increase, will only take another 120 payments to pay off the loan Total payments made = 530.06*120 + (530.06 + 325.40)*120 = 166,262.40 -> interest = 66,262.40X = 530.06 B 120 = 81,068.47 With increase, will only take another 120 payments to pay off the loan Total payments made = 530.06*120 + (530.06 + 325.40)*120 = 166,262.40 -> interest = 66,262.40

34. Loan Amortization and SFs

35. Loan Amortization and SFs Sinking Funds (continued): OLB for SF method = L – SF Balance Pt = SF balancet – SF balancet-1 = interest earned on SF in periodt + deposit to SF at time t Interest payments are being made, but interest is also being accumulated in the sinking fund It = Net interest paid at time t = i(L) – j(st-1¯|j ) For PV of payments under the amortization and SF methods to be equivalent: If interest rate for interest pmts = i, interest rate for SF = j: 1/ an¯|i* = i + 1/ sn¯| j OLB is purely theoretical, since funds are separate If asked, what is the equivalent interest rate under amortization method?…OLB is purely theoretical, since funds are separate If asked, what is the equivalent interest rate under amortization method?…

36. Loan Amortization and SFs Interest Payment = Net Interest Paid Since i=j=8%, Net/Outstanding Loan Balance under both is the sameInterest Payment = Net Interest Paid Since i=j=8%, Net/Outstanding Loan Balance under both is the same

37. Example 17 The Act Sci Club borrows $25,000 at an effective annual interest rate of 12%. It has the following options for repayment: i) Annual amortization method, with payments made at year end for 10 years ii) Paying annual interest at year end and building up a sinking fund (earning 7%/year) by making level annual payments at year end, to pay off the loan at the end of 10 years. Determine the absolute value of the difference between the total annual payment under Option i) and Option ii). Amortization: payment of $4424.60 Sinking Fund: payment of (3000 + 1809.43) = 4809.43 -> difference of 384.83Amortization: payment of $4424.60 Sinking Fund: payment of (3000 + 1809.43) = 4809.43 -> difference of 384.83

38. Bonds Bond Terminology Coupon (Fr): generally fixed sum of money paid regularly to the bondholder Number of Periods (n): number of coupons paid Face/Par Value (F): used to calculate coupon payments Coupon rate (r): % of F given as a coupon each period Nominal annual coupon rates are generally provided, so we must convert to the effective coupon rate per period Coupon=Fr Yield rate (i): effective rate/period at which CFs are discounted Nominal annual yield rates are generally provided—again, convert Redemption Value (C): future value of the bond at expiry Unless otherwise stated, C = F (bond matures/redeemable at par) i.e. a $1000 bond with 8% semiannual coupons maturing in 10 years at par Most bonds have semiannual coupons $1000 bond means F = $1000, not C; 8% semi-annual coupons means 4% coupons per periodMost bonds have semiannual coupons $1000 bond means F = $1000, not C; 8% semi-annual coupons means 4% coupons per period

39. Bonds Bond Pricing at Issue Price = Coupon* an¯|i + PV(Redemption Value) = (Fr) an¯|i + Cvin, where vi is determined using effective yield rate per period Premiums and Discounts If Price of bond > Redemption Value: Premium Define g as Fr/C: then if g > i, Premium = Price – C = (Cg-Ci) an¯|i If Price of bond < Redemption Value: Discount Define g as Fr/C: then if g < i, Discount = C – Price = (Ci – Cg) an¯|i If g = i and F = C, Price of bond = Redemption Value: Par P and D are oppositesP and D are opposites

40. Bonds Book Value (directly after coupon payment) Same concept as Outstanding Loan Balance (Bt) Bt = Fr an-t¯|i + Cvin-t B0 = Price Bn = C Amount of Amortization of Premium/Discount Same concept as principal repaid (Pt), but demonstrates the reduction (premium) or increase (discount) in Book Value over time Pt = (Cg – Ci)vn-t+1 Dt = (Ci-Cg)vn-t+1 Bt = Bt-1 – Pt Bt = Bt-1 + Pt It = Fr – Pt = iBt-1 It = Fr + Pt = iBt-1 Amounts for amortization of premium or discount follow a geometric progression (divide by v to get from Pt to Pt+1) Pt reduces Book Value down to Redemption Value, since interest earned < amount of coupon, so part of coupon goes towards reducing principal owed and part towards payment of interest—vice versa for discountPt reduces Book Value down to Redemption Value, since interest earned < amount of coupon, so part of coupon goes towards reducing principal owed and part towards payment of interest—vice versa for discount

41. Example 18 Jeffrey purchases a $1000 par value, 10 year, 5% bond with semi-annual coupons. Jeffrey pays a price of P1. After receiving his coupon payment at the end of six years, Jeffrey sells the bond to Aaron at a price of P2. Aaron retains the bond to maturity. The yield rate on Jeffrey’s investment is 7% convertible semi-annually, and the yield rate on Aaron’s investment is 6% convertible semi-annually. What is P1? P2 = 25*(a angle 8 at 3%) + 1000/1.03^8 = 964.90 P1 = 25*(a angle 12 at 3.5%) + 964.90/1.035^12 = 880.14P2 = 25*(a angle 8 at 3%) + 1000/1.03^8 = 964.90 P1 = 25*(a angle 12 at 3.5%) + 964.90/1.035^12 = 880.14

42. Example 19 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. Can’t use normal formula for Pt since i,g not constant -> instead use B8 – B7 (book value is increasing since discount) B7 = 7954.82, B8 = 8052.56 -> accumulation of discount = 97.74Can’t use normal formula for Pt since i,g not constant -> instead use B8 – B7 (book value is increasing since discount) B7 = 7954.82, B8 = 8052.56 -> accumulation of discount = 97.74

43. Bonds Callable Bonds Can be redeemed by the borrower (issuer of bond) before the normal maturity date Price of a callable bond should be calculated using the worst possible redemption date for the bondholder (modify n accordingly): Earliest date for bond selling at Premium Forfeit some coupons, which are greater than interest payments Latest date (maturity) for bond selling at Discount Interest payments > coupons received Figure out if selling at Premium/Discount Then calculate price using n at appropriate dateFigure out if selling at Premium/Discount Then calculate price using n at appropriate date

44. Bonds Book Value between coupon payments Use Bt, book value of bond on coupon date prior to redemption date Dirty Price: sale price of a bond between coupon payments Dt+k = Bt(1+i)k, where k is a fraction of a period, 0<k<1 i.e. if semi-annual coupon date is July 23rd and sale is on November 3rd, k = (8+31+30+31+3)/(8+31+30+31+30+31+23) = 103/184 (Semipractical) Clean Price: quoted price of a bond between coupon payments Ct+k = Bt(1+i)k – k(Fr) = Dt+k – k(Fr)

45. Example 20 A $1000 bond redeemable at $1050 has 7.5% semi-annual coupons and matures on July 1, 2017. Find the actual selling price of this bond on November 15, 2013, and the price that would be quoted in a financial newspaper on the same date, based on a nominal annual yield of 5.80% compounded semiannually. Book Value at July 1, 2013 = $1,099.70 k = 137/184 -> Dirty Price = $1,123.36 -> Clean Price = Dirty Price – 137/184*37.5 = $1,095.44Book Value at July 1, 2013 = $1,099.70 k = 137/184 -> Dirty Price = $1,123.36 -> Clean Price = Dirty Price – 137/184*37.5 = $1,095.44