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ACTSC 231 Final Review

ACTSC 231 Final Review. Introduction. Jeffrey Baer 3B Actuarial Science Work terms at Manulife and Towers Watson Waterloo SOS President, May 2009 – Aug 2010 5 th review session of the term. Outline. 1 . Growth of money Equations of value and fund performance Annuities

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ACTSC 231 Final Review

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  1. ACTSC 231 Final Review

  2. Introduction • Jeffrey Baer • 3B Actuarial Science • Work terms at Manulife and Towers Watson • Waterloo SOS President, May 2009 – Aug 2010 • 5th review session of the term

  3. Outline 1. Growth of money • Equations of value and fund performance • Annuities 4. Loan amortization and sinking funds 5. Bonds 6. Spot/forward rates 7. Duration/price sensitivity/immunization

  4. Growth of Money • Accumulation Functions: • a(0) = 1 and a(t) ≥ a(0) for all t ≥ 0 • Future Value (FV/AV at time t) = a(t) * Present Value (PV) • AK(t) is the FV at time t of $K investment made at time 0 = K * a(t) • a(t) means that our accumulation starts at time 0! • Only works for money invested at (or discounted to) t0 • i.e. money invested at t1 cannot be accumulated to t5 using a(4) Example: Let a(t) = t2 + 1, t>=0. What is the accumulated value at time 2 of deposits of $1 at time 0 and $2 at time 1?

  5. Compound interest • a(t) = (1 + i)t , t ≥ 0 • Pays interest on balance earned so far • a(t)*a(s) = a(t + s) Example: Edward invests $100 for 2n years at 8% compound interest per year, and then reinvests the proceeds for another n years at x% compound interest. Jacob invests $100 for 2n years at x% compound interest per year, and then reinvests the proceeds for another n years at 10% compound interest. Calculate x (x>0) if Edward and Jacob have the same amount of money after 3n years.

  6. Effective Rate of Interest • The amount of interest payable over a period as a proportion of the balance at the beginning of the period • i[n-1, n] = in = • Compound: i

  7. Effective Rate of Interest Lian deposits $4,300 into an account on March 1, 1998. The bank guarantees that the annual effective rate for a balance under $5,000 is 3.5% and for a balance over $5,000 is 5%. Suppose that there are no other deposits or withdrawals except for a withdrawal of $1,000 on March 1, 2003 and a deposit of $500 on March 1, 2004. Find his account balance on March 1, 2006.

  8. Discount Rate • Effective rate of discount: d = • Compound discount: a(t) = (1 – d)-t • (1+i)t = (1-d)-t • d= i/(1 + i) i = d/(1 – d)

  9. Present Value • Discount Functions: • v(t) = 1/a(t) • PV = FV * v(t) • Compound: v(t)= (1+i)-t = • v = v(1) = 1/(1 + i) = 1 – d We can use either discount functions or accumulation functions to get PVs or AVs!

  10. Nominal Rates • Not effective interest rates • Cannot be used directly for PV/AV calculations! • Convertible/compounded mthly: must divide the nominal rate by m to get an effective mthly rate: • (1 + i) = (1 + i(m)/m)m • (1 – d) = (1 – d(m)/m)m

  11. Force of Interest • Force of Interest • δt = or • a(t) = e ; v(t) = e- • Used for continuous compounding • Compound interest: δ = ln(1+i) => constant • a(t) = eδt ; v(t) = e-δt • Example: • If the monthly nominal discount rate d(12)is 5%, calculate δ, d(4) and i(1/2).

  12. Interest Rates Example 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 δ3/4.

  13. Inflation • Measures purchasing power of currency • Rate of inflation r[t1,t2] = , where Q(t) = value of index (i.e. CPI) at time t • Real rate of return j = • Effects of inflation have been removed (“inflation-adjusted” return on an investment)

  14. Equations of Value • PV of cash inflow = PV of cash outflow Example: A single payment of $800 is made to replace 3 payments: $100 in 2 years, $200 in 3 years, and $500 in 8 years. When should the payment of $800 be made, given the annual effective rate of 5%?

  15. Reinvestment rates • Often times, two or more interest rates involved in a transaction • Consider all cash flows to get total yield on investment Example: Chengguo makes a one time investment of $200 in an account earning an annual effective rate of 5%. If the annual interest payments from the account are reinvested in another account earning 10%, what is Chengguo’s annual yield at the end of 5 years?

  16. Fund Performance • Exact (usually involves using quadratic formula) : • Simple interest approximation (DWRR): • Basic midpoint approximation: • Time weighted rate of return (from s to t):

  17. Yield Rates Example Irene’s investment account has a balance of $1000 at the beginning of the year. On Feb. 1, the account had a balance of $1050 and Irene deposited an additional $200. On Jul. 1, the balance was $1400 and Irene deposited an additional $X. On Aug. 1, the balance was $1800 and Irene withdrew $700. At the end of the year, Irene had $900 in her investment account. If Irene’s dollar-weighted rate of return was 0.096, find her time-weighted rate of return.

  18. Level Annuities • An annuity is a regular series of payments • Annuity immediate: payments are made at the end of the year • Annuity due: payments are made at the beginning of the year • Three components: amount of payment, interest/yield rate (i), length (n)

  19. Level Annuities • PV of $S/year for n years at ann. eff. interest rate i: • Immediate: PV = S*an¯| = Sv + Sv2 + Sv3 + … + Svn = Sv(1 + v + v2 + … + vn-1) = Sv(1-vn)/(1-v) Since v/(1-v) = i : = S(1-vn)/i • Due: PV = S*än¯|= S(1-vn)/d • AV of $S/year for n years, int rate i, at time n: • Immediate: AV = S*sn¯| = S(1+i)n-1 + S(1+i)n-2 + … + S(1+i) + S = S(1 + (1+i) + … + (1+i)n-1) = S(1 – (1+i)n)/(1 – (1+i)) = S[(1+i)n – 1]/i • Due: FV = S* n¯| = S[(1+i)n – 1]/d

  20. Level Annuities • Relationship between due and immediate: • än¯|= S(1-vn)/[i/(1+i)] = (1+i)S(1-vn)/i= (1+i)an¯|(same with s) • än¯| = 1 + an-1¯| sn¯| = 1 + n - 1¯| • Usage of the accumulation function: • If the last payment was just made, the AV should be calculated using an immediate annuity • If there are n deposits at annual intervals, with the last deposit just made, the AV = sn¯|, regardless of how the time diagram is labelled • i.e. if deposits of $1 are made at the beginning of the year for five years starting two years from now, what annuity symbol represents the AV right after the last deposit? • Use the TVM functions on your financial calculator to calculate unknown interest rates, if necessary • Make sure you reset your calculator!

  21. Deferred Annuities • Deferred Annuities: • A deferred annuity begins payments at some time t ≠ 0 or 1 • PV of deferred annuity starting at time t = vt-1(an¯|) or vt(än¯|) • Example: • At time t = 0, Batman 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, Batman 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.

  22. Other Annuities • Perpetuities: • A perpetuity is an annuity with payments lasting forever • AV of a perpetuity is infinite! • PVperp. immediate= a∞¯| = 1/iPVperp. due = ä∞¯| = 1/d • Continuous Annuities: • ān¯| = PV of $1 paid continuously throughout the year for n years • = • = (1-vn)/ δ = (i/ δ)* an¯| • an¯| < ān¯| < än¯| <=> d <d(m) < δ < i(m) < i • sbarn¯| = [(1+i)n – 1]/ δ • PV of continuous annuity at rate f(t) per year:

  23. Continuous Annuity Examples 1. Payments are made into an account continuously at a rate of 8Y +tY per year, for 0≤ t≤10. At time T = 10, the account is worth $20,000. Find Y if the account earns interest according to a force of interest δt= 1/(8 + t) at time t, for 0 ≤ t ≤ 10. 2. Given δt = 2/(10 + t), t>=0, calculate a4¯| .

  24. Increasing Annuities • Arithmetic Progression: • i.e. payment of 5 at t1 , 8 at t2 , 11 at t3 , etc. • In general, for annuity immediate with first payment P and constant increase of Q each period: • PV = P(an¯|) + Q(an¯| - nvn)/i • AV = P(sn¯| ) + Q(sn¯| - n)/i • When P = Q = k: • PV = k(Ian¯|) = k(än¯| - nvn)/i AV = (Isn¯|) = k( n¯| - n)/i • PV increasing perpetuity = k(Ia∞¯|) = k/(id) • In general, for annuity due with first payment P and constant increase of Q each period: • PV = P(än¯|) + Q(an¯| - nvn)/d • AV = P( n¯| ) + Q(sn¯| - n)/d • When P = Q = k: • PV = (Iän¯|) = k(än¯| - nvn)/d AV = (I n¯|) = k( n¯| - n)/d • PV perpetuity = k(Iän¯|) = k/d2

  25. Increasing Annuities • Geometric Progression: • Consider an annuity immediate with payments increasing by a constant factor of 1+k (i.e. 1 at t1, 2 at t2, 4 at t3, etc. if k=1) • For interest rate i, initial payment of X: • PV = X[1 – [(1+k)/(1+i)]n]/(i-k) • If i = k : PV = nXv • If i ≠ k : PV = (Xv)än ¯|j , where j = (i-k)/(1+k) • AV = PV(1+i)n PV or AV due = PV or AV immediate*(1+i)

  26. Increasing Annuities Example Example: Tiger Woods purchases an increasing annuity immediate for $80,630 that makes annual payments for 20 years to his 20 mistresses as follows: (i) For the first 10 years, the first payment is P and each subsequent payment is P more than the previous one (i.e. P, 2P. . . 10P for t = 1, 2, . . . , 10;) and (ii) For the remaining 10 years, the first payment is 10P(1.05) and each subsequent payment is 5% larger than the previous one (i.e. 10P(1.05), 10P(1.05)2, . . . ,10P(1.05)10 for t = 11, 12, . . . , 20). If the annual effective rate of interest is 7%, then determine the value of P.

  27. Different Int. and Pmt. Periods • Use: a(m)n¯| = i/i(m) * an¯| ä(m)n¯| = d/d(m) * än¯| s(m)n¯| = [(1+i)n-1]/i(m) (m)n¯| = [(1+i)n-1]/d(m) Or… just convert interest rates

  28. Different Int. and Pmt. PeriodsExample Santa takes out a 25-year $200,000 mortgage on his workshop with monthly payments at 5% compounded semi-annually. If Santa had taken out a 25-year $200,000 mortgage with weekly payments at 5% compounded semi-annually instead, how much less would he have paid after two years? (Assume payments can be non-integral.)

  29. Loan Amortization • Loan Terminology • Principal (L): Balance of the loan at t0 • Outstanding Loan Balance (Bt): Remaining balance of the loan / principal not yet paid at time t • Loan Payments and Outstanding Loan Balance • Loan payments are an annuity! (Principal = Payment * an¯|) • Each payment Xt= interest paid (It) + principal repaid (Pt) • It = i*Bt-1 • Pt = Bt-1 – Bt • Bt can be calculated in two ways: • Retrospective (backward): Bt = L(1+i)t – Xst¯| • Prospective (forward): Bt = X an-t¯|

  30. Loan Amortization Example • Example: Cyntha takes out a 20-year $25,000 amortized loan with payments made at the end of each year at an annual effective rate of 8%. Calculate the regular payment, the outstanding balance after the 10th payment, and the principal paid on the 11th payment. What would be the regular payment and OLB after the 5th payment if the interest rate is 8% for the first 10 years, and 7% thereafter?

  31. Sinking Funds • Sinking Funds: • Instead of payments consisting of part principal, part interest: • Interest payments are paid on the initial loan principal, which remains constant—hence interest payments remain constant • Separate deposits made into a “sinking fund” (SF) eventually accumulate to the initial loan value • Interest rate for interest pmts (i) may differ from SF interest rate (j) • Total payment per period = interest pmt + SF pmt = i(L) + L / sn¯|j • 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 )

  32. Sinking Fund Example A yacht owner pays back a loan of $20,000 using a sinking fund. He makes interest payments at the end of each year for 10 years, using an annual effective interest rate of 7%. He deposits $2,000 annually for the first 5 years and $1,000 annually for the next 4 years into the sinking fund account (all payments are made at the end of each year). If the sinking fund account earns an annual effective interest rate of 3%, what is his payment at the end of the 10thyear to pay off the loan? Show the first two years of the amortization schedule under the sinking fund and amortization (assuming end of year payments at 7%) methods.

  33. Amortization Schedules Amortization Method: Sinking Fund Method:

  34. 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 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 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: “Par Value”) i.e. a $1,000 bond with 8% semi-annual coupons maturing in 10 years at par

  35. Bonds • Bond Pricing at Issue • Price = PV(Coupons) + 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, Price of bond = Redemption Value (“Sold at Par”)

  36. Bond Pricing Example A $1,500 4% 12-year bond is sold to yield 5% convertible semi-annually. The discount on the bond is $100. Find the redemption amount of the bond.

  37. Bonds • Callable Bonds • Can be redeemed by the borrower 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 call 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

  38. Callable Bond Example A $1,000 bond with semi-annual coupons at rate 4% per year matures at the end of 10 years. The bond is callable at $1,050 at the ends of years 4 through 6, at $1,025 at the ends of years 7 through 9, and at $1,000 at the end of year 10. Find the maximum price that an investor is willing to pay to obtain a yield rate of 5% convertible semi-annually.

  39. Financial Analysis • Spot Rate (rt): • Annual effective yield rate for an investment of length t years made at t0 (e.g. zero-coupon bond) • i.e. if 3-year spot rate is 6%, can invest $1 at time 0 at 6% effective/year and receive 1.063 at time 3 • Forward Rate (ft,t+n) • Annual effective yield rate earned on an investment made t years from now for the period t -> t+n • ft, t+n= • i.e. if 2-year forward rate from time 4 (f4,6) is 6%, we can make an investment 4 years from now that will earn 6% annual effective from time 4 to time 6

  40. Forward Rate Example If the 1-year spot rate is 2% and the price of a 4-year $100 zero coupon bond is $84.69, calculate the 3-year forward rate from time 1.

  41. Duration • Macaulay Duration • Represents average timing of an asset’s cash flows weighted by the PV of each cash flow • -P’(δ)/P(δ) or -d/dδ[ln P(δ)] , where P(δ) = asset price = [Σ(Ct*vt*t)]/[Σ(Ct*vt)] = [Σ(Ct*vt*t)]/(Price of Asset) • Single Cash Flow at time n: MacD = n • Level annuity immediate: MacD = (Ia)n¯| / an¯| • Modified Duration • -P’(i)/P(i) • ModD= v(MacD)

  42. Duration and Price Sensitivity • Duration of a Portfolio • If a portfolio contains multiple assets with known durations and present values: • Portfolio duration = Σ(Asset duration*PV asset)/(PV portfolio) • Weighted average of individual asset durations, with weights = asset prices • Price Sensitivity • Change in price of asset due to change in force of interest (Δ) • New price= Original Price – (MacD)(Original Price)(Δ) • Higher duration → More Price Sensitive • If interest rates increase, what happens to the price of our asset?

  43. Duration Example LeBron repays a loan with two payments. One year from today, he pays $1,000, and two years from today, he pays another $1,000. Assuming the annual effective rate of interest is 10%, find the following: (a) The Macaulay duration of the loan. (b) The new PV of his loan payments if the yield decreases to 9%.

  44. Redington Immunization • Redington Immunization: • Protects against small changes in interest rate • Given a portfolio of assets and liabilities, immunized if: 1. PV Assets = PV Liabilities 2. Duration of Assets = Duration of Liabilities 3. Convexity of Assets > Convexity of Liabilities

  45. Immunization Example Manulife needs to make three payments of $100 at times 2, 4, and 6. Manulife plans to meet these obligations with an investment plan that produces asset cash flows of X at time 1 and Y at time 5. The effective rate of interest is 10%. Determine X and Y so that the investment plan has the same present value and duration as the liability cash flows.

  46. Questions Questions? Good Luck! jeffreybaer@gmail.com

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