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IABE – 2009 Annual Conference Las Vegas , 18-21 October 2009

Variable Annuity and Its Application in Bond Valuation Budi Frensidy Faculty of Economics, University of Indonesia. IABE – 2009 Annual Conference Las Vegas , 18-21 October 2009. Introduction. Variable annuity differs from growing annuity In a growing annuity, the growth is in percentage

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IABE – 2009 Annual Conference Las Vegas , 18-21 October 2009

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  1. Variable Annuity and Its Application in Bond ValuationBudi FrensidyFaculty of Economics, University of Indonesia IABE – 2009 Annual Conference Las Vegas, 18-21 October 2009 Budi Frensidy - FEUI

  2. Introduction • Variable annuity differs from growing annuity • In a growing annuity, the growth is in percentage • In a variable annuity, the growth or the difference is in nominal amount such as Rp 2 million or –Rp 100,000 • Like growing annuity, we also have a specific equation, albeit longer, to calculate the present value • Because it is time saving, the equation is very valuable for the scholars and the financial practitioners as well Budi Frensidy - FEUI

  3. Introduction (2) • Variable annuity can be used when a business owner plans to pay off his debt with decreasing installments every period • It can also be used for an employee who feels convenient with increasing installments of his home ownership loan to be in line with his growing salary • Last, variable annuity can be applied to value bonds of which the principal is paid off in equal amounts periodically, along with the diminishing periodic interest • A set of illustrations with gradual difficulty and the logics of the equation are given Budi Frensidy - FEUI

  4. Example 1 • A Rp 60 million loan with 10% interest can be paid off in three annual installments. The payment for the principal is the same for each installment that is one third of the initial loan or Rp 20 million. Make the schedule of the loan installments • Interest expense for the first year = 10% x Rp 60 milion = Rp 6 million • Interest expense for the second year = 10% x Rp 40 milion = Rp 4 million • Interest expense for the third year = 10% x Rp 20 milion = Rp 2 million Budi Frensidy - FEUI

  5. Example 1 (2) • Difference– Rp 2 million– Rp 2 million • The above schedule for loan payment actually fulfils a variable annuity with n = 3, interest rate (i) = 10%, beginning installment or first payment (a1) = Rp 26 million, and nominal difference (d) of -Rp 2 million • This constant difference is the key to prove that the present value of the cash flows is Rp 60 million namely: • (Rp 22 million – Rp 2 million) + (Rp 24 million – 2 x Rp 2 million) + (Rp 26 million – 3 x Rp 2 million) = 3 x Rp 20 million Budi Frensidy - FEUI

  6. Example 1 (3) • Another way to get the above result is by using a short-cut equation. Notice that the difference (-Rp 2 million) = the principal paid per period x periodic interest rate or - d = periodic principal payment x i Periodic principal paid = -d/i Total principal paid = number of periods x periodic principal paid Total initial loan = n x (-d/i) = -nd/i = 3 x – (– Rp 2 million)/10% = Rp 60 million Budi Frensidy - FEUI

  7. Example 2 • Calculate the present value of the following annual cash flows if the discount rate is 10% p.a.: Rp 46 million, Rp 44 million, and Rp 42 million The schedule of the cash flows can be divided into two series: Budi Frensidy - FEUI

  8. Example 2 (2) • How can we get such two series? • First, we must get the principal paid per period which is –d/i or – (– Rp 2 million)/10% = Rp 20 million • So, the cash flows for series 2 is 20 million + 10% (60 million), 20 million + 10% (40 million), 20 million + 10% 920 million) or Rp 26 mil, Rp 24 mil, Rp 22 mil • From this result, we can calculate the present value of the loan which is n (– d/i), which is 3 x Rp 20 million = Rp 60 million (from Example 1) • Based on these results, we can compute cash flows for series 1 which is the difference of the total installment and cash flow series 2 • The series 1 cash flow is Rp 20 million, derived from Rp 46 million minus Rp 26 million or Rp 44 million minus Rp 24 million or Rp 42 million minus Rp 22 million Budi Frensidy - FEUI

  9. Example 2 (3) • Thus, the present value of the above cash flows is the present value of series 1 which is Rp 49,737,039,8 and the present value of the second series which is Rp 60 million, based on the computation in Example 1. The total present value becomes Rp 109,737,039,8 • The present value of series 1 can be computed using the present value equation for the ordinary annuity with the periodic payment or PMT or A = Rp 20 million, n = 3, and i = 10% Budi Frensidy - FEUI

  10. Example 2 (4) • Notice that it is a Rp120 million loan with 3 principal payments of Rp 20 million each plus 5% periodic interest • If we use 5% discount rate, the PV is exactly Rp 120 million Budi Frensidy - FEUI

  11. Example 3: Decreasing Variable Annuity • Unlike other annuities, variable annuity requires that we divide the cash flows between series 1 and series 2 • Calculate the present value of the following cash flows, if it is known that i = 10% Budi Frensidy - FEUI

  12. Example 3: Decreasing Variable Annuity (2) • First, we calculate the principal paid per period which is –d/i = Rp 10,000/10% = Rp 100,000 • So, the present value of series 2 cash flows is n (-d/i) = 16 (Rp 100,000) = Rp 1,600,000 • Based on this result, we can compute the first installment of series 2 which consists of the principal payment and its accumulated interest. In illustration 3, the amount is Rp 100,000 for periodic principal payment and i (-nd/i) or 10% (Rp 1,600,000) = Rp 160,000 for the interest • Therefore, the series 1 cash flow is Rp 360,000 – Rp 100,000 – Rp 160,000 = Rp 100,000 Budi Frensidy - FEUI

  13. Example 3: Decreasing Variable Annuity (3) • The total present value = PV of Series 1 + PV of Series 2 • PV = PV of ordinary annuity Rp 100,000 for 16 years at 10% + Rp 1,600,000 • PV = Rp 782,370.86 + Rp 1,600,000 = Rp 2,382,370.86 • And the complete series 1 and 2 cash flows are: Budi Frensidy - FEUI 13

  14. Example 3: Decreasing Variable Annuity (4) Budi Frensidy - FEUI

  15. PV Equation for Variable Annuity • In addition to the PV equation for the series 2 cash flows (-nd/i), there is also a short-cut equation to get the series 1 cash flows • First, we must understand that each installment consists of series 2 cash flow which is the principal payment & its accumulated interest and series 1 cash flow namely the fixed annuity • So, the cash flow for series 1 is the first installment amount (a1) minus the principal payment (-d/i) and minus the first interest payment (i x (-nd/i)) or -nd. Notice that –nd/i is the total initial loan Budi Frensidy - FEUI

  16. PV Equation for Variable Annuity (2) • If we denote the series 1 cash flow by A, then A = • Therefore, the present value for this series is: PV = or PV = A • Finally, if we combine PV of series 1 cash flows and PV of series 2 cash flows, we get the complete PV equation PV = Budi Frensidy - FEUI

  17. Example 4: Increasing Variable Annuity Calculate the present value of the cash flows Rp 22 million next year that rises Rp 2 milion every year for 4 times if the relevant discount rate is 10% p.a. i = 10% n = 4 d = Rp 2 million a1 = Rp 22 milion First, we will find out the periodic cash flow for series 1: A = Rp 22 million + + 4 (Rp 2 million) A = Rp 22 million + Rp 20 million + Rp 8 million A = Rp 50 million Budi Frensidy - FEUI

  18. Example 4: Increasing Variable Annuity (2) So, the series 1 and series 2 cash flows become: • PV of series 1 cash flows is PV of ordinary annuity with A = Rp 50 million namely Rp 158,493,272.3 • Whereas, PV of series 2 is -Rp 80 million • Thus, PV of the above cash flows is Rp 158,493,272.3 + (-Rp 80,000,000) = Rp 78,493,272.3 Budi Frensidy - FEUI

  19. The Application in Bond Valuation • One of the applications of variable annuity is to value the fair price of bonds • The valuation of a bond always involves two kinds of interest rates i.e. the bond coupon rate and the investor’s expected yield • The cash flow patterns for bond repayment are also two. First, bonds that pay only the coupon periodically and the principal at the maturity date. Second, bonds that pay off the pricincipal in equal amounts every period, plus the accrued periodic interest • The principal balance of the bond payable in the second group will decline from one period to another period and the amount of the accrued periodic interest decreases as well Budi Frensidy - FEUI

  20. Example 5: Bond Valuation • A corporation issues a US$ 100,000 bond with 4% annual coupon. The bond will be repaid in 20 equal principal payment every year-end, $ 5,000 each plus the accrued interest. Calculate the fair price of the bond if an investor requires 10% yield for this bond. n = 20 i = 10% d = 4% x $ 5,000 = $ 200 a1 = $ 5,000 + 4% ($ 100,000) = $ 9,000 Budi Frensidy - FEUI

  21. Example 5: Bond Valuation (2) Budi Frensidy - FEUI

  22. Example 5: Bond Valuation (3) PV= PV= PV= PV= US$ 65,540.69 Budi Frensidy - FEUI

  23. Example 5: Bond Valuation (4) Schedule of series 1 and series 2 of the bond Budi Frensidy - FEUI

  24. Summary • We have a short-cut mathematical equation to calculate the present value of a variable annuity • A variable annuity is defined as an annuity that grows at a certain nominal amount (d) every period. The difference (d) between two successive periods can be positive or negative • Compared to the other fourteen formulae, the present value equation for the variable annuity is the hardest and the longest • The present value of a variable annuity is always the sum of two series, series 1 and series 2 Budi Frensidy - FEUI

  25. THANK YOU Budi Frensidy - FEUI

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