1. Budi Frensidy - FEUI1 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

2. Budi Frensidy - FEUI2 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

3. Budi Frensidy - FEUI3 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

4. Budi Frensidy - FEUI4 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

5. Budi Frensidy - FEUI5 Example 1 (2) Year 1Year 2Year 3 Amount of installmentRp 26 millionRp 24 millionRp 22 million 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 (a 1 ) = 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

6. Budi Frensidy - FEUI6 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

7. Budi Frensidy - FEUI7 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: Year 1Year 2Year 3 Series 1Rp 20 million Series 2Rp 26 millionRp 24 millionRp 22 million Rp 46 millionRp 44 millionRp 42 million

8. Budi Frensidy - FEUI8 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

9. Budi Frensidy - FEUI9 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%

10. Budi Frensidy - FEUI10 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

11. Budi Frensidy - FEUI11 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% YearInstallmentYearInstallment 1Rp 360,0009Rp 280,000 2Rp 350,00010Rp 270,000 3Rp 340,00011Rp 260,000 4Rp 330,00012Rp 250,000 5Rp 320,00013Rp 240,000 6Rp 310,00014Rp 230,000 7Rp 300,00015Rp 220,000 8Rp 290,00016Rp 210,000

12. Budi Frensidy - FEUI12 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

13. Budi Frensidy - FEUI13 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:

14. Budi Frensidy - FEUI14 Example 3: Decreasing Variable Annuity (4) YearInstallmentSeries 1Series 2 1Rp 360,000Rp 100,000Rp 260,000 2Rp 350,000Rp 100,000Rp 250,000 3Rp 340,000Rp 100,000Rp 240,000 4Rp 330,000Rp 100,000Rp 230,000 5Rp 320,000Rp 100,000Rp 220,000 6Rp 310,000Rp 100,000Rp 210,000 7Rp 300,000Rp 100,000Rp 200,000 8Rp 290,000Rp 100,000Rp 190,000 9Rp 280,000Rp 100,000Rp 180,000 10Rp 270,000Rp 100,000Rp 170,000 11Rp 260,000Rp 100,000Rp 160,000 12Rp 250,000Rp 100,000Rp 150,000 13Rp 240,000Rp 100,000Rp 140,000 14Rp 230,000Rp 100,000Rp 130,000 15Rp 220,000Rp 100,000Rp 120,000 16Rp 210,000Rp 100,000Rp 110,000

15. Budi Frensidy - FEUI15 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 (a 1 ) 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

16. Budi Frensidy - FEUI16 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 =

17. Budi Frensidy - FEUI17 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 a 1 = 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

18. Budi Frensidy - FEUI18 Example 4: Increasing Variable Annuity (2) So, the series 1 and series 2 cash flows become: YearCash FlowsSeries 1Series 2 1Rp 22 millionRp 50 million-Rp 28 million 2Rp 24 millionRp 50 million-Rp 26 million 3Rp 26 millionRp 50 million-Rp 24 million 4Rp 28 millionRp 50 million-Rp 22 million 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

19. Budi Frensidy - FEUI19 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

20. Budi Frensidy - FEUI20 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 a 1 = $ 5,000 + 4% ($ 100,000) = $ 9,000

21. Budi Frensidy - FEUI21 Example 5: Bond Valuation (2) YearPrincipal PaymentInterest ExpenseTotal 1$ 5,000$ 4,000$ 9,000 2$ 5,000$ 3,800$ 8,800 3$ 5,000$ 3,600$ 8,600 4$ 5,000$ 3,400$ 8,400 5$ 5,000$ 3,200$ 8,200 6$ 5,000$ 3,000$ 8,000 7$ 5,000$ 2,800$ 7,800 8$ 5,000$ 2,600$ 7,600 9$ 5,000$ 2,400$ 7,400 10$ 5,000$ 2,200$ 7,200 11$ 5,000$ 2,000$ 7,000 12$ 5,000$ 1,800$ 6,800 13$ 5,000$ 1,600$ 6,600 14$ 5,000$ 1,400$ 6,400 15$ 5,000$ 1,200$ 6,200 16$ 5,000$ 1,000$ 6,000 17$ 5,000$ 800$ 5,800 18$ 5,000$ 600$ 5,600 19$ 5,000$ 400$ 5,400 20$ 5,000$ 200$ 5,200

22. Budi Frensidy - FEUI22 Example 5: Bond Valuation (3) PV = PV = US$ 65,540.69

23. Budi Frensidy - FEUI23 Example 5: Bond Valuation (4) Schedule of series 1 and series 2 of the bond YearSeries 1Series 2Total 1$ 3,000$ 6,000$ 9,000 2$ 3,000$ 5,800$ 8,800 3$ 3,000$ 5,600$ 8,600 4$ 3,000$ 5,400$ 8,400 5$ 3,000$ 5,200$ 8,200 6$ 3,000$ 5,000$ 8,000 7$ 3,000$ 4,800$ 7,800 8$ 3,000$ 4,600$ 7,600 9$ 3,000$ 4,400$ 7,400 10$ 3,000$ 4,200$ 7,200 11$ 3,000$ 4,000$ 7,000 12$ 3,000$ 3,800$ 6,800 13$ 3,000$ 3,600$ 6,600 14$ 3,000$ 3,400$ 6,400 15$ 3,000$ 3,200$ 6,200 16$ 3,000 $ 6,000 17$ 3,000$ 2,800$ 5,800 18$ 3,000$ 2,600$ 5,600 19$ 3,000$ 2,400$ 5,400 20$ 3,000$ 2,200$ 5,200

24. Budi Frensidy - FEUI24 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

25. Budi Frensidy - FEUI25 THANK YOU