budi frensidy - feui1 variable annuity and its application in bond valuation budi frensidy faculty...

Budi Frensidy - FEUI 1

Variable Annuity and Its Application in Bond Valuation

Budi Frensidy Faculty of Economics, University of Indonesia

IABE – 2009 Annual ConferenceLas Vegas, 18-21 October 2009


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


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


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


Example 1 (2)Year 1 Year 2 Year 3

Amount of installment Rp 26 million Rp 24 million Rp 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 (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


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


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 millionThe schedule of the cash flows can be divided into two series:

Year 1 Year 2 Year 3

Series 1 Rp 20 million Rp 20 million Rp 20 million

Series 2 Rp 26 million Rp 24 million Rp 22 million

Rp 46 million Rp 44 million Rp 42 million


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


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%


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


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%

Year Installment Year Installment

1 Rp 360,000 9 Rp 280,000

2 Rp 350,000 10 Rp 270,000

3 Rp 340,000 11 Rp 260,000

4 Rp 330,000 12 Rp 250,000

5 Rp 320,000 13 Rp 240,000

6 Rp 310,000 14 Rp 230,000

7 Rp 300,000 15 Rp 220,000

8 Rp 290,000 16 Rp 210,000


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



•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:



Year Installment Series 1 Series 2

1 Rp 360,000 Rp 100,000 Rp 260,000

2 Rp 350,000 Rp 100,000 Rp 250,000

3 Rp 340,000 Rp 100,000 Rp 240,000

4 Rp 330,000 Rp 100,000 Rp 230,000

5 Rp 320,000 Rp 100,000 Rp 220,000

6 Rp 310,000 Rp 100,000 Rp 210,000

7 Rp 300,000 Rp 100,000 Rp 200,000

8 Rp 290,000 Rp 100,000 Rp 190,000

9 Rp 280,000 Rp 100,000 Rp 180,000

10 Rp 270,000 Rp 100,000 Rp 170,000

11 Rp 260,000 Rp 100,000 Rp 160,000

12 Rp 250,000 Rp 100,000 Rp 150,000

13 Rp 240,000 Rp 100,000 Rp 140,000

14 Rp 230,000 Rp 100,000 Rp 130,000

15 Rp 220,000 Rp 100,000 Rp 120,000

16 Rp 210,000 Rp 100,000 Rp 110,000


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


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 =

ndi

da1

i

)i1(1 n

nd

i

da1

i

)i1(1 n

i

)i1(1 n

nd

i

da1 i

nd


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 = 4d = Rp 2 milliona1 = 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 millionA = Rp 50 million

%10

million 2 Rp


Example 4: Increasing Variable Annuity (2)

So, the series 1 and series 2 cash flows become:

Year Cash Flows Series 1 Series 2

1 Rp 22 million Rp 50 million -Rp 28 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


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


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


Example 5: Bond Valuation (2)Year Principal Payment Interest Expense Total

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


Example 5: Bond Valuation (3)

i

)i1(1 n

nd

i

da1

i

nd

%10

%)101(1 20

)200( 20

%10

200000,9

%10

)200( 20

4,000 000,31.0

1.11 20

PV =

PV =

PV =

PV = US$ 65,540.69


Example 5: Bond Valuation (4)Schedule of series 1 and series 2 of the bond

Year Series 1 Series 2 Total

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 $ 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


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


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budi frensidy - feui1 variable annuity and its application in bond valuation budi frensidy faculty...

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