Robert D. Brown III Incite! Decision Technologies, LLC 678-947-5997 rdbrown@incitedecisiontech

1. Bond Portfolio Valuation with Analytica™ Robert D. Brown III Incite! Decision Technologies, LLC 678-947-5997 rdbrown@incitedecisiontech.com

2. Yucca Mountain Projecthttp://www.lumina.com/casestudies/BechtelSAIC.htm • Simulate value of a bond portfolio over a long horizon • Portfolio used to hedge against fluctuations in operating profit of facility • Buy bonds with positive profit • Sell bonds on shortfall • Given the current fee structure, will the account go bankrupt? If so, when? Giving You Confidence to Decide

3. Bond Cash Flow The purchase of a bond results in M coupon payments of value C every period until the maturity date, T, followed by a return of principal, the par value, P, at the maturity date P C1 C2 … … … CM-2 CM-1 CM Time 1 2 T-2 T-1 T Giving You Confidence to Decide

4. Bond Value or Price • The value of a bond at any point in time before the maturity date is the sum of the present values, at market interest rates rt, of the remaining coupon payments and principal where 1≤ t ≤ T • Interest rates change stochastically • For Yucca Mtn., interest rates simulated by an extension of Ibbotson’s model, developed by Bob Kenley Giving You Confidence to Decide

5. Portfolio Value The value of a portfolio of bonds at any point in time before the maturity date is the sum of the values of the N bonds in the portfolio at the point in time of the valuation Giving You Confidence to Decide

6. Illustration • A bond is purchased in 2002 with a par value of $1000, a coupon rate of 5%, a maturity of 10 years • For simplicity, assume the interest rate remains fixed at 5% over the horizon of the bond • The coupon payments will be a series of ten payments equal to 5% * $1,000 = $50 from 2002 through 2011 • In each Year, the holder sees a series of remaining coupon payments with a present value Giving You Confidence to Decide

7. Illustration The perceived value of a bond’s coupon streams in each Year is found by summing the present value of the coupon payments across Valuation Year Giving You Confidence to Decide

8. Illustration • The perceived value of a bond’s principal in each Year is found by calculating the present value of the principal using the remaining years until the principal is realized • The value of the principal converges over time to the face value of the bond Giving You Confidence to Decide

9. Illustration • The time value of the bond is the sum of the present value of the coupon stream and the principal value • The total value of the portfolio is the sum of the above calculations for each bond C = + P = Giving You Confidence to Decide

10. Conceptualizing the Problem in Analytica • Coupon Rate = 5% • Maturity = 10 Yrs • Profit and Interest Rates vary stochastically Giving You Confidence to Decide

11. Positive profits represent par values of bonds purchased Profit*(Profit>0) Giving You Confidence to Decide

12. Interest rates vary randomly in time This is variation is for illustration purposes only. It does not represent a real interest rate evolution. Giving You Confidence to Decide

13. Set up the coupon payments • Coupon_payment = Coupon_rate * Par_value • Example: Coupon_payment = 5% * $480.1K = $24.01K each year until maturity • Need to distribute coupon payments across Year beginning in the year bond purchased • Need two indexes with equal length and elements • Coupon_payment = (Year>=Purchase_year And Year<=(Purchase_year+Maturity-1))*Coupon_rate*Par_value[Year=Purchase_year] Giving You Confidence to Decide

14. Calculate the remaining present value of each coupon stream • Need another index with equal length and elements as other two • Coupon_present_value = (Valuation_year>=Year And Valuation_year<=(Purchase_year+Maturity-1))*Coupon_payments/(1+Interest_rates)^(Valuation_year-Year) • This plane is orthogonal to the prior plane Giving You Confidence to Decide

15. Calculate the remaining value of each coupon stream • Summing the values in the prior plane across Valuation_year gives the total remaining value of each coupon stream in each year and reduces the Valuation_year index from the hypercube • Remaining_coupon_value = Sum(Coupon_present_value, Valuation_year) • Summing the values in this result across Purchase_year gives the total value of all coupon streams in each year and reduces the Purchase_year index from the hypercube • Portfolio_total_coupon_value = Sum( Remaining_coupon_value, Purchase_year) Giving You Confidence to Decide

16. Repeat the process for the par value • Since the principal payments occur only once, the third index is not needed; this effort requires one less dimension • Spread the par value over the life of the bond • Principal_payments = (Year>=Purchase_year And Year<=(Purchase_year+Maturity-1)) * Par_value [Year=Purchase_year] • Next calculate the present value of the future principal payment for each year remaining in the life of the bond • Principal_payment_present_value = Principal_payments/(1+Interest_rates)^(Maturity-(Year-Purchase_year)-1) • Find the value of all the principal payments for all the bonds • Portfolio_total_principal_value = Sum(Principal_payment_present_value, Purchase_year) Giving You Confidence to Decide

17. Conceptualizing the Problem in Analytica Run Index Valuation Year Year Year Coupon Value Principal Value Purchase Year Purchase Year Four dimensions Three dimensions Giving You Confidence to Decide

18. Bond Portfolio Valuation with Analytica™ Questions? Giving You Confidence to Decide