Robert D. Brown IIIIncite! Decision Technologies, LLC

678-947-5997rdbrown@incitedecisiontech.com

Bond Portfolio Valuation with Analytica™

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?

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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

C1 C2 … … … CM-2 CM-1 CM

P

Time

1 2 T-2 T-1 T

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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€

Vt =C1

(1+ rt )T−tt

T

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥+

P

(1+ rt )T−t

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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

€

Bt =i=1

N

∑ Vt,i

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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

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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

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€

∑

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

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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 =

+

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Conceptualizing the Problem in Analytica

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• Coupon Rate = 5%• Maturity = 10 Yrs• Profit and Interest Rates vary stochastically

Positive profits represent par values of bonds purchased

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Profit*(Profit>0)

Interest rates vary randomly in time

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This is variation is for illustration purposes only. It does not represent a real interest rate evolution.

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]

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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

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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)

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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)

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Conceptualizing the Problem in Analytica

Year

Valuation Year

Purc

hase

Year

Coupon Value

Principal Value

Year

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Purc

hase

Year

Run Index

Four dimensions Three dimensions

Bond Portfolio Valuation with Analytica™

Questions?

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