Question

You are buying a $10,000 car, and your dealer gives you three options for payment plans.

• Pay $3,640 each year for the next three years: D1 = 3, 640 , D2 = 3, 640 , D3 = 3, 640

• Pay a $467 interest payment each year for 3 years, and then the $10,000 principal back: D1 = 467, D2 = 467, D3 = 10, 467

• Pay $1,850 in year 1, $3,700 in year 2, and $5,550 in year 3: D1 = 1, 850 , D2 = 3, 700 , D3 = 5, 550.

These payment plans can be thought of as three fixed-income securities, with different payment patterns, all with present value equal to $10,000. Assume your credit is so good that these securities are free of credit risk, so these security prices satisfy the no-arbitrage pricing formulas. Calculate the zero-coupon interest rates r1, r2, r3 implied by the prices of these securities.

Answer 1

3640/(1+r1)+3640/(1+r2)^2+3640/(1+r3)^3=10000

467/(1+r1)+467/(1+r2)^2+10467/(1+r3)^3=10000

1850/(1+r1)+3700/(1+r2)^2+5550/(1+r3)^3=10000

Let 1/(1+r1) be x, 1/(1+r2)^2 be y and 1/(1+r3)^3 be z

The equations become

3640x+3640y+3640z=10000

467x+467y+10467z=10000

1850x+3700y+5550z=10000

Solving these equations simultaneously we get

x=129401/134680,y=61599/67340,z=3173/3640

x=129401/134680
=>1/(1+r1)=129401/134680
=>r1=134680/129401-1=4.08%

y=61599/67340
=>1/(1+r2)^2=61599/67340
=>r2=(67340/61599)^(1/2)-1=4.56%

z=3173/3640
=>1/(1+r3)^3=3173/3640
=>r3=(3640/3173)^(1/3)-1=4.68%