Question

If i2(2020) denotes the annual interest rate on 2-year bonds issued in 2020 and i1(2020) and i2(2021) represent interest rates on 1-year bonds purchased in 2020 and 2021, the equilibrium relationship between the 2-year bond and the two 1-year bonds is___________.

Answer 1

Answer:

i2(2020) x 2/(1+ discounting rate) =  i1(2020) + i2(2021)/(1+ discounting rate)

Explanation:

Coupon received from 2-year bonds issued in 2020 = Face value * i2(2020) x 2 years

Coupon from 1-year bonds purchased in 2020 = Face value * i1(2020) * 1 year

Coupon from 1-year bonds purchased in 2021 = Face value * i2(2021) * 1 year

PV of 2-year bonds issued in 2020 = (Face value * i2(2020) x 2)/(1+ discounting rate)^2

PV of coupon from 1-year bonds purchased in 2020 = Face value * i1(2020)/(1+ discounting rate)

PV of coupon from 1-year bonds purchased in 2021 = Face value * i2(2021)/(1+ discounting rate)^2

As in equilibrium relationship, the present value (PV) of coupon received from 2-year bonds issued in 2020 = PV of coupon from 1-year bonds purchased in 2020 + PV of coupon from 1-year bonds purchased in 2021

⇔ (Face value * i2(2020) x 2)/(1+ discounting rate)^2 = Face value * i1(2020)/(1+ discounting rate) + Face value * i2(2021)/(1+ discounting rate)2

⇔  (i2(2020) x 2)/(1+ discounting rate)^2 = i1(2020)/(1+ discounting rate) + i2(2021)/(1+ discounting rate)2

⇔  i2(2020) x 2/(1+ discounting rate) =  i1(2020) + i2(2021)/(1+ discounting rate)