Question

Need explanation how these values are found using algebra, answers needing to be a percentage; "r" is the forward rate:  

1 year coupon: $97=100/(1+r₁)¹

2 year coupon: $94= 100/(1+r₂)²

3 year coupon: $93= 100/(1+r₃)³

4 year coupon: $90= 100/(1+r₄)⁴

Answer 1

Fwd rate , It is the rate that expected to be in furture for that particular year and estimated now itself based on diffferent YTMs.

Year1 Fwd Rate = [ 100 / 97 ] - 1

= 1.0309 -1

= 0.0309 i.e 3.09%

Year 2 Fwd Rate = [(1+Year 2 YTM)² / (1+Year 1 YTM) ] - 1

Year 2 YTM = [100 /94] ^1/2 - 1

= 1.0314 - 1

= 0.0314 i.e 3.14%

Year 2 Fwd Rate = [(1+0.0314)² / (1+0.0309) ] - 1

= [(1.0314)² / (1.0309) ] - 1

= [1.0638 /1.0309 ] - 1

= 1.0319 - 1

= 0.0319 i.e 3.19%

Year 3 Fwd Rate = [(1+Year 3 YTM)³ / (1+Year 2 YTM)² ] - 1

Year 3 YTM = [100 /93] ^1/3 - 1

= 1.0245 - 1

= 0.0245 i.e 2.45%

Year 3 Fwd Rate = [(1+0.0245)³ / (1+0.0314)² ] - 1

= [(1.0245)³ / (1.0314)² ] - 1

= [1.0753 /1.0638 ] - 1

= 1.0108 - 1

= 0.0108 i.e 1.08%

Year 4 Fwd Rate = [(1+Year 4 YTM)⁴ / (1+Year 3 YTM)³ ] - 1

Year 4 YTM = [100 /90] ^1/4 - 1

= 1.0267 - 1

= 0.0267 i.e 2.67%

Year 4 Fwd Rate = [(1+0.0267)⁴ / (1+0.0245)³ ] - 1

= [(1.0267)⁴ / (1.0245)³ ] - 1

= [1.1112 /1.0753 ] - 1

= 1.0333 - 1

= 0.0333 i.e 3.33%