Seminar 3 Discount Rates & Yield Curves (Lecture Notes Recap) 1) Spot bond – Zero coupon bond (One payment at maturity)
FV FV Time Time Yr 4 (Maturity Year)
Yr 0
2) Annuity bond – Constant cash flow at regular intervals
Yr 0
CF
CF
CF
CF
Yr 1
Yr 2
Yr 3
Yr 4
Time
3) Par bond – Pays coupons and Principal amount
FV
Yr 0
CF
CF
CF
CF
Yr 1
Yr 2
Yr 3
Yr 4
What are the similarities among the 3 bonds above? Spot bond + Annuity bond = Par Bond
Time
Spot Yields to Annuity Yields Time Spot Annuity Par C (YTM) 1 5 5 5 2 6 5.65 5.97 5.97 3 7.5 6.524 7.36 7.36 Figures indicated in RED are calculated in the below steps before imputing in. For the first year, cash flows on both types of bonds are identical. Therefore, Annuity Yield for 1st Year is also 5%.
Yr 0 Discount the annuity yields to present value.
CF
CF
Yr 1
Yr 2
Time
Calculate the PV of the 2-year annuity which gives $1 each year : 1 PV = =
(1+ S1) 1 (1+ 0.5)
1 +
(1+ S2) 2 +
1 (1+ 0.6)
2
PV (Annuity) = 1.8424 Calculate the Returns on Annuity for year 1 and 2 (R2) at the bond price of $1.8424 1 1 + PV (Annuity) = 2 (1+ R2) (1+ R2) R2 = 5.65%
Yr 0 Discount the annuity yields to present value. 1 PV = (1+ S1) Substitute S1= 5 S2 = 6 S3 = 7.5
CF
CF
Yr 1
Yr 2
Yr 3
1 +
(1+ S2)
PV of Annuity = 2.6473 1 PV (Annuity) = + (1+ R3) R3 = 6.524%
CF
1 2
+
(1+ S3)
1 (1+ R3)
3
1 2
+
(1+ R3)
3
Time
Spot Bond to Par Bond For the first year, cash flows on both types of bonds are identical. Therefore, Par Yield for 1st Year is also 5%. a) Determine the 2 year par yield (ie, YTM for a 2 year par bond) :
P
Yr 0
C
C
Yr 1
Yr 2
Time
Px = 100 (Price of bond = 100, or the face value because it is a par bond) 100 = C/(1+S1) + (C+P)/(1+S2) 2 100 = C/ (1+0.05) + (C+100)/(1+0.06) 2 C = 5.97 Where P = 100 S1 = 5% S2 = 6% Therefore, for par bond, coupon rate = YTM. Therefore, 2 year par yield = C = 5.97 b) Determine the 3-year par yield (ie, YTM for a 3 year par bond) : Px = 100 100 = C/(1+S1) + C = 7.36 Where P = 100 S1 = 5% S2 = 6% S3 = 7.5%
C/(1+S2) 2 + (C+P)/(1+S3) 3
Therefore, for par bond, coupon rate = YTM. Therefore, 3 year par yield = C = 7.36
Par Bond to Spot Bond Figures indicated in RED are calculated in the below steps before imputing in. For the first year, cash flows on both types of bonds are identical. Therefore, Spot Yield for 1st Year is also 6%. Time (t) 1 2 3
Spot 6% 7.035% 9.22%
Par 6 7 9
a) Determine the 2-year spot yield, ie, S2 from a 2-year par bond :
P
Yr 0
C
C
Yr 1
Yr 2
Time
Px = 100 (Price of bond = 100, or the face value because it is a par bond) Px = C/(1+S1) + (C+P)/(1+S2) 2 100 = 7/(1+0.06) + (7+100)/( 1+S2) 2 where Px = 100, C = 7 (2-year par bond’s YTM = 7% = coupon rate), P = 100 S1 = 6% Therefore, S2 = 7.035
a) Determine the 3-year spot yield, ie, S3 from a 3-year par bond :
P
Yr 0
C
C
C
Yr 1
Yr 2
Yr 3
Time
Px = 100 (Price of bond = 100, or the face value because it is a par bond) Px = C/(1+S1) + C/(1+S2) 2 + (C+P)/(1+S3) 3 100 = 9/(1+0.06) + 9/( 1+0.0735) 2 + (9 + 100)/( 1+S3) 3 where Px = 100, C = 9 (3-year par bond’s YTM = 9% = coupon rate), P = 100, S1 = 6%, S2 = 7.035%. Therefore, S3 = 9.22%