Question

Consider a bond with a $1000 face value, with a 20 year maturity and a coupon rate of 8% which pays coupons with a semi-annual frequency. The 4th coupon payment will be received in 1 second. Calculate   the value of the bond if the YTM is 4%, 6%, 8%, and 10% (APR with semi-annual compounding). Explain why there is a negative relation between bond price and YTM.

Answer 1

Given Information:

Face Value = $10000, Maturity 20 years, Coupon rate= 8%, Semi Annualy, Coupon payment= 80

Question specifically tells that 4th payment is about to receive. So that means 2 year has passed and Now 18 year has left for Bond to mature. Accordingly no . of payments left= 18x2=36.

Calculation of Bond value Valuation of an asset is Present value of all future cash flows, so does for Bond.

i) AT 4 % YTM

PV of Future cash inflows( from investor point of view) 80 every 6 month and 1000 at Maturity. And rat ehere to be divided by 2 as semi annualy.

(80 x Cumulative discount factor @ 2%% for 36 time period) + (1000 x Present Value factor@2%% at 36 time)

Calculate the abobe with use of Present value tables. instead use financial calcultor to solve this.

For texas BA 2, Put value.N=36, I/Y=2, PMT=80, Fv=1000 Compute PV

Answer: 2529.33

ii) At 6%

Instead of 2% Use 3 % in Above Calculation.

Answer= 2091.61

iii)

ii) At 8%

Instead of 3% Use 4 % in Above Calculation.

Answer= 1756.33

ii) At 10%

Instead of 4% Use 5 % in Above Calculation.

Answer= 1496.41

Why there is negative relation between YTM and Bond Price

1. YTM is Yield to maturity and Given YTM in question is the market yield in same type of other bonds available. Suppose you are the investor you know market yield is only 4% and there is bond in market of value 1000 giving 8% coupon payment semi annually which is way more than the market yield, then why not there be a rush to buy that bond who is providing tremendous yield to investor.When people start buying that bond, price of bond will increase and people continue to buy till it reaches 2529(at 4% market yield) or 2091( at 6% market yield) because at that level investor would be indifferent because both will provide the same yield at in market. No one want to miss the opportunity so in this way markets are linked. Also if such minor opportunity available investor grab that and process known as Arbitrage.

2. Second reason for there inverse relation is simply the mathematics calculations. Present Value factor are bigger at 4% and tend to get smaller at every decrease in %. So while calculation present value of cash flows, Value comes at 4% is higher than 6% because PV factors of 4% are also higher than 6%.