Question

What is the relationship between price and yields for fixed income securities? There is a government bond with a par value of $1000, and %4 semi- annual coupon rate. The bond was issued 6 years ago when the market interest rates are 8%. (The yield to maturity is %8). The remaining maturity of this bond is 1,5 years. Make comment about the price of this bond.

a- If somehow, the market interest rates remain unchanged (ytm unchanged)

b- If the market interest rates rise (ytm goes up)

c- If the market interest rates go down. (ytm goes down)

Answer 1

Solution:

There is an "inverse" relationship between the price and yields of fixed income securities, which means if the price of the fixed income securities increases the yield decreases and vice-versa.

Present value of the bond=

C= coupon rate, will compute semi annually as mentioned in question

YTM= yield to maturity

N= number of period

P=Par or face value of the bond

A) when the market rates remain unchanged for 6 years.

Coupon rate= 2% (semi annual) of $1000=$20

YTM= 4% (semi annual)

P= Par value= $1000

N= 12 (6 months=1 period and total 6 years= 72 months= 12 periods)

PV= Present value of the bond 6 years ago

PV of the bond= 20/(1+4%)^1 + 20/(1+4%)^2 + 20/(1+4%)^3 + 20/(1+4%)^4 + 20/(1+4%)^5 + 20/(1+4%)^6 + 20/(1+4%)^7 + 20/(1+4%)^8 + 20/(1+4%)^9 + 20/(1+4%)^10 + 20/(1+4%)^11 + 20/(1+4%)^12 + 1000/(1+4%)^12

PV= 20(0.9615+0.924556+0.888+0.85+0.82+0.79+0.759+0.73+0.70+0.67+0.64+0.62)+624.5

PV=20(9.385)+624.5

PV= $812.29

If the market rates remain unchanged, it means the par or face value of the bond is higher than present value of the bond, which implies YTM is greater than the coupon rate.

B) If the market rates(YTM) increase, then the price of the bond decreases as there is an inverse relationship between them,also increasing market rates would increase the denominator in the formula, thus would decrease the value of the bond.

C) If the market rates(YTM) decrease, then the price of the bond increases with respect to the present value of the bond calculated.