Question

First, consider a 10 year bond with a coupon rate of 7% and annual coupon payments. Draw a graph showing the relationship between the price and the interest on this bond. The price should be on the y- axis and the interest rate on the x-axis. To compute the various prices, consider interest rates between 2% and 12% (use 0.5% increments). So your x-axis should go from 2%, then 2.5% ... until 11.5% and then 12%.

Is the relationship linear (i.e. is the slope constant)? Start at 7%. If interest rates go up or down by 0.5% is the price changing by the same amount? What type of relationship do we observe between prices and interest rates (liner, concave, convex or something else)?

Answer 1

Price of the bond is calculated by solving the following equation:

Here r = interest rate

so if r = 2% then substituting the value of r in the above equation, we can calculate the price or create the following schedule:

Year	CF	Discount Factor	Discounted CF
1	$       70.00	1/(1+0.02)^1=	0.980392157	0.980392156862745*70=	$       68.63
2	$       70.00	1/(1+0.02)^2=	0.961168781	0.961168781237985*70=	$       67.28
3	$       70.00	1/(1+0.02)^3=	0.942322335	0.942322334547044*70=	$       65.96
4	$       70.00	1/(1+0.02)^4=	0.923845426	0.923845426026514*70=	$       64.67
5	$       70.00	1/(1+0.02)^5=	0.90573081	0.905730809829916*70=	$       63.40
6	$       70.00	1/(1+0.02)^6=	0.887971382	0.887971382186192*70=	$       62.16
7	$       70.00	1/(1+0.02)^7=	0.870560179	0.870560178613914*70=	$       60.94
8	$       70.00	1/(1+0.02)^8=	0.853490371	0.853490371190112*70=	$       59.74
9	$       70.00	1/(1+0.02)^9=	0.836755266	0.836755265872658*70=	$       58.57
10	$ 1,070.00	1/(1+0.02)^10=	0.8203483	0.820348299875155*1070=	$     877.77
				Price= Sum of all Discounted CF	$ 1,449.13

Similarly we can substitute the value of r and calculate the price at different interest rates and get the following data table:

Cost of capital	Price
2.00%	$     1,449.13
2.50%	$     1,393.84
3.00%	$     1,341.21
3.50%	$     1,291.08
4.00%	$     1,243.33
4.50%	$     1,197.82
5.00%	$     1,154.43
5.50%	$     1,113.06
6.00%	$     1,073.60
6.50%	$     1,035.94
7.00%	$     1,000.00
7.50%	$         965.68
8.00%	$         932.90
8.50%	$         901.58
9.00%	$         871.65
9.50%	$         843.03
10.00%	$         815.66
10.50%	$         789.48
11.00%	$         764.43
11.50%	$         740.45
12.00%	$         717.49

Using this data table, we can calculate the price:

• As it is not a straightline, the relationship is not linear, i.e. slope is not constant.
• Starting at 7% when the interest rates go up or down by 0.5% the price is not changing by the same amount, as we can see in the data table and graph above
• The relationship is convex to the origin.