Question

The price of a zero-coupon bond with maturity 1 year is $943.40. The price of a zero-coupon bond with maturity 2 years is $898.47. For this problem, express all yields as net (not gross) rates. Assume the face values of the bonds are $1000.

Assuming the liquidity preference theory is valid and the liquidity premium in the second year is 0.01, what is the expected short rate in the second year?

Assuming that the expectations hypothesis is valid, what is the expected price of the 2 year bond at the beginning of the second year?

Answer 1

A	B	C	D	E	F	G	H	I	J	K	L
2
3		Spot rate for 1 and 2 Years can be calculated as follows:
4
5		Maturity (Years)	Price	Face Value	Spot rate
6		1	$943.40	$1,000	6.00%	=(E6/D6)^(1/C6)-1
7		2	$898.47	$1,000	5.50%	=(E7/D7)^(1/C7)-1
8
9		Calculation of short rate using liquidity premium theory:
10		As per liquidity premium theory,
11
12		n- year rate = Liquidity Premium + [(f1 + f2+ …+fn)/n]
13
14		Where f1, f2, …,fn are n-years forward rate.
15
16		Given the following data:
17		2- Year Rate	5.50%
18		1- Year Rate	6.0%
19		Liquidity Premium	1.00%
20
21		Assuming forward rate in year 2 is f2.
22
23		As per liquidity premium theory,
24		2- year rate = Liquidity Premium + (f1 + f2)/n
25		or
26		5.50% = 1%+(6%+f2)/2
27
28		Solving the above equation f2 can be found as below:
29
30		f2	3.00%	=((D17-D19)*2)-D18
31
32		Hence short rate in second year will be	3.00%
33
34		Calculation of short rate using expectation theory:
35
36		Using pure expectation theory, Expected interest rate starting from year i for one year can be calculated as follows:
37		[1+E(ir1)]=(1+1Ri)i / (1+1Ri-1)i-1
38
39		Where 1Ri is the i year tresury rate.
40
41		Given the following data:
42		1 year treasury rate (1R1 )	6.00%
43		2 year treasury rate (1R2 )	5.50%
44
45
46		Now 1 year treasury rate starting at year 2 can be calculated as follows:
47		[1+E(2r1)]	=(1+1R2)2 / (1+1R1)1
48			=(1+5.5%)2 / (1+6.0%)1
49			1.050007	=((1+D43)^2)/((1+D42)^1)
50
51		E(2r1)	5.00%	=D49-1
52
53		Hence short rate in the 2nd year is	5.00%
54