5
Theory of Real Interest Rates
• Real interest rates are determined by supply and demand of funds in the economy. • 3 factors in determining real interest rates:
Where b is the bond holding, u’>0 and u”<0
8
• The optimality condition is
u '(c0 ) (1 r ) u '(c1 ) or ( for c1 c0 dc ) u '(c0 ) 1 c0u "(c0 ) dc r 1 (1/ 1) u '(c1 ) u '(c0 ) c0
3
Forms of Interest Rates
• In this market, this information on the time value of money is given in several different forms:
– Spot interest rates – Price of discount bonds (e.g., zero-coupon bonds and STRIPS) – Prices of coupon bonds – Yield-to-maturity (an average of spot interest rates) – Forward interest rates
n
2
Term Structure of Interest Rates
• Our objective is to value riskless cash flows. • Given the rich set of fixed-income securities traded in the market, their prices provide the information needed to value riskless cash flows at hand.
No ! r is the
discount rate that cannot be used for so long period
Ct Let NPV P (1 r)t 0 0 t 1
n
n Ct Ct (1 r )t (1 r )t t 1 t 1 t
CHAPTER TWO: Time Value of Money and Term Structure of Interest
1
Discounted Cash Flow Formula
?
Yes ! r is the
expected rate of return, i.e., the n mean of the Ct P discount rates for PV 0 t (1 rt ) t 1 difhis information is expressed depends on the particular market.
4
Determination of Interest Rate
• Four basic factors
Capital production ability —— the more the capital’s expected return, the higher the interest rates and vice versa. 2. Uncertainty of capital production ability —— the more the uncertainty, the higher the risk premium required and the higher the interest rates and vice versa. 3. Time preference of consumption —— the stronger preference to current consumption, the higher the risk premium required and the higher the interest rates and vice versa. 4. Risk aversion —— the more the risk aversion, the higher the risk premium required and the lower the riskfree interest rates. 1.
Thus, the real interest rate is given by 1 c0u "(c0 ) dc r (1/ 1) [ ] u '(c0 ) c0
Relative risk aversion coefficient
9
Nonlinear technology
Time 1
– Has endowment of ( e0, e1) – Faces a bond market with interest rate r.
7
• He maximizes his utility over his consumption now and later:
max u (c0 ) u (c1 ) s.t. c0 e0 b c1 e1 (1 r )b
u2 u1
– Aggregate endowments – Aggregate investment opportunities – Aggregate preferences for different consumption path
6
• Consider a representative investor: