Expectations: The basic tools

Relation between nominal and real interest rates

Need to adjust the nominal interest rate to take into account expected inflation. One plus the real interest equals the ratio of 1 plus the nominal interest rate, divided by 1 plus the expected rate of inflation (1+rt) = (1+it)/(1 + πet+1) When the nominal interest rate and the expected inflation are not too large: rt = it - πet+1 The real interest rate is (approximately) equal to the nominal interest rate minus expected inflation.

Constant interest rates and payments forever

$Vt = $z/i

Present value w/ constant interest rate

$Vt = $zt + [1/(1+it)]$zet+1 + [1/(1+it)^2]$zet+2+ ... the present value is a weighter sum of current and expected future payments, with weights that decline geometrically through time.

Implications in short run and medium run

Higher money growth leads to lower nominal interest rates in the short run but to higher nominal interest rates in the medium run. Higher money growth leads to lower real interest rates in the short run but has no effect on real interest rates in the medium run. In the medium run, the real interest rate returns to the natural interest rate, rn. It is independent of the rate of money growth.

Expected present discounted value

Value of a sequence of future payments that is the value today of an expected sequence of payments. 1 dollar this year is worth 1+it dollars next year. 1 dollar next year is worth 1/(1+it) dollars this year 1/(1+it) is the present discounted value of 1 dollar next year. 1/(1+it) - discount factor one-year nominal interest rate, it, is sometimes called the discount rate. Because the nominal interest rate is always positive, the discount factor is always lass than 1: A dollar next year is worth less than 1 dollar today. The present discounted value of 1 dollar two years from now is equal to 1/(1+it)[(1+it)(1+it+1)] dollars.

Fisher effect/ hypothesis

In the medium run, the nominal interest rate is equal to the natural real interest rate plus the rate of money growth. So an increase in money growth leads to an equal increase in the nominal interest rate. i = rn + gm Evidence found in study of eight Latin American countries - expected inflation and nominal interest rates hold on average, they are not close to one another in any one country at any one time.

Real interest rates

Interest rates expressed in terms of a basket of goods.

Nominal interest rates

Interest rates expressed in terms of dollars (or, more generally, in units of the national currency).

Looking at the IS-LM model

IS Firms, in deciding how much investment to undertake, care about the real interest rate: Firms produce goods. They want to know how much they will have to repay, not in terms of dollars but in terms of goods. Y= C(Y-T) + I(Y,r)+G LM Demand for money is dependent on the nominal interest rate. The opportunity cost is what they give up by holding money rather than bonds. The opportunity cost of holding money is equal to the difference between the interest rate from holding bonds minus the interest from holding money. M/P = YL(i)

General formula of EPDV

Present discounted value of sequence of payments: $Vt = $zt + [1/(1+it)]$zt+1 + {1/(1+it)[(1+it)(1+it+1)]}$zt+2+ ... The more distant the payment, the smaller the discount factor, and thus the smaller today's value of that distant payment. Actual decisions based on expectations of future payments rather than on actual values for these payments. $Vt = $zt + [1/(1+it)]$zet+1 + {1/(1+it)[(1+it)(1+it+1)]}$zet+2+ ...

Implications from relation between nominal and real interest rate

When expected inflation = 0, the nominal and the real interest rates are equal. Because expected inflation is typically positive, the real interest rate is typically lower than the nominal interest rate. For a given nominal interest rate, the higher the expected rate of inflation, the lower the real interest rate.

Other terminology

ex-ante - the real interest rate ex-post - the realized real interest rate Despite the large decline in nominal interest rates, borrowing was actually more expensive in 2006 than it was in 1981. This is due to the fact that inflation (and, with it, expected inflation) has steadily declined since the early 1980s.