Interest Rates



Understanding Interest Rates

One period real interest rate (in terms of goods)

[pic]

This tells us that one unit of commodity to be received tomorrow is less valuable to you than one unit of commodity received today. You will be just indifferent between one unit of goods today and 1+rt units of goods tomorrow, i.e., the discounted present value of 1+rt units of goods tomorrow is equal to one unit of goods today.

Thus, the real interest rate is the intertemporal price of commodities. It affects the intertemporal choices of consumers, workers and firms.

One period nominal interest rate (in terms of money)

[pic]

Again, this says that $1 of cash flow to be received tomorrow is less valuable to you than $1 received today. Thus, the discounted present value of cash flow $(1+it) tomorrow is equal to $1 today.

Thus, we have

[pic]

where [pic]

Then,

[pic].

[pic]

Since [pic] is small, the approximate measure is

[pic]

Since we can not observe [pic] at time t, therefore we have to make a forecast on the value of [pic], that is [pic].

Thus, we have

[pic]

This is Fisher Equation.

What we observe at time t is only the nominal rate [pic], which takes account of the expected inflation rate and the ex ante real interest rate.

The real rate of interest may reflect the desired real rate by the lender (due to time preference), or the real borrowing cost by the borrower (marginal productivity).

How do we measure the expected inflation rate?

1) Survey

Median Expected Price Change Next 12 Months

[pic]

Source: Survey of Consumers, University of Michigan

2) Treasury Inflation-Protected Securities (TIPS)

Treasury Inflation-Protected Securities (TIPS), introduced in 1997, are bonds issued by the US Treasury which provide investors with protection against inflation. Its returns are indexed to inflation, thus promising bondholders a sure real return over the life of the bond. Unlike nominal Treasury bonds, the real return of a TIPS is fixed at auction time. The face value and the coupon payments are indexed to the inflation rate.

An example: You purchase a $10,000 TIPS bond at 3% today. Note that the TIPS auctioned at 3% reflects the expected inflation rate at this time, which is supposed to be zero.

In year 1, you receive two semi-annual payments totaling $300.

Suppose at the end of year 1, the CPI is 5%. So the principal of the bond goes up by 10% to $10,500.

In year 2, the total interest payments will be 3% of the new principal, or $315.

When the TIPS matures, you will receive either the current principal value or the original TIPS amount whichever is higher.

TIPS are useful in determining the expected inflation. The difference between the yield of normal Treasury bonds and the yield of TIPS is a good indicator of the market's inflation expectation.

[pic] = Expected inflation rate = yields of the 10-year Treasury bonds - yields of TIPS

[pic]

Suppose the 10-year nominal rate for a nominal T-bond is 4.4%, and a TIPS guarantees a 2.6% real return. This implies a TIPS-derived expected inflation rate of 1.8%.

However, there are two factors that affects the accuracy of this measure [pic]:

1. TIPS provide insurance from inflation risk ([pic] over-estimates the true expected inflation rate)

Nominal bond holders should be compensated for inflation risk, say, 0.5%. This says that TIPS derived expected inflation rate adjusting for inflation risk premium should be (4.4% - 0.5%) - 2.6% = 1.3%.

2. TIPS are less liquid ([pic] under-estimates the true expected inflation rate)

Imagine that if TIPS is as liquid as Treasury bonds, then the investors will be willing to hold TIPS at a rate lower than 2.6%. Suppose the liquidity premium is 0.9%. Thus, the TIPS-derived expected inflation rate adjusting for liquidity premium should be 4.4% - (2.6% - 0.9%) = 2.7%.

Thus, the TIPS-derived expected inflation rate adjusting for both inflation risk premium and liquidity premium should be 1.8% - 0.5% + 0.9% = 2.2%.

[pic][pic]

D’Amico, Kim, and Wei (2007)

[pic]

The ex post real rate of interest depends on the realized inflation rate:

[pic]

For example, given a quote [pic], and the public forecast [pic], lenders and borrowers are willing to lend and borrow at a real rate of 7% which is based on their considerations on time preference and real cost of borrowing, respectively.

If the realized inflation rate is higher than expected, then the realized rate is lower than 7%. That is, the real cost of borrowing by the borrowers and the real purchasing power of the sum received by the lender is lower than expected. If realized inflation is very high, say [pic], the real interest rate becomes negative.

Evidence shows that nominal rates and real rates do not co-move, suggesting that the realized inflation rate often turned out to be very different from what the public had anticipated.

[pic]

Real and Nominal Interest Rates (Three-Month Treasury Bill), 1953–2008

The Measures of Interest Rates

Yield to maturity (殖利率)

The measure of yield to maturity (YTM) is based on calculating the discounted present value (PV) of the proceeds generated by a security.

Yield to maturity is defined to be the interest rate that equates the PV of payments received from an asset with its value today (price).

Note that the price of the security is determined by the demand and supply of the market.

Consider 4 types of credit market instruments

1. Simple loans: Bank commercial and industrial (C&I) loans, consumer loans.

P = [pic], where P = the instrument’s value (price) today, CF = future cash flow (including the principle).

Example: you borrow from a bank $100 today and are contracted to repay $110 one year from now.

Thus, the current value (price) of the loan contract is $100, and the cash flow includes the principle $100 and interest payment $10. This implies that the YTM satisfies $100 = [pic], i.e., i=10%.

2. Fixed-payment loans: mortgages, auto loans.

Example: you purchase a new home and take a $1,000,000 mortgage which lasts 10 years. You are required to pay $9,500 each month. What is the implied interest rate? The YTM implied by this mortgage satisfies

$1,000,000 = [pic].

Note that the interest rate computed above is a monthly rate. You have to multiply it by 12 to obtain the annual rate.

If the banker tells you that the annual mortgage rate is 8%, how much should you pay each month for the next 10 years?

$1,000,000 = [pic]

3. Coupon bonds: Long term Treasury bonds, corporate bonds. See the article三大債券利率狂飆.doc.

Example: You purchase a 5-year corporate bond, with a face value F=$1,000 and annual coupon payment c=$100. Suppose the value (price) of the bond sold today P= $1,000. Then, the YTM implied by the bond satisfies

$1,000 = [pic]

It is easy to see that i=10%. This is because the price of the bond today is equal to the face value, P=F.

The coupon rate is defined to be

[pic]=10%.

Note that in this case, when P=F, then i=c.

You can check that if P=1,100 > F, then i=8.48% < c; and if P=990 < F, then i=10.25% > c.

We conclude that the price of bond and interest rate are negatively correlated.

The Consol: A Coupon Bond without Maturity

A consol is a perpetual bond that was issued in 1749 by the British government to convert a number of outstanding debt issues into a promise to pay 3% interest to the note holders forever.

The holder of a 100 pound consol today receives 2.5 pounds a year in interest. Since it will never be redeemed, the holder can only sell it to another party at a price that depends on the perceived average long term interest rate.

The long life of the consol provides a consistent history of long-term interest rates in Britain. During the Napoleonic wars when the survival of the country was at stake, the rate reached 6.3%. After the crisis, the rate declined almost continuously throughout the nineteenth century. The price reached a high point in 1896-1898 when the consol yielded only 2.25%. The consol is sometimes referred to as a “stock”, perhaps because of its similarity to the preferred stock of a blue chip company.

Example: Suppose a $100 consol pays $2.5 per year. The YTM of a consol satisfies

$100 = [pic]

This says that i=2.5%. In general, P = [pic]. That is

[pic].

Historically, when calculators were unavailable, the measure C/P was used to approximate the YTM, which was called the current yield ([pic]).

We have two observations:

(1) As a coupon bond has longer maturity, its interest rate is close to C/P, thus the current yield is a good approximate of the YTM.

(2) As the price of a coupon bond is close to its face value, then the YTM is close to the coupon rate. Since

[pic].

Again, in this case, the current yield is a good approximate of the YTM.

4. Discount bonds: Short-term commercial papers, Treasury bills, 中央銀行定期存單 (CD or NCD; 30, 91, 182天期)

Short-term financial instruments are issued at a discount. There are no coupon payment during the life of the instrument. The difference between the face value and the price of instrument is interest payment.

Example: A bank issues a 6-month commercial paper with face value F=$1,000 today, sold for P=986. The YTM implied by the instrument satisfies

[pic].

Thus, i=1.42%. Note that this is semi-annual rate. The annual yield of the commercial paper is i=2.84%. During the sub-prime crisis the bank urgently need cash and can only sell newly-issued 6-month commercial paper for P=$955. What is the implied yield for the buyer? Again, i=4.712%. The annual yield of the buyer of this commercial paper is i=9.424%.

Rat of Returns

[pic],

where the rate of return of a coupon bond can be expressed as the sum of the current yield and the rate of capital gain (loss).

The rate of return equals the YTM only if the holding period equals the time to maturity, i.e., you do not sell the bond before it matures.

Example: You purchase a 2-year bond with face value F=1,000 and annual coupon payment $100 at a price P=1,100.

a) The implied TYM is [pic]. That is i=4.6%. (You can check that if you do not sell the bond before the bond matures, your rate of return is

[pic]=4.6%,

if i=4.6%.)

b) Suppose after one year, after receiving a coupon payment $100, you need cash and sell the bond for P=$900. What’s your rate of return for holding the bond for one year?

[pic].

c) What is the rate of return for the buyer of the bond when the bond matures?

[pic].

Volatility of Bond Returns

Prices and returns for long-term bonds are more volatile than those for shorter-term bonds. Therefore, longer-term bonds have a higher interest-rate risk.

[pic]

One-Year Returns on Different-Maturity 10%-Coupon-Rate Bonds When Interest Rates Rise from 10% to 20%

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