Lecture 2.2 The meaning of interest rates
Before we can proceed further with the study of money and banking, it is useful to obtain a more precise understanding of what the term ‘interest rates’ means. In this lecture, it is shown that a concept known as the yield to maturity is one of the most useful measures of the interest rate.
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Source: Mishkin Ch 4
Learning objectives
Distinguish the four main types of interest rate structure in a loan contract
Show how to calculate the present value of future cash flows and the yield to maturity for each type
Recognize the distinctions between yield to maturity, current yield, rate of return, and capital gain
Interpret the distinction between real and nominal interest rates
Preliminary observation …
In normal times, interest rates are materially above zero, because people prefer money now over money later
Currently interest rates are close to zero for policy reasons
The examples we will be studying generally assume that interest rates are positive
Measuring interest rates
Present value: a dollar paid to you one year from now is less valuable than a dollar paid to you today.
Why: (1) a dollar deposited today can earn interest and become $1 x (1+i) one year from today. (2) In not having access to $1 today, you are constraining your opportunities to spend in the next year.
Note: Convention used in this analysis is that the interest rate i is expressed as a decimal (thus, a 10 percent interest rate is 0.10).
Present value: a simple example
$100 today is worth …
Equivalently …
A future payment of $110 in a years’ time is worth (has a present value of) $100 today
So, for a one-year payment:
Present Value = cash flow/(1+i)
Simple present Value: Definition and example
Example: I am due to receive a cash payment of $1000 in a year’s time. The interest rate is 3%.
The present value of this future payment is $1000/(1.03) = $970.87
PV (present value)
100/(1+i)2
100/(1+i)n
Why the present value calculation is useful: we cannot directly compare payments scheduled at different points in the timeline
Simple present value, continued
Four types of credit-market instruments
The concept of present value applied to any type of credit market instrument.
There are four types we will consider:
Simple loan (borrow money and repay with interest)
Fixed-payment loan (interest and repayment combined in a series of equal payments)
Coupon bond (regular coupon payment, then repayment of face value)
Discount bond (zero-coupon bond)
Yield to maturity of a debt security
Yield to maturity: the interest rate that equates the present value of the total cash flow payments received from a debt instrument with its value today (present value = price)
In a security market, we know the stream of future payments (CF)
If we know PV, we can infer i
alternatively, if we know i we can calculate PV
Stream of future payments (CF)
Present value of future payments (PV)
Interest rate (i)
Yield to maturity on a simple loan
Fixed-payment loan (most housing loans are of this type)
Coupon bond
Holder receives a stream of fixed coupon payments (C) then repayment of face value of the loan (F) at maturity
Most government bonds are of this type
Re: comment on next slide.
If F=1 and C=i, then P=1.
Coupon bond
When the coupon bond is priced at its face value, the yield to maturity equals the coupon rate (C/F).
The price of a coupon bond and the yield to maturity are negatively related.
The yield to maturity is greater than the coupon rate when the bond price is below its face value.
See formula from previous slide.
Coupon bond, continued
Consol or perpetuity: a bond with no maturity date; it does not repay principal but pays fixed coupon payments forever
For coupon bonds, this equation gives the current yield, an easy-to-calculate approximation to the yield to maturity.
Discount bond (zero-coupon bond)
A security has a face value of $1000
It matures in 1 year’s time
It is trading in the market at a price of $980
Then i = (1000 – 980)/980 = 2.04%
The distinction between interest rates and returns
Concept: If you sell a security before it matures, the price may have changed
Rate of return: coupons earned plus change in price during the period held
The distinction between interest rates and returns
The return equals the yield to maturity only if the holding period equals the time to maturity.
A rise in interest rates is associated with a fall in bond prices, resulting in a capital loss if time to maturity is longer than the holding period.
The more distant a bond’s maturity, the greater the size of the percentage price change associated with a given interest-rate change.
Implication: long-term bonds are a more risky investment than short-term bonds
Relationship between price, yield and term to maturity
(stylised representation)
Stream of future payments
Security price
Security price
Security price
Distinction between interest rates and returns, continued
When rate of interest on new bonds goes up to 20%, old bond’s secondary price (price of a 10% coupon rate bond) changes to make the yield on the old bond 20%:
Maturity and volatility of bond returns: interest-rate risk
Prices and returns for long-term bonds are therefore more volatile than those for shorter-term bonds.
There is no interest-rate risk for any bond whose time to maturity matches the holding period. (But there is still an opportunity cost/gain if the market yield changes during that period
If the price falls after you buy the bond, it means you could have bought it more cheaply
Important implication: today’s market price depends on expected future price (subject of a future lecture)
Distinction between real and nominal interest rates
Nominal interest rate is simply the actual reported interest rate: that is, without any adjustment for inflation.
Real interest rate is adjusted for changes in price level so it more accurately reflects the cost of borrowing in terms of purchasing power
Ex ante real interest rate is adjusted for expected changes in the price level.
Ex post real interest rate is adjusted for actual changes in the price level.
Simple example of a real interest rate calculation
Nominal interest rate (R) = 5%
Term to maturity = 1 year
Expected one-year inflation rate when loan is made = 3.5%
Actual inflation during the life of the loan = 3%
Ex ante real interest rate (r) = 1.5%
Ex post real interest rate (r) = 2%
Rationale: inflation reduces the real value (purchasing power) of the required loan repayment
Fisher equation
Real and nominal interest rates, United States
The real cash rate in Australia is currently below zero
Let i = .10
In one year: $100 X (1+ 0.10) = $110
In two years: $110 X (1 + 0.10) = $121
or $100 X (1 + 0.10)2
In three years: $121 X (1 + 0.10) = $133
or $100 X (1 + 0.10)3
In n years
$100 X (1 + i)n
Let i = .10
In one year: $100 X (1+ 0.10) = $110
In two years: $110 X (1 + 0.10) = $121
or $100 X (1 + 0.10)
In three years: $121 X (1 + 0.10) = $133
or $100 X (1 + 0.10)
In n years
$100 X (1 + i)
PV = today’s (present) value
CF = future cash flow (payment)
= the interest rate
PV = amount borrowed = $100
CF = cash flow in one year = $110
= number of years = 1
(1 + ) $100 = $110
= 0.10 = 10%
For simple loans, the simple interest ra
yield to maturity
The same cash flow payment every period
throughout
the life of the loan
LV = loan value
FP = fixed yearly payment
= number of years until maturity
LV = . . . +
1 + (1 + )(1 + )(1 + )
Using the same strategy used for the fix
ed-payment loan:
P = price of coupon bond
C = yearly coupon payment
F = face value of the bond
= years to maturity date
P = . . . +
1+(1+)(1+)(1+)(1
= nominal interest rate
= real interest rate
= expected inflation rate
When the real interest rate is low,
there are greater incentives to borrow a
nd fewer incentives to lend.
The real inter
est rate is a better indicator of the in
centives to
borrow and lend.
201720122007200219972022 -4 -2 0 2 4 6 8 % -4 -2 0 2 4 6 8 % AustralianCashRateand90-dayBillYield 90-daybillyield Realcashrate* Cashrate target * Calculatedusingaverageofyear-endedweightedmedianinflation andyear-endedtrimmedmeaninflation. Sources:ABS;AFMA;ASX;RBA
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