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ECON 310 009 Money & Banking J. Scott Sperling Summer 2001 [email protected]
Interest Rates: Calculation†
1 Simple and Compound Interest∗
Before we get into “the” interest rate, I just want to take a couple of minutes and play with interest rates.
We begin with some common terms and calculations from the realm of fixed-income investments. The amount of the investment is called the principal. The “fixed-income” from the investments is called interest. The interest per unit of principal per unit of time is called the interest rate.
Most commonly interest rates are quoted in dollars per year per dollar of principal. These units can be written: $/(y$). The dollar units cancel, so this interest rate has units of one over years. Similarly, if the interest rate is apples per day per apple borrowed, the apple units will cancel, and the units of the interest rate will be one over days. In general, the units of an interest rate are one over some unit of time. When the unit of time is a year, we say that an interest rate is an annual interest rate.
If the unit of time is not mentioned, then it will almost always be an annual interest rate. Interest rates that are quoted in some specific unit of time can be converted to any other unit of time via a simple linear transformation. For example, a daily interest rate of x% corresponds to an annual interest rate of (365)(x)%.
Example:My credit card has an APR of (annualized percentage rate) of 16.8%. What is the daily interest rate? R =
Under simple interest the interest is earned on the amount of the principal only. In this case, after n years the value of the investment will be:
Vs(n) = RP n + P, (1)
where V is the value of your investment, s indicates that we are calculating simple interest, P is the principal of a fixed-income investment and R is the annual rate of interest.
For example, suppose you make an investment of $5000 at a 4.5% simple annual interest rate. After two years the value of your investment will be:
Vs(2) = (0.045)($5000)(2) + $5000 = $5, 450. †If you find any errors or have any questions about these notes, please email me at [email protected]. ∗Once again, special thanks to Andrew Sellgren.
It is much more common for interest to be compounded annually. In this case, at the end of each year, that year’s interest will be added to the principal, so the investment will earn interest on the interest. The first year will be just like simple interest, since none of the interest will yet be compounded. Accordingly, the value after the first year will be (where a denotes that we are compounding annually): Va(1) = RP + P = (1 + R)P.
After the second year, the value will be:
Va(2) = RVa(1) + Va(1) = R(1 + R)P + (1 + R)P = (1 + R)^2 P.
Similarly, after n years, the value will be:
Va(n) = (1 + R)nP. (2)
Of course, this formula works only on integral numbers of years. For non-integral numbers, you round down to the nearst year n, compute Va(n), and use that in the simple-interest formula (1) for the fraction of the last year.
Using the previous example, you invest $5000 at a 4.5% annual interest rate, but this time interest compounds annually. After two years the value of your investment will be:
Va(2) = (1 + 0.045)^2 ($5000) = $5460. 13.
Notice that the investment is worth less under simple interest than under compound interest, since under compounding you earn about $10 of interest on the first year’s interest.
The above reasoning for compounding annually applies to compounding more frequently. The only catch is that the interest rate needs to be quoted in terms of the same time interval as the compounding. If R is an annual interest rate, and interest is to compound t times per year, then the value of an investment after n years will be:
Vt(n) =
( 1 +
R
t
)tn P. (3)
We return to our example again, this time supposing that interest compounds daily. After two years, the value will be:
V 3 65(2) =
( 1 +
)(365)(2) ($5000) = $5470. 84.
As we compound more and more frequently, we arrive at the expression for continuous compounding:
Vc(n) = lim t→∞
( 1 +
R
t
)tn P = eRnP.
Returning for the last time to the example, assuming continuous compounding, after two years the value of the investment will be:
Vc(2) = e(0.045)(2)($5, 000) = $5, 470. 87.
When economists say “interest rate” they are generally talking about the yield to maturity, because they consider it the most accurate measure of interest rates.
3.1 Calculating the Yield to Maturity
3.1.1 Simple Loan
Suppose you make a $100 loan to your neighbor and a year from now you receive a repayment of $110. Let i denote the yield to maturity. The present value of future payments must equal today’s value of the loan (see footnote 1), so:
1 + i solving for i, $100(1 + i) = $ $100 + $100i = $ $100i = $110 − $ i =
Notice: for simple loans, the simple interest (10% in the above example) equals the yield to maturity.
3.1.2 Fixed-Payment Loan
To find the yield to maturity of a fixed payment loan, all we need to do is, by definition, sum the present value of all future payments using the equation for present value (Equation 4).
Let LV =the loan value, F P =the fixed yearly payments and n =the number of years to maturity. At the end of year one you make a payment of F P and the present value is F P/1 + i, at the end of year two the present value of your payment is F P/(1 + i)^2 and so on. So:
LV =
F P
1 + i
F P
(1 + i)^2
F P
(1 + i)^3
F P
(1 + i)n^
Notice that it would take a lot of work to solve this equation for i, which is why there are present value tables which do the work for you (or you could also program a calculator or use a financial calculator to solve for i).
3.1.3 Coupon Bond
So let’s keep doing the same thing as above, all you have to do to find the yield to maturity for a coupon bond is sum the present value of each of the coupon payments plus the present value of
the face value of the bond at maturity (remember, you get paid the coupon rate every year plus when the bond expires you get the face value back). Let P =the price of the coupon bond, C = the yearly coupon payment, F = the face value of the bond, and n = years to maturity date, then:
P =
C
1 + i
C
(1 + i)^2
C
(1 + i)^3
C
(1 + i)n^
F
(1 + i)n^
Again, it would be difficult to solve this equation for i, but there are tables and calculators to help you find the solution. To give you an idea, Table 1 lists the yield to maturity on a 10-year 10% coupon bond with a face value of $1000.
Bond Price i $1200 7.13% 1100 8. 1000 10. 900 11. 800 13.
Table 1: Yield to maturity on a 10-year, 10% coupon bond, par = $1000.
There are three very important facts that you can see from Table 1:
When the bond price equals par (equals face value), yield to maturity is equal to the coupon rate.
There is a negative relationship between the bond price and the interest rate (yield to maturity), i.e. ↑ P →↓ i.
The yield to maturity is greater than the coupon rate when the bond price is below par.
3.1.3.1 Consols There is a special type of coupon bond called a consol (aka a perpetuity) which is a bond with no maturity date and no repayment of principal, but pays a fixed coupon payment forever. The formula is quite simple, because it reduces Equation 6 to:
P =
C
i
rewritten in terms of i:
i =
C
P
Notice again, as i increases, P decreases.
Notice that this is the same as Equation (8) for consols. Therefore, the current yield for a consol is the same as its yield to maturity. When a coupon bond has a long maturity, it is very much like a consol and thus the current yield is a good approximation of the yield to maturity. However, when the term to maturity is short, the current yield becomes a worse approximation of yield to maturity. You should also note that as the price of the bond gets closer to the par value, the current yield also becomes a better approximation of yield to maturity. In fact, when the price of the bond is equal to its par value, the current yield is the yield to maturity. Note also that the current yield is also inversely related to price and when the current yield increases so does yield to maturity.
3.2.2 Yield on a Discount Basis
Yield on a discount basis applies to discount bonds, most notably U.S. Treasury bills. It is calculated as follows:
idb =
F − P
F
×
days to maturity
where idb = yield on a discount basis, F = face value of the discount bond, and P = purchase price of the discount bond. Note that because this formulation uses percentage gain on the face value and considers the year to be 360 days, it underestimates the yield to maturity. This underestimate is greater the longer the days to maturity. Finally, as with the current yield, yield on a discount basis and yield to maturity are positively related.
3.3 Rate of Return
Interest rates are not the whole story when it comes to figuring out how well you do when holding a security. In order to find this out we want to find out the rate of return. The rate of return is the payments to the owner of a security plus the change in its value. Note, the return on a bond will not necessarily be equal to the interest rate on that bond. Generally, the rate of return from time t to time t + 1 is expressed as:
RET =
C + Pt+1 − Pt Pt RET = return from holding the bond fromttot + 1 Pt = price of the bond at timet Pt+1 = price of the bond at timet + 1 C = coupon payment.
Notice that this equation can be split into two separate terms:
RET =
C
Pt
Pt+1 − Pt Pt
C/Pt should look familiar, it is ic, the current yield (Equation (11)). The second term is the rate of capital gain, i.e. the change in the bond’s price relative to the purchase price.
There are a few key things you should know about the relationship of the rate of return and the interest rate:
- Only bonds with time to maturity equal to their holding period have a rate of return equal to the yield to maturity.
- If interest rates rise (prices fall), returns fall (if holding period is shorter than term to matu- rity).
- The more distant the maturity of a bond, the greater the size of the percentage change in price associated with an interest-rate rise (i.e. its more volatile).
- The more distant the maturity of a bond, the lower the rate of return when interest rates rise.
- A bond’s returns can be negative if interest rates rise.
Prices and returns for long-term bonds are more volatile than those of short-term bonds. This means that there is more risk in long-term bonds. When risk is affected by changes in the interest rate, we call it interest-rate risk (we will talk about this in more detail later).
3.4 Real and Nominal Interest Rates
Lastly, a quick word on real versus nominal interest rates. We have been discussing nominal interest rates so far. If we want to talk about the “true” cost of borrowing, however, we need to use the real interest rate. The real interest rate is the nominal interest rate adjusted for inflation. Formally:
ir = i − πe. (14)
In words, the real inflation rate is equal to the nominal interest rate minus expected inflation. Why expected inflation? If you are holding a one-year bond, you will want to know what your real interest rate is when the bond expires, a year from now. You can’t know what the inflation rate will be in a year, so you use expected inflation. Suppose you a have bond which yields 5% and you expect inflation to rise 2% by the end of the year, then the real interest rate on your bond is ir = 5% − 2% = 3%.
When the real interest rate is low, it is less costly to borrow money, so there are greater incentives to borrow and fewer incentives to lend.
