Learning Unit #9

Interest Rates

Objectives of Learning Unit #9

• Yield to Maturity• Rate of Return• Real and Nominal Interest Rates

Yield to Maturity

• Yield to maturity: The interest rate that equates the present value of payments received from a debt instrument with its value today.– Given cash flows in future from a debt instrument such as

coupon bond and fixed payment loan, a present value of the cash flows can be computed with a choice of an interest rate. Depending on your choice of interest rate, the present value could be high or low.

– If the present value is exactly equal to the value (price) of the debt instrument today, then that interest is the yield to maturity.

Yield to Maturity as Interest Rate

• There are many interest rate formula available. Only “yield to maturity” is considered as the interest rate in economics and finance.

• In finance, the yield to maturity is called “internal rate of return.”

• All other formula to compute interest rates are approximations of actual interest rate and they are NOT the interest rate!

• Remember when we say “interest rate,” it means the yield to maturity!

Example of Yield to Maturity

• You borrow $100 today and promise to pay back $110 one year later.– A future cash flow is $110 one year later.– A value of instrument (simple loan) is $100 today.

• Cash flows on timeline should look like

0 1

$100

$110

Example of Yield to Maturity

• A present value of future cash flow is

iPV

1110$

• Equate the present value to the simple loan today:

iPVtodayValue

1110$100$

• Solve for i: i = 10%

Another Example of Yield to Maturity

• You borrow $100 today and promise to pay back $121 two years later. Cash flows on timeline looks as

0 1 2

$121

$100

• Equate a present value of cash flow with the value today.

2)i1(121$PV100$TodayValue

• Solve for i: i = 10%

Yield to Maturity Formula

• Formula #4: i = (FV/PV)1/n - 1

This formula can be applied only on simple loan or discount bond where there is only one cash flow in future.

• Ex. A 2-year simple loan with $100 principal and $121 payback.

Here, PV=$100, FV=$121, and n=2, so the yield to maturity is

%101.01100$121$i

21

Yield to Maturity of Bonds• When there are more than one cash flows in future,

you cannot use Formula #4.• Instead, you may use a business calculator, Microsoft

Excel, or simply try-and-error.

• Ex. A coupon bond with 5% coupon rate and 2 year maturity is priced at $950 today.

• Cash flows on timeline look like

0 1 2

$50 $1,050

$950

Yield to Maturity of Bonds

• Equate a present value of future cash flows with a value of bond today:

2)i1(1050$

i150$PV950$todayValue

• Solve for i: i = 7.8%

• If you have a business calculator, simply provide cash flows including today’s value and year, then compute for IRR (internal rate of return).

Trial-and-Error MethodYou can find an approximate yield to maturity by trial-and-error method.– First, use the formula of current yield to make an initial guess: current yield

= $50/$950 = 5.2%– Second, use the rate of return formula to make another guess: rate of return

= ($50 + $1050 - $950)/$950 = 15.8%– An actual yield to maturity is between these two numbers, so let’s start with

i = 10% (middle point). Then, PV = $913.22 < $950 = value today, it is too small. To make PV higher, i must be lower (an inverse relationship).

– Choose i = 7%, then PV = $963.84 > $950, so i is little too small, and i should be little greater than 7%.

– Choose i = 8%, then PV = $946.50 < $950, it is very close, and i should be little less than 8%.

– Choose i = 7.8%, then PV = $949.93 ≈ $950, it is a good approximation!– So, an approximate yield to maturity is 7.8%.

Using Microsoft Excel

You can use Microsoft Excel to compute a yield to maturity.– First, input cash flows: -$950, $50, $1050

Notice that the value today is negative!– Then, use IRR function under “Financial formula”

where you select three numbers (cells) within the formula.

– You should get 7.796% as a result.

Other Measures of Interest RatesThe yield to maturity is the most accurate measure of interest rates. Two approximation measures of interest rates are• Current Yield• Yield on A Discount Basis

Current Yield• You cannot use the yield to maturity formula #4 on

any debt instrument with more than one future cash flows.

• Instead, you can use the current yield to approximate a yield to maturity.

Formula #5 (Current Yield): i = C/P C: Annual coupon payment

P: Price of console today

• Current yield is an approximation, it is NOT a yield to maturity or the interest rate.

Console• Console: A perpetual coupon bond with no maturity.

A console has a face value and a coupon rate like a regular coupon bond. Its issuer pays a fixed annual coupon payment each year and forever.

• Console were issued and traded in U.K. many many years ago. No one issues console anymore.

• The longest maturity of Treasury security is 30 years, while the longest maturity of corporate bonds is 100 years. Who wants to have console? Who can promise to pay forever?

Console and Current Yield• For a perpetual fixed payment like console, the

current yield is exactly equal to the yield to maturity. So, you can use the current yield formula to compute a yield to maturity on console.

• A console is priced at $1000 today and provides $100 annual coupon payments forever. The yield to maturity of the console is its current yield, that is,

%101000$100$i

• This is a special case. For any other fixed maturity bonds (even with 100 years maturity) or variable cash flows, you cannot use the current yield formula to compute the yield to maturity!

Console alike in the U.S. • Console is not traded in the U.S., so why is it useful

to know?• There are some financial instrument which acts like

console. Do you know any financial instrument which does not have maturity and promises to pay a fixed amount each year?

• Preferred stocks issued by regulated monopoly such as Duke Power.– Preferred stocks like common stocks do not have maturity.– Preferred stocks promise to pay a fixed dividend as long as

issuing corporations make profits.– Regulated monopoly like Duke Power is almost guaranteed

to make profits each year by the government.– Utility companies like Duke Power is expected to continue

its business almost forever.

Current Yield on Preferred Stock

• Because some preferred stocks act like console, you can use the current yield formula to compute a yield to maturity on the preferred stocks.

• Ex. A preferred stock costs $90 today and pays $6 dividend every year. The yield to maturity on the preferred stock is

i = $6/$90 = 6.7%

Yield on A Discount Basis

• Yield on a discount basis is used to approximate an interest rate on U.S. Treasury bills.

• U.S. Treasury bills were first issued in 1929. How could traders know their interest rates? Did they have business calculators or Microsoft Excel? Of course not. They used the yield on a discount basis to find quickly approximate interest rates with paper and pen!

Formula of Yield on A Discount Basis

maturitytodays360x

FPFi

F: Face value P: Purchase price

Formula #6:

• U.S. Treasury bills have maturities of 1 month (4 weeks), 3 months (13 weeks), 6 months (26 weeks), and one year (52 weeks). How many days in one month, 3 months, 6 months, or one year approximately?

• U.S. Treasury bills has face values of $100, $1,000, or $10,000 (sold in increments of $100).

Example of Yield on A Discount Basis

%95.4364360x

000,10$500,9$000,10$i

• Ex. A T-bill has one year maturity, $10,000 face value, and is sold at $9,500.

• Note that a number of days to maturity is 364, which is 52 weeks (one year T-bill).

Yield to Maturity and Saving

• As the yield to maturity formula takes into account all cash flows from a debt instrument, it implicitly assumes that a buyer of the instrument to hold it until its maturity and to receive all cash flows.

• The yield to maturity formula is useful to evaluate a return an investor may receive from a security if he holds it until maturity.

• However, if an investor sells the debt instrument before its maturity, the yield to maturity formula will not tell how much return he gets from the debt instrument.

Rate of Return

• If an investor sells a security before its maturity, he can evaluate how much return he earned by holding the security for a given period by using “Rate of Return” formula.

• The rate of return takes into account both purchase price and sales price of a security and any payments between.

Formula of Rate of Return

C + Pt+1 - Pt R = —————— x 100 Pt

R: Rate of ReturnPt: Price of bond in year tPt+1: Price of bond in year t+1C: Total coupon payment between year t and

year t+1

Formula #6:

Use of Rate of Return Formula• The rate of return formula is often used to

evaluate a return from investment on a particular security by a saver.

• In general, in year t, a saver purchased a security at Pt and in t+1 (any time after year t) he sells it at Pt+1.

• For example, a saver purchased a security at $950 (Pt) last year (t), has received $80 annual coupon payment (C), and sells it at $980 (Pt+1) today (t+1).

Example of Rate of Return

You bought a bond at $990 last year, received $50 annual coupon payment, and just sold at $1,000 today.

R = (50+1,000-990)/990 x 100 = 6.06%

Components of Rate of ReturnThe rate of return formula can be decomposed into two terms (sources of returns):

C + Pt+1 - Pt C Pt+1 - PtR = —————— = ——— + ———— Pt Pt Pt

• The first term C/Pt is a current yield.It tells a part of return coming from coupon payments.

• The second term Pt+1-Pt/Pt is a rate of capital gain.It tells the other part of return coming from a change in a price of security over time. If there is no change in price, this term will be zero, and a saver gains only from coupon payments.

Example of Decomposition of Rate of Return

You bought a bond at $990 last year, received $50 annual coupon payment, and just sold at $1,000 today.

R = 50/990 + (1,000-990)/990 = 5.05% + 1.01% = 6.06%• Of 6.06% rate of return, 5.05% comes from

coupon payment, while 1.01% comes from an increase in price of bond over one year period.

Capital Gain

• Capital gain: You sell a security at higher price than the price at which you bought.

• Rate of capital gain: (Pt+1 - Pt)/Pt

• Ex. You bought a bond at $990 last year, received $50 coupon payment, and just sold at $1,000 today.

RCG = (1,000-990)/990 = 1.01%

Capital Loss

• You may not always have a capital gain. Often a price of financial instrument or asset falls, then you loose its value and end up to sell at well lower price than what you paid for (Capital loss).

• Will you get a capital gain or capital loss from your textbook when you sell it? How about your car?

• Often, prices of financial instruments such as stocks and bonds fall over a certain period of time and investors are suffered from capital losses.

Rate of Return on Investment• You can use the rate of return formula on any

financial instrument and assets.– Example: Stock – Michelle purchased Apple stock

at $396.75 on August 1, 2011 and sold it at $614.32 on July 19, 2012, and received no dividend over one year period.

– Example: House – Mr. Jones purchased a house at $100,000 in 1991 and sold it at $158,000 this year.

– Example: Textbook – you purchased a textbook at $120 at the beginning of the semester and plan to sell it at $40 at the end of the semester.

Internal Rate of Return

• One problem of the rate of return formula is that it does not take into account of time span of the investment. It does not matter how long a saver holds it, the resulting rate of return is the same.

• The internal rate of return takes into account cash flows on timeline, and is the most accurate measurement of rate of return on investment.

• First, draw cash flows on timeline, including purchase price (as outflow) and sales price (as inflow). Then, apply the business calculator or Microsoft Excel to compute the internal rate of return.

Example of Internal Rate of Return

Steven purchased a U.S. Treasury bond at $960 in 2013, received an annual coupon payment of $80 in 2014 and 2015, and sold it at $990 in 2015. The cash flows on timeline should look

2013 2014 2015

$80 $80+$990

$960

2)i1(990$80$

i180$960$

Like the yield to maturity, you solve for i: i = 9.8%

Real and Nominal Interest Rates

• Real interest rate: The interest rate that is adjusted for the inflation rate.

• Formula #7 (Fisher Equation): i = r + i: Nominal interest rater: Real interest rate: Inflation rate

Inflation and Value of Money• As the price level increases (inflation), a value of future

cash flows (purchasing power of money in future) decreases.– A dollar in future can buy less than a dollar today.

• An interest rate that a borrower promises to pay tells a lender how much a principle increases over time or how much dollar he will pay back.– 5% interest rate means your $100 will increase its value to

$105 next year.• Due to inflation, future cash flows will not buy as many

goods and services as they could without inflation, so a lender is actually getting less than 5% in terms of value. – An interest rate promised by a borrower does not promise

whether you can buy more or less in future from that payment.

Inflation and Interest Rate• If you can get 20% interest rate on your saving, will it

be a good deal? What will happen if an inflation rate is 50%?– Your $100 can buy 50 Big Macs (at $2 each) today.– If you loan your $100 at 20% annual interest rate, you will

get $120 next year.– With 50% inflation rate, a price of Big Mac will increase to

$3 (= $2 x 1.5).– Then, your $120 future cash flow can buy 40 Big Macs (=

$120/$3) next year.– Are you really getting 20% return from your loan or

loosing it?• What it really matters is not how much dollar ($120)

you get in future, but how much (40 Big Macs) you can buy from that cash flow in future.

Real Interest Rates

• Real interest rate: The interest rate that is adjusted for the inflation rate.

• Formula #7 (Fisher Equation): i = r + i: Nominal interest rater: Real interest rate: Inflation rate

• A real interest rate on a security tells you how much more goods and services you can purchase in future out of payments from that security.

Example of Real Interest Rate

You bought a 1-year CD at 5% (nominal) interest rate last year. During the last one year, the inflation rate was 3%. How much is a real interest rate on the CD? i = 5% and = 3%

5% = r + 3% r = 2%

Although you get 5% more cash from this CD than what you put in last year, its value decreased by 3%, so you can actually purchase only 2% more goods and services this year than last year.

Nominal vs. Real Interest Rate• Both lenders and borrowers must be concerned

with the real interest rate rather than the nominal interest rate.– Even if a nominal interest rate is high, if an inflation

rate is also high, the real interest rate may be low.

• Lenders want high real interest rate, while borrowers want low real interest rate.– Higher the real interest rate, more the lenders are

willing to lend their funds.– Lower the real interest rate, more the borrowers are

willing to borrow funds.

Examples of Real and Nominal Interest Rates

• You loaned $100 to your brother at 10% interest rate one year ago, and he returns $110 today. Last one year, the inflation rate was 10%. Did you gain from this investment (loan) to your brother? Did your brother loose from this loan from you? How much was a real interest rate?

• If you expect an inflation rate will be 3%, are you willing to loan your funds at 1%, 3%, or 5% of nominal interest rate? How much will be a real interest rate?

Real Interest Rate, Nominal Interest Rate, and Inflation

• Because we live in an economy with continuous inflation, a nominal interest rate is always greater than a real interest rate.

• Inflation rate changes from year to year, so a difference between a nominal interest rate and a real interest rate also changes.

Figure 1: Real and Nominal Interest Rates, 1953-2011

A vertical distance between a blue line (nominal interest rate) and a brown line (real interest rate) is an inflation rate (as indicated in a red arrow) on this chart.

Inflation Rate and Actual Real Interest Rate• When a saver purchases a bond or loans a fund, he knows

future cash flows and a nominal interest rate, but uncertain how much those future cash flows actually worth (how much goods and services you can buy with cash flows).

• A saver must make a guess on inflation rate (i.e. 3%). If he wants 5% real interest rate, then he must ask 8% nominal interest rate on a bond.

• However, no one can predict future inflation rate precisely, so he will get more or less actual real interest rate in reality.– If an actual inflation rate happens to be 3%, then he will get 5% real

interest rate as he expected.– If an actual inflation rate happens to be 1%, then he will get 7% real

interest rate– If an actual inflation rate happens to be 6%, then he will get only 2%

real interest rate

Inflation-Indexed Bonds

• Inflation-Indexed bonds guarantee a real interest rate by adjusting coupon and principal payments for changes in price level (inflation).– Indexed bonds have a fixed real interest rate, but nominal

interest rate varies as price level in economy changes.– TIPS (Treasury Inflation Protection Securities) are

inflation-indexed bonds issued by the U.S. government.– Ex. A borrower guarantees 5% real interest rate on $100

loan. A lender will receive $108 (8% nominal interest rate) if an actual inflation rate is 3%, or she will receive $112 (12% nominal interest rate) if an actual inflation rate is 7%.

Example of Bond Quotations on Barron’s

• Dow Jones’ publishes quotes of U.S. Treasury securities (Treasury bills, notes, and bonds) and corporate bonds every day on the Wall Street Journal (online only) and weekly on Barron’s. – See “How to Interpret Bond Quotations of

Barron’s” on Blackboard

Disclaimer

Please do not copy, modify, or distribute this presentation without author’s consent.

This presentation was created and owned byDr. Ryoichi Sakano

North Carolina A&T State University