Rate of Return

Lesson 2 How Time Value of Money Affects Returns

Time Value of Money

Aim: What does the Time Value of Money do

for the value of our investments over the long term?

Do Now: Explain why, if someone owed you

money, you’d rather receive it now than, say, five years from now.

Do Now answer: If you receive it now, you will have the opportunity to invest it and earn money. In five years, it could grow to a lot more than the original amount.

Time Value of Money

Time Value of MoneyThe idea that a dollar received today is worth more than a dollar received in the future. A dollar received today can either buy something or be reinvested, earning additional income from return or interest.

Time Value of Money is the principle that drives the expected return model.

Present Value FormulaPresent Value Formula: The Present Value Formula

determines how much a sum of money to be

received in the future would be worth today. The

Future Value of money is discounted using an

interest rate.

PV = Present Value of the $

FV = Future Value of the $

i = interest rate that can be earned

n = number of interest periods

Discounting to Present Value

Discounting: Finding the Present Value of an investment by taking the Future Value of that investment and reducing it annually by the interest rate (ie: a rate you think you can earn).

Applying Time Value of Money1) Congratulations! You have won a cash prize of $20,000!

You have two options. Which is better?

The choice is clear: most people will choose to take the money today.

2) Now, what if the options are:

Or

Here, the choice is less clear. The option you would choose is based on how much interest you think you can earn each year on $20,000 received today

A. Receive $20,000 now B. Receive $20,000 in 3 years

A. Receive $20,000 now B. Receive $23,000 in 3 years

Applying Time Value of MoneySituation 1: Assume you can earn 5% per year. After three years you will have $23,152.50 so you would take the $20,000 today rather than the $23,000 in the future. We arrived at this by manipulating the PV formula to solve for FV:

FV = PV * (1+i)n

Using I = 5%, N = 3 and PV = 20,000 we get:

FV = 20,000(1+.05)3 = $23,152.50

Taking the $20,000 today would be the best option!

Applying Time Value of MoneySituation 2: Now assume the interest rate is 4%, compounded annually. After three years, you will have $22,497.28. If you can only earn 4%, you would rather take the offer of $23,000 in 3 years. The calculation:

FV = PV*(1+i) n

FV = 20,000(1+.04)3 = $22,497.28

Again, taking the $23,000 in three years is best!

Compound vs. Simple Interest

Simple Interest:• In the simple interest calculation, periodic bond

coupon or stock dividend payments are not assumed to be reinvested and, thus, do not generate additional interest.

The formula for simple interest is:

Interest = Principal*Interest Rate*Time

Compound vs. Simple InterestIn the compound interest calculation, the coupons and dividend payments are assumed to be reinvested. They generate additional interest after they are paid to the investor because the investor reinvests the payments. Compound interest causes your money to grow at a faster rate

The formula for compound interest is: M = P(1+i)n

M = The amount generated (inc. the initial principal)P = Principal, the initial investmenti = Interest raten = Number of interest periods

Compound vs. Simple InterestExample: I.N. Vestor wants to know how much he will make in 3

years if he puts $10,000 into his savings account today. The

savings account interest rate is 4% compounded annually. How

much will I.N. Vestor have?

Simple Interest Solution: 4% of $10,000 is $400. Each year

when the interest is paid, I.N. Vestor takes it from the account

and spends it. $10,000 * 4% * 3 years = $1,200 in total interest

collected.

Compound Interest Solution: i = 4% n = 3 P = 10,000

M= 10,000 (1+.04) 3= $11,248.64

I.N. Vestor will have $11,248.64 after 3 years, $1,248.64 of which is compound interest

Return

Expected/Required Return: Return an investor requires or expects for assuming a certain amount of risk with the investment. The expected return is based on future expected cash flows for a set period of time.– Holding period in the expected return calculation

is based on the period beginning today (n=0) and ending at an established time in the future.

Expected Rate of Return for Stocks

Components of Stocks’ Expected Return Future expected dividends. Expected price of stock at the end of the holding

period.

Unlike bonds, stocks have no maturity date. To recover principal, an investor must sell stock in the stock market.

Expected Rate of Return Formula - Stocks

P0 = price of the stock todayi = expected rate of returnD1 = dividend year 1D2 = dividend year 2D3 = dividend year 3

= price of the stock in future

P0 = + + + … +

Expected Rate of Return Formula - Stocks

Example: I.N. Vestor buys 1 share of stock XYZ at $1,000 per share. It pays a $20 per year dividend. He believes the price of the stock after 1 year will be $1,200. What is the expected rate of return?

Solution: 1,000 = _20_ + 1200

(1+i)1 (1+i)1

1000 (1+ i) = 1220

1000i = 220

i = 0.22 = 22%The expected rate of return is 22%

Expected Rate of Return for BondsComponents of Bonds’ Expected Return

CouponA company that issues a bond

must pay the coupon (interest) payments to the

bondholder.

Principal The original investment.

Bond Cash Flows

A company that issues a bond must pay the coupon payment (interest), usually twice per year, to the bondholder over the term of the bond.

When the bond matures, the issuer repays the principal to the bondholder.Long-term bonds have higher coupon rates than short-term bonds. This is because of risk. The longer the term, the greater the chance of default.

Bond Price Formula

P0 = + + + … +

• C = Coupon payment (usually semi-annually) in dollars, not %n = Number of periods i = Yield to maturity (expected return)M = Value at maturity, or par value (typically $1,000 per bond)

Example

On January 1, 2013, I.N. Vestor buys a new 5% 2-yr $1,000 face value. The interest rate in the market is 3%.

Assuming that I.N. Vestor holds the bond to maturity, he will receive two $25 coupon payments each year for two years, as follows: 6/1/2013, 12/1/2013, 6/1/2014, and the last one on 12/1/2014.

At the maturity date, December 31, 2014, I.N. Vestor receives his principal investment of $1,000 plus the last coupon of $25.

Over the two-year period, I.N. Vestor has received $25 four times plus his original investment at maturity of $1,000 = 1,000 + 100 = $1,100.

Cash Flow of Bonds

Yield to Maturity: This is the rate of return that an investor would earn if he bought the bond at its current market price and held it until maturity.

In our example, a bond is paying 5% interest on its face value. But with a drop in interest rates, investors are now willing to earn just 3% on a bond investment.

A 5% bond can be “turned into” a 3% investment by asking the buyer to pay more than face value for it.

0 1 2 3 4

Coupon 1 = $25 Coupon 1 = $25 Coupon 1 = $25

$1000 principal repayment Coupon 1 = $25

Investor pays $1000 for bond

6/1/11 12/1/11 6/1/12 12/1/12

Cash Flow of Bonds

Solution using Formula:i = 3% = 1.5% (because of semi-annual coupons)n = number of periods = 4 (2 years*2 payments/year)C = 25 M = par value = 1,000P = 25/(1+.015)1+ 25/(1+.015)2+ 25/(1+.015)3+ 25/(1+.015)4+ 1000/(1+.015)4

P = 24.63054187 + 24.2665437 + 23.9079248+23.55460576+942.1842303P = $1,038.54

Formula for a Bond’s Yield to Maturity (Expected Return)

YTM = C + F - P

n

F + P

2

C = Coupon Payment F = Face Value of Bond P = Price of Bond N = years until maturity

Example• Example: Suppose you are looking at a $1,000

face, 5% bond with 5 years to maturity that the seller wants $1,050 for it. If you buy it, what is the YTM?

C= 50

F= 1,000

P= 1,050

n= 5

YTM = C+ (F-P)/n

(F+P)/2

YTM = 50 + (1,000-1,050)/5

(1,050+1,000)/2

YTM = 3.90%

The yield to maturity is 3.90%

Solution YTM = Yield To Maturity

Lesson Summary 1 of 21. Why would you rather have $1,000 today than

$1,000 five years from now?

2. What do we call growing money forward through time, where we are able to earn interest on prior interest?

3. What do we call the opposite, taking a sum of money owed in the future and reducing its value as it’s brought back to the present?

4. Which investment, a stock or a bond, has cash flows for a finite period of time?

Lesson Summary 2 of 25. Between stocks and bonds, which is

easier to value, and why?

6. What are some of the terms we use for the rate at which we discount a series of future cash flows?

7. What does the Time Value of Money do for the value of our investments over the long term?

Web Challenge #1Challenge: Find a bond screener/listing site such as at Yahoo Finance or FINRA. Using the coupon rate, the number of years to maturity and the current market price shown (this will be % of face value), use the formula to calculate the Yield to Maturity (YTM). Compare it to YTM shown. How close is your calculation? What do you think can account for any difference?

Web Challenge #2Challenge: Interest rates have come down significantly since the Great Recession that ended around 2009. This was one of the Federal Reserve’s steps to help the economy. Corporations that have wanted or needed to borrow money since then have been the beneficiary because they can now borrow at very low rates. Research five A-rated (or higher) corporate bonds that have been issued before 2008 and five that have been issued after 2009. Describe what you found. Was it as we expected? If not, do you have any ideas why?

Web Challenge #3Q: What’s the best way to invest in bonds even though we don’t know whether interest rates will go up and down? A: Experts say that the best way to do it is

with a strategy called bond “laddering”. Challenge: Research laddering. Answer:

What is it? What does it allow bond investors to accomplish? If there are any criticisms of this strategy, what are they?