# Bond Pricing and Risk-Money and Banking-Handouts, Lecture notes for Banking and Finance. Amity Business School

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Money & Banking – MGT411 VU Lesson 10

Bond Pricing BOND PRICING & RISK

Real Vs Nominal Interest Rates Risk Characteristics Measurement

Bond Pricing

A bond is a promise to make a series of payments on specific future date. It is a legal contract issued as part of an arrangement to borrow The most common type is a coupon bond, which makes annual payments called coupon payments The percentage rate is called the coupon rate The bond also specifies a maturity date (n) and has a final payment (F), which is the principal, face

value, or par value of the bond The price of a bond is the present value of its payments To value a bond we need to value the repayment of principal and the payments of interest

Valuing the Principal Payment

A straightforward application of present value where n represents the maturity of the bond Valuing the Coupon Payments: Requires calculating the present value of the payments and then adding them; remember, present

value is additive Valuing the Coupon Payments plus Principal Means combining the above

Payment stops at the maturity date. (n)

A payment is for the face value (F) or principle of the bond

Coupon Bonds make annual payments called, Coupon Payments (C), based upon an interest rate,

the coupon rate (ic), C=ic*F

A bond that has a \$100 principle payment in n years. The present Value (PBP) of this is now:

P F \$100 BP (1 i ) n (1 i ) n

If the bond has n coupon payments (C), where C= ic * F, the Present Value (PCP) of the coupon payments is:

PCPC

(1 i )1 C

(1 i ) 2 C

(1 i ) 3

...... C

(1 i ) n

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Money & Banking – MGT411 VU Present Value of Coupon Bond (PCB) = Present value of Yearly Coupon Payments (PCP) + Present Value of the Principal Payment (PBP)

PCBPCP

PBP

C (1 i)1

C (1 i) 2

C (1 i) 3

......

C (1 i) n

F (1 i) n

Note: The value of the coupon bond rises when the yearly coupon payments rise and when the interest

rate falls Lower interest rates mean higher bond prices and vice versa. The value of a bond varies inversely with the interest rate used to discount the promised payments

Real and Nominal Interest Rates

So far we have been computing the present value using nominal interest rates (i), or interest rates expressed in current-dollar terms

But inflation affects the purchasing power of a dollar, so we need to consider the real interest rate (r), which is the inflation-adjusted interest rate.

The Fisher equation tells us that the nominal interest rate is equal to the real interest rate plus the expected rate of inflation

Fisher Equation:

i = r + e

Or r = i - πe

Figure: Nominal Interest rates, Inflation, and real interest rates

20

15

10

5

0

-5

-10

1979 1 1985 1 1 1994 1997 2000 2003

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N om

in al

In te

re st

R at

e (%

)

Money & Banking – MGT411 VU Figure: Inflation and Nominal Interest Rates, April 2004

30 Turkey •

25

20 Brazil • Russia

15 •

45º line

10 South Africa

UK 5 US

0 5 10 15 20 25 30

Inflation (%)

Risk

Every day we make decisions that involve financial and economic risk. How much car insurance should we buy? Should we refinance the home loan now or a year from now? Should we save more for retirement, or spend the extra money on a new car? Interestingly enough, the tools we use today to measure and analyze risk were first developed to

help players analyze games of chance. For thousands of years, people have played games based on a throw of the dice, but they had little

understanding of how those games actually worked Since the invention of probability theory, we have come to realize that many everyday events,

including those in economics, finance, and even weather forecasting, are best thought of as analogous to the flip of a coin or the throw of a die

Still, while experts can make educated guesses about the future path of interest rates, inflation, or the stock market, their predictions are really only that—guess.

And while meteorologists are fairly good at forecasting the weather a day or two ahead, economists, financial advisors, and business gurus have dismal records.

So understanding the possibility of various occurrences should allow everyone to make better choices. While risk cannot be eliminated, it can often be managed effectively.

Finally, while most people view risk as a curse to be avoided whenever possible, risk also creates opportunities.

The payoff from a winning bet on one hand of cards can often erase the losses on a losing hand. Thus the importance of probability theory to the development of modern financial markets is hard

to overemphasize. People require compensation for taking risks. Without the capacity to measure risk, we could not

calculate a fair price for transferring risk from one person to another, nor could we price stocks and bonds, much less sell insurance.

The market for options didn't exist until economists learned how to compute the price of an option using probability theory

We need a definition of risk that focuses on the fact that the outcomes of financial and economic decisions are almost always unknown at the time the decisions are made.

Risk is a measure of uncertainty about the future payoff of an investment, measured over some time horizon and relative to a benchmark.

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Money & Banking – MGT411 VU

Risk can be quantified. Characteristics of risk

Risk arises from uncertainty about the future. Risk has to do with the future payoff to an investment, which is unknown. Our definition of risk refers to an investment or group of investments. Risk must be measured over some time horizon. Risk must be measured relative to some benchmark, not in isolation. If you want to know the risk associated with a specific investment strategy, the most appropriate

benchmark would be the risk associated with other investing strategies

Measuring Risk

Measuring Risk requires: List of all possible outcomes Chance of each one occurring The tossing of a coin What are possible outcomes? What is the chance of each one occurring? Is coin fair? Probability is a measure of likelihood that an even will occur Its value is between zero and one The closer probability is to zero, less likely it is that an event will occur. No chance of occurring if probability is exactly zero The closer probability is to one, more likely it is that an event will occur. The event will definitely occur if probability is exactly one Probabilities can also be expressed as frequencies

Table: A Simple Example: All Possible Outcomes of a Single Coin Toss

PossibilitiesProbabilityOutcome #1 1/2 Heads #2 1/2 Tails

We must include all possible outcomes when constructing such a table The sum of the probabilities of all the possible outcomes must be 1, since one of the possible

outcomes must occur (we just don’t know which one) To calculate the expected value of an investment, multiply each possible payoff by its probability

and then sum all the results. This is also known as the mean. Case 1 An Investment can rise or fall in value. Assume that an asset purchased for \$1000 is equally likely to fall to \$700 or rise to \$1400

Table: Investing \$1,000: Case 1

PossibilitiesProbabilityPayoffPayoff ×Probability #1 1/2 \$700 \$350 #2 1/2 \$1,400 \$700

Expected Value = Sum of (Probability times Payoff) = \$1,050

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Money & Banking – MGT411 VU Expected Value = ½ (\$700) + ½ (\$1400) = \$1050

Case 2

The \$1,000 investment might pay off

\$100 (prob=.1) or \$2000 (prob=.1) or \$700 (prob=.4) or \$1400 (prob=.4)

Table: Investing \$1,000: Case 2

PossibilitiesProbabilityPayoffPayoff ×Probability #1 0.1 \$100 \$10 #2 0.4 \$700 \$280 #3 0.4 \$1,400 \$560 #4 0.1 \$2,000 \$200

Expected Value = Sum of (Probability times Payoff) = \$1,050

Investment payoffs are usually discussed in percentage returns instead of in dollar amounts; this allows investors to compute the gain or loss on the investment regardless of its size

Though both cases have the same expected return, \$50 on a \$1000 investment, or 5%, the two investments have different levels or risk.

A wider payoff range indicates more risk.