Econ Interest Rates

4 types of credit market instruments

1. simple loan 2. fixed payment loan 3. coupon bond 4. discount bond

Rate of return (return)

Accurately measures how well a person does by holding a bond or other security over a particular period of time. The return on a bond will not necessarily equal the interest rate on that bond The return on a bond held from time t to time t + 1 can be written as R = (C + Pt+1 - Pt)/Pt R = return from holding the bond from time t to time t + 1, Pt = price of the bond at time t, Pt+1 = price of the bond at time t + 1, C = coupon payment You can split the equation into two terms: R = (C/Pt) + (Pt+1 - Pt)/Pt The first term is the current yield ic (the coupon payment over the purchase price) The second term is the rate of capital gain, or the change in the bond's price relative to the initial purchase price where g = rate of capital gain. You can write the equation as R = ic + g This shows that the return on a bond is the current yield plus the rate of capital gain Returns will differ from the interest rate especially if the price of the bond experiences sizable fluctuations, which then produce substantial capital gains or losses.

Real IR

Adjusted for inflation by subtracting expected changes in the price level (inflation) so that it more accurately reflects the true cost of borrowing Ex ante real IR bc it is adjusted for expected changes in the PL Most important to economic decisions and typically what financial economists mean when they make references to the real IR

Q: What is the after-tax real interest rte for a person in a 30% tax bracket on a bond yielding 10%? The expected inflation is 20%.

After tax real IR = ATRI = i(1-T)-π^e = .1(1-.3) - .2 = -.13 *When inflation is really high, higher than the nominal rate, real earnings will be negative. However, because you are paying taxes on your nominal interest rates the tax bracket takes the after tax real interest rate even lower.

Fixed-payment loan

Also called a fully amortized loan Lender provides the borrower with an amount of funds which must be repaid by making the same payment every period consisting of part of the principal and interest for a set number of years Installment loans like car loans and mortgages are frequently the fixed payment type

Increase in IR associated with a ______ in bond prices

An increase in IR is associated with a fall in bond prices resulting in capital losses on bonds whose terms to maturity are longer than the holding period

Q: Consider a coupon bond that has $1,000 par value and a coupon rate of 10%. The bond is currently selling for $1,150 and has eight years to maturity. What is the bond's yield to maturity? Use a coupon bond calculator to solve.

Answer: 7.44%

Present value (or present discounted value)

Based on the commonsense notion that a dollar of cash flow paid to you one year from now is less than the dollar value of a dollar paid to you today

Q: Suppose a bond has a 10% coupon rate, a $1000 par value, and matures in 2 years. Assume semiannual compounding and market rates of interest are 12%. What is the price of the bond?

Because it is only two years we can solve by hand. C = 10%, F = 1000, matures in 2 years, i = 12% p = [(100/2)/(1+.06)] + [(100/2)/(1+.06)^2] + [(100/2)/(1+.06)^3] + [(100/2)/(1+.06)^4] = 1000/(1.06)^4 = 47.17 + 44.50 + 41.98 + 39.60 + 792.10 = 965.35

Discount (zero coupon) bond

Bought at a price below its face value (at a discount) and the face value is repaid at the maturity date Unlike coupon bonds: no interest payments, it just pays off the face value Ex. a discount bond with a face value of $1,000 might be bought for $900 and in a years time the owner would be repaid the face value of $1,000 U.S. Treasury Bills, U.S. savings bonds, and long-term zero-coupon bonds are examples

Simple loan

Commercial loan to business Lender provides borrow with a principal amount of money which is repaid at maturity date with interest

Coupon bond equation

Coupon Bond Same strategy as fixed payment loans = equate today's value with present value, bc coupon bonds also have multiple cash flow payments, the PV = sum of PVs of all the coupon payments plus the PV of the final payment of the FV of the bond P = C/(1+i) + C/(1+i)^2 + C/(1+i)^3 + ... C/(1+i)^n + F/(1+i)^n Most coupon bonds actually make coupon payments on a semiannual basis rather than once a year P = price of coupon bond, C = yearly coupon payment, F = face value of the bond, n = years to maturity date To solve using a financial calculator:n = years to maturity, FV = face value of the bond, i = annual IR, PMT = yearly coupon payments, Then push the PV button

Coupon bonds (semi-annual)

Coupon bonds (semi-annual) P = [(C/2)/(1+i) ] + [ (C/2)/(1+i)^2 ] + [ (C/2)/(1+i)^3 ] + ... [ (C/2) /(1+i)^n ] + [F/(1+i)^n] n = 2n

Q: What is the current yield for a bond that has a par value of $1000 and coupon interest rate of 10.95%? The current market price for the bond is $921.01?

Current yield Par value = 1000, i = 10.95%, P = $921.01, ic = C/P ic = 109.5/921.01 = 1.89%

Cash flows

Different debt instruments have very different streams of cash payments to the holder, with very different timing. Thus, first we need to understand how to compare the values of different types of debt instruments before we can see how interest rates are measured.

Discount bond

Discount bond = Zero coupon bond Ex. US treasury bills, US savings bonds, Long-term zero coupon bonds

Fixed payment loan

Equate today's value of the loan with its present value Bc fixed payment loans have more than one cash flow payment, the PV of the fixed payment loan is calculated as the sum of the PVs of all cash flows Suppose the loan is 1000 and the yearly cash flow payments are 85.81 for the next 25 years 1,000 = 85.81/(1+i) + 85.81/(1+i)^2 + 85.81/(1+i)^3 + ... 85.81/(1+i)^25 LV = FP/(1+i) + FP/(1+i)^2 + FP/(1+i)^3 + ... FP/(1+i)^n To find the yearly payment for the loan using a financial calculator: n = number of years, PV = amount of the loan (LV), FV = amount of the loan after n years, i = annual interest rate then push the PMT button = fixed yearly payment (FP)

Paper loss

If Irving does not sell the bond, the capital loss is often referred to as a "paper loss." This is a loss nonetheless because if he had not bought this bond and had instead put his money in the bank, he would now be able to buy more bonds at their lower price than he presently owns.

If Pete borrows 100 from his sister and next year she wants 110 back, what is the yield to maturity?

If Pete borrows 100 from his sister and next year she wants 110 back, what is the yield to maturity? PV = CF/(1+i)^n 100 = 110/(1+1), i = .1 or 10% For simple loans the simple IR equals the yield to maturity

What would make RoR small?

If the purchase price (denominator) is large.

Rate of capital gain

If we rearrange the RoR equation we can get: R = (C/Pt) + [(Pt+1-Pt)/Pt] *R = ic + g* ic = current yield, g = [(Pt+1-Pt)/Pt] or the change in the bond's price relative to the purchase price also known as the rate of capital gain

Fixed payment loan = fully amortized loans

Installment loans like cars or mortgages Funds are repaid by making the same payment each period Part if for principal, part of interest coupon bond US treasury bonds + notes and corp. bonds payments every year until maturity date At maturity, a specified final amount (face or par value) is due Identified by: Issuing corp. or gov, Maturity date. Coupon rate, Amount of yearly coupon payments Expressed as a percent of the bonds face value

Fixed payment loan Q: You decide to purchase a new car and need a $30,000 loan. You take out a loan from the bank that has an interest rate of 7%. What is the yearly payment to the bank to pay off the loan in 5 years?

LV = [ FP/(1+i) ] + [ FP/(1+i)^2 ] + [ FP/(1+i)^3 ] + [ FP/(1+i)^n ] LV = loan value, i = IR, FP = fixed cash flow payment, n = number of years until maturity Solve using excel by typing: =PMT(rate, Nper, PV, [FV], [type] PMT = payment, rate = IR, Nper = number of periods, PV = present value, FV = future value (Cash balance you want after the last payment is made, likely to be zero), type can be ignored here-leave at zero

Perpetuity

Means: without end Special case of coupon bond A perpetual bond with no maturity date and no repayment of principal, makes a fixed coupon payment of $C forever A flow of money that will be received on a regular basis without a specified ending date Used to calculate the present value of an annuity Formula = PV = C/R C = the amount of the cash flow, R = rate or discount rate The concept of perpetuity is a basic concept used in the dividend discount pricing model for securities as well as real estate valuation Ex. British security called the Consol Perpetuities are also called Consols Pc = C/Ic Pc = price of consol, c = yearly payment, Ic = yield to maturity Rearranged: Ic = C/Pc

Yield to maturity

Most important way to calculate IRs The IR that equates the PV of cash flows received from a debt instrument with its value today Financial economists consider it the most accurate measure of IRs The key to the calculations is equating today's value of the debt instrument with the value of all its future cash flow payments

Real vs. Nominal interest rates

Nominal IR is the IR that is named/reported Real interest rates is adjusted for inflation Also called: Ex ante real IR = π^e, ex post real IR Fisher equation = i = ir + π^e i = nominal rate, ir = real rate, π^e = expected rate of inflation

Coupon bonds (annual)

P = [C/(1+i) ] + [ C/(1+i)^2 ] + [ C/(1+i)^3 ] +... [ C/(1+i)^n ] + [F/(1+i)^n] = PV of each coupon payment + PV of final payment (face value) P = price of the coupon bond, C = yearly coupon payment, F = face value of bond, n = years to maturity date, i = yield to maturity

Simple loan Q: You borrow $2500 and next year you have to pay $2,750 back. What is the yield to maturity (IR) on this loan?

PV = CF/(1+i)^n CF = Cash flow/Future value 2500 = 2750/(1+i) i = .1

Coupon bond

Pays the owner of the bond a fixed interest payment (coupon payment) every year until the maturity date, when a specified final amount (face value or par value) is repaid Named bc the bondholder used to obtain payment by clipping a coupon off the bond and sending it to the bond issuer A coupon bond with $1,000 face value might pay you a coupon payment of $100 per year for 10 years and at the maturity date repay you the face value amount of $1,000 (the face-value of a bond is usually in $1,000 increments) Identified by three pieces of info: Corporation or gov agency that issues the bond Maturity date of the bond The bond's coupon rate, the dollar amount of the yearly coupon payment expressed as a percentage of the face value of the bond In the example, if the bond has a yearly coupon payment of $100 and a face value of $1,000, the coupon rate is $100/$1,000 = .10 or 10% Capital market instruments such as U.S. Treasury bonds and notes and corporate bonds are examples

Q: What is the price of a perpetuity that has a coupon of $50 per year and a yield to maturity of 2.5%? If the yield to maturity doubles, what will happen to its price?

Pc = 50/.025 = $2000 Pc = 50/.5 = $1000 When the yield doubles, the price of the perpetuity is cut in half

Conclusion from Table 3.2 analysis

Prices and returns are more volatile for long-term bonds because they have higher interest rate risk IR risk is the riskiness of an asset's return that results from changes in the IR No IR risk for any bond whose immaturity equals holding period But short term bonds may have reinvestment risk if the investor's holding period is longer than the term to maturity of the bond

Question: A 10 year, 7% coupon bond with a face value of $1000 is currently selling for $871.65. Compute your RoR if you purchase the bond now and sell it next year for $880.10.

R = [70 + (880.10 - 871.65)]/871.65 = 0.09 or 9%

Reinvestment risk

Refers to the risk that a bond's future coupons will have to be reinvested at a lower IR The longer the bond's maturity and the higher IR the higher the reinvestment risk Zero coupon bonds have no reinvestment risk If an investor's holding period is longer than the term to maturity of the bond, the investor is exposed to a type of IR risk called reinvestment risk Occurs bc the proceeds from the short term bond need to be reinvested at a future uncertain IR Suppose an investor has a holding period of 2 years and purchases a 1000 1 year 10% coupon rate bond at face value and then another one at the end of the first year, if the IR rises to 20% at the end of the year he earns more so if he has a holding period longer than the term to maturity and the IR rises, he benefits and vice versa

Simple loan

Simplest form-lender provides the borrower with an amount of funds called the principal that must be repaid to the lender at the maturity date along with additional payment for interest In the case of a simple loan, the interest payment divided by the amount of the loan is a natural and sensible way to measure the IR 100 * (1+i)^n Many money market instruments are this type like commercial loans to businesses

Perpetuity or consol

Special case of a coupon bond Perpetual bond with no maturity date and no repayment of principal that makes fixed coupon payments of $C forever Pc = C/ir P = price of the perpetuity, C = yearly payment, ir = yield to maturity of the perpetuity As i goes up, the price of the bond falls I = C/Pr When a coupon bond has a maturity of 20 years or more it is close to a perpetuity

Current yield

Special case of the coupon bond When a coupon bond has a really long term to maturity (20 years or more) it acts like a perpetuity The discounted value of the cash flows more than 20 years out are practically zero That means the current yield is very close to the yield to maturity

Cash flow

Streams of cash payment to the holder of a debt instrument → we are interested in the CF not just the FV

Indexed bonds

TIPS Used to protect investors from the impact of inflation Also used to derive expected inflation rates

Return on a bond

Tells you how good an investment it has been over the holding period Equal to the yield to maturity only in one special case: when the holding period and maturity of the bond are identical

Ex post real interest rate

The IR that is adjusted for actual changes in the price level Describes how well a lender has done in real terms after the fact

Future value

The amount the sum will be worth at a future date when allowed to earn interest at the prevailing rate Formula = FV = (PV)(1+i)^n Q: i = 6% FV = 20M n = 10 A: PV = 200 mil /(1.06)^10 = 112 mil

Duration

The average lifetime of a debt security's stream of payments Just because two bonds have the same term to maturity doesn't mean they have the same IR risk A long term discount bond with 10 years to maturity makes all of the payments at the end of the 10 years but coupon bond with 10 years to maturity makes payments before the date We might guess that the coupon bond's effective maturity (the term to maturity that accurately measures interest-rate risk) is shorter than zero-coupon discount bonds and it's true

Additive property of duration

The duration of a portfolio of securities is the weighted average of the durations of the individual securities with the weights reflecting the proportion of the portfolio invested in each The greater the duration of a security, the greater its interest-rate risk.

Rate of Return (RoR)

The earnings an asset generates in excess of its initial cost Calculated based on the cash flows generated and can include a capital gain element Can be negative Usually express as x% per year Formula for a single period = (final asset value - initial asset value) / initial asset value Used to compare similar assets to determine which is the better investment Can be measured for any type of asset including collectibles like fine art

The more distant a bond's maturity

The more distant a bond's maturity, the greater the size of the price change associated with an interest-rate change The more distant a bond's maturity, the lower the rate of return that occurs as a result of the increase in the IR Even though a bond has a substantial initial interest rate, its return can turn out to be negative if interest rates rise.

What bond's return equals the initial YTM

The only bond whose return equals the initial yield to maturity is one whose time to maturity is the same as the holding period

Rate of Return

The payments to the owner plus the change in the security's value, expressed as a fraction of its (the security's) purchase price Return from time t to t+1 is denoted by R = [C + (Pt+1 - Pt)]/Pt R = return from holding bond from t to t+1, Pt = price of the bond at time t, Pt+1 = price of the bond at time t+1, C = coupon payment

Discounting the future

The process of calculating today's value of dollars received in the future PV = CF/(1+i)^n

Fisher equation

The real interest rate is more accurately defined by the Fisher equation which states that nominal IR i = the real IR ir + expected rate of inflation π^e i = ir + π^e

IR risk

The risk that investments already held will lose market value if new investments with higher interest rates enter the market Affects value of bonds more directly than stocks Risk to all bond holders Prices of longer-maturity bonds respond more dramatically to changes in IRs This helps explain why prices and returns for long-term bonds are more volatile Prices changes of 20% are common for 20 yr maturity bonds Long-term bonds are therefore quite risky The riskiness of an asset's return that results from changes in IR is IR risk Short term debt instruments don't have this risk-bonds with a maturity as short as the holding period have no IR risk This is bc the price at the end of the holding period is already fixed at the face value so the change in IRs can have no effect on the price at the end of the holding period so the return will equal the yield to maturity

Present value

To compare sums from different times The amount need today to yield a given future sum at prevailing IRs Formula = PV = (FV)/(1+i)^n n = number of period, i = IR Explains why investment falls when IR rises

Q: The duration of a $100 million portfolio is 10 years. $40 million in new securities are added to the portfolio, increasing the duration of the portfolio to 12.5 years. What is the duration of the $40 million in new securities?

Total in portfolio = $140 D = duration of the $40 million in new securities 12.5 = [10*(100/140)] + [D*(40/140)] 12.5 = 7.1425 + 0.2857D D = 18.75 We increased our interest rate risk by adding the new securities.

TIPS

Treasury inflation protected securities Treasury securities that make adjustments for inflation as reflected in the consumer price index Effective way to eliminate inflation risk Have a fixed IR that is paid semi-annually, the inflation adjustment is also made on a semi-annual basis The adjustment is made to the bond's par value rather than the IR Protects the bond's interest rate and face value from inflation Some tax concerns: The internal revenue service considers an adjustment to a security's face value as taxable income Because of this most investors prefer to own TIPS in mutual funds, or, even better, in their tax-deferred retirement accounts

Discount bond-Zero coupon bond

US Treasury Bills, US saving bonds, long term 0 coupon bond Bought at less than the face value Face value repaid at maturity date Makes no interest payments Earnings = face value (price they paid)

How long have real IRs been observable in the US

Until recently, real interest rates in the United States were not observable because only nominal rates were reported. This all changed in January 1997, when the U.S. Treasury began to issue indexed bonds, bonds whose interest and principal payments are adjusted for changes in the price level

The distinction between IRs and RoRs

What if the IR changes and what if we sell a bond before it matures?

How can we quantify the risk bonds face when IRs change?

What is the capital gain or loss? Depends on effective maturity (duration) Average lifetime of a debt security's stream of payments IR risk increases with duration

Nominal IR

What we have been calling the IR making no allowance for inflation

Discount

When a bond sells for less than its par value

Premium

When market price exceeds par value

When the IR is higher, the cash payments in the future are discounted ______ heavily and become _______ important in present-value terms relative to the total present value of all the payments

When the IR is higher, the cash payments in the future are discounted more heavily and become less important in present-value terms relative to the total present value of all the payments

What happens when the IR is low

When the IR is low there are more incentives to borrow and fewer to lend

Discount bond equation

Yield-to-maturity calculation is similar to a simple loan i = (F-P)/P, F = face value of the bond, P = current price of the discount bond In other words the yield to maturity equals the increase in price over the year divided by the initial price In normal circumstances, investors can earn positive returns from holding the securities and so they sell at a discount, meaning the current price of the bond is below the face value, so F-P should be positive and the yield to maturity should be as well, but not always like in Japan and other places For a discount bond, the yield to maturity is negatively related to the current bond price, the same conclusion we reached for a coupon bond

Q: What is the yield to maturity on a $1000 face-value discount bond maturing in one year that sells for $800?

i = (F-P)/P, basically the increase in value over one year divided by the initial amount: F-P should be greater than zero except in some circumstances, i is inversely related to P F = face value, P = current price F = 1000, P = 800, i = (1000-800)/800 = 25%

Q: What is the real IR on a bond yielding 10%? The expected inflation is 20%.

i = ir + π^e 10 = ir + 20 Real interest rate is -10%

Q: What is the ex ante real IR if the nominal IR is 9% and the expected inflation rate is 7%?

i = ir +π^e 9 = 2+7 The ex ante real IR = 2%

Longer term to maturity =

longer duration

The price of a coupon bond and the yield to maturity are ______

negatively-related: that is, as the yield to maturity rises, the price of the bond falls. If the yield to maturity falls, the price of the bond rises.

Four types of Credit Market Instruments

simple loan, fixed payment loan, coupon bond, discount bond. These four types of instruments require payments at different times: simple loans and discounts bonds make payment only at their maturity dates, whereas fixed payment loans and coupon bonds have payments periodically until maturity. To calculate the values of each we use present value to help us measure IRs.

The yield to maturity is greater than the coupon rate when

the bond price is below its face value.

All else being equal when IRs rise

the duration of a coupon bond falls

A higher coupon rate on a bond,

the shorter the duration

When the coupon bond is priced at its face value,

the yield to maturity equals the coupon rate.