CFS Chapter 4- The meaning of Interest Rates

What is the most accurate measure of interest rates?

Yield to maturity (YTM).

What is yield to maturity (YTM)?

The rate that equates the present value of future cash flows with the current price.

What is the present-value formula?

PV = CF / (1 + i)^n

What happens to present value when the interest rate rises?

Present value falls.

What are the 4 credit-market instruments?

1. Simple loans

2. fixed-payment loans

3. coupon bonds

4. discount bonds

What is a simple loan?

One lump-sum payment of principal plus interest at maturity.

What is a fixed-payment loan?

A loan repaid with equal payments over its life.

What is a coupon bond?

A bond providing periodic coupon payments + face value at maturity.

What is a discount bond?

A bond sold below face value that pays no periodic interest.

What type of instrument is a Treasury bill?

A discount bond.

What is the key relationship between bond prices and interest rates?

They move in opposite directions.

When does YTM equal the coupon rate?

When the bond sells at its face value.

If a bond sells below face value, how does YTM compare to its coupon rate?

YTM is greater than the coupon rate.

If a bond sells above face value, how does YTM compare to its coupon rate?

YTM is less than the coupon rate.

What is a consol (perpetuity)?

A bond paying a fixed coupon forever with no maturity date.

What is the price formula for a consol?

P = C / i

What is the YTM formula for a consol?

i = C / P

What is a bond's rate of return?

The total gain or loss from coupons and changes in bond price.

What are the two components of bond return?

Current yield and the rate of capital gain.

When does a bond's return equal its YTM?

When the holding period equals the bond's maturity.

What is interest-rate risk?

The risk that interest-rate changes cause bond prices and returns to change.

Which bonds have greater interest-rate risk?

Long-term bonds.

What is reinvestment risk?

The risk that future interest or principal payments must be reinvested at uncertain rates.

When is there no interest-rate risk over the holding period?

the holding period = the bond's maturity

What happens to the capital gain rate when bond prices decrease?

The capital gain rate becomes negative.

What happens to the capital gain rate when bond prices increase?

The capital gain rate becomes positive.

What is the approximate Fisher equation?

i = r + πᵉ

How do you calculate the approximate real interest rate?

r = i − πᵉ

What is an ex ante real interest rate?

• Real interest rate

• calculated using expected inflation

What is an ex post real interest rate?

The real interest rate calculated using actual inflation.

What happens to the real interest rate when inflation rises and nominal rates stay constant?

The real interest rate falls.

Why are real interest rates important?

They show:

• real purchasing-power cost of borrowing

• benefit of lending

What happens when the real interest rate is negative?

• Borrowers may benefit

• lenders lose purchasing power in real terms

Why can the rate of return differ from a bond's stated interest rate?

• Because the bond's price can change

• creating a capital gain or loss

What is the price of a coupon bond?

P = C/(1+i) + C/(1+i)² + ... + C/(1+i)ⁿ + F/(1+i)ⁿ

What does each variable in the coupon-bond price formula represent?

• P = price

• C = annual coupon

• i = YTM

• n = years to maturity

• F = face value

What happens to a coupon bond's price when its YTM increases?

• price decreases

• because future cash flows have a lower present value

What happens to a coupon bond's price when its YTM decreases?

Its price increases because future cash flows have a higher present value.

What information is needed to calculate a coupon bond's YTM?

• Bond price

• coupon payment

• face value

• years to maturity

How do you calculate the YTM of a one-year discount bond?

YTM = (Face value − Price) / Price.

What happens to a discount bond's YTM when its price increases?

YTM decreases.

What happens to a discount bond's YTM when its price decreases?

YTM increases.

When does a discount bond have a negative YTM?

current price > face value

What is the relationship between maturity and bond-price sensitivity?

• The farther the maturity

• the greater the percentage price change

• from an interest-rate change

What happens when the holding period is shorter than a bond's maturity?

• bond has interest-rate risk

• because its future selling price is uncertain

What happens when a bond's holding period exceeds its maturity?

• investor faces reinvestment risk

• because payments must be reinvested at uncertain rates

What happens to a bond's return when interest rates rise?

• Bond prices fall

• creating a capital loss

• and lowering the bond's return

What happens to a bond's return when interest rates fall?

• Bond prices rise

• creating a capital gain

• and increasing the bond's return

How do you calculate the YTM of a fixed-payment loan?

LV = FP/(1+i) + FP/(1+i)^2 + . . . + FP/(1+i)^n

Can the YTM of a fixed-payment loan usually be solved by hand?

No. A financial calculator is typically needed because i appears in multiple terms.

How do you calculate the annual payment on a fixed-payment loan?

FP = LV [(i / 1) - (1+i)^-n]

What is the difference between YTM and current yield?

• YTM considers total cash flows over the bond's life

• Current yield: annual coupon payment ➗ current price

What is a paper loss on a bond?

• Capital loss

• caused by a price decline that has not been realized

• because the bond has not been sold

What is the after-tax real interest rate?

The real interest rate after accounting for income taxes and expected inflation.

How do you calculate the after-tax nominal interest rate?

After-tax nominal rate = nominal rate × (1 − tax rate).

How do you calculate the after-tax real interest rate?

After-tax real rate ≈ nominal rate × (1 − tax rate) − expected inflation.

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If YTM is less than the coupon rate, price is above face value.

What clues tell me a question is asking for present value?

Look for "worth today," "present value," or future payment(s) and an interest/discount rate.

What formula do I use for one future payment?

PV = CF/(1+i)^n.

What formula do I use when there are several future payments?

Discount each payment separately and add them: PV = Σ CF_t/(1+i)^t.

How do I recognize a payment that needs no discounting?

If the payment is made immediately/today, it is at t = 0, so PV equals the payment.

What clues tell me to use the one-year coupon-bond YTM formula?

Look for:

• one year to maturity

• a coupon

• face value

• current price

Use i = [(C+F)/P] − 1

What clues tell me to use the Fisher equation?

Look for:

• nominal interest, real interest, and expected inflation.

• Approx.: nominal = real + expected inflation.

What clues tell me actual inflation matters?

Look for "actual inflation" or "ex post." Use actual inflation to determine the realized real interest rate.

How do I know whether a bond is selling at a premium or discount?

• Price > face value = premium.

• Price < face value = discount.

A question gives me a bond's current price and expected future price. What do I calculate and how?

Calculate the capital gain rate:

• (Future Price − Current Price) / Current Price.

• If future price is higher, it is a capital gain; if lower, it is a capital loss.

A question asks for the ex ante real interest rate. Which inflation number do I use?

Use expected inflation: Ex Ante Real Rate ≈ Nominal Rate − Expected Inflation.

A question asks for the ex post real interest rate. Which inflation number do I use?

Use actual inflation. Ex Post Real Rate ≈ Nominal Rate − Actual Inflation.

A question gives me face value, coupon rate, YTM, and years to maturity and asks for the bond's current price. What do I do?

• Step 1 find the dollar coupon: Face Value × Coupon Rate.

• Step 2 P = C/(1+YTM) + C/(1+YTM)^2 + C/(1+YTM)^3 + FV/ (1+YTM)^3

What is the YTM formula for a simple loan?

i = [(CF/PV)^(1/n)] − 1

• amount received today as PV

• single future repayment as CF

How do I calculate total bond return?

Total Return = Current Yield + Capital Gain Rate, or Return = [Coupon + (Future Price − Current Price)] / Current Price.

How do I calculate the capital gain rate?

Capital Gain Rate = (Future Price − Current/Purchase Price) ➗ Current/Purchase

How do I calculate the YTM of a one-year coupon bond?

• Step 1 calculate the coupon: C = Face Value × Coupon Rate.

• Step 2 use i = [(C + Face Value) / Current Price] − 1.

How do I handle a payment received immediately in a PV question?

A payment received immediately is t = 0, so it is NOT discounted. Its present value equals the full payment.

How do I calculate the nominal interest rate when the real rate and expected inflation are given?

Use the Fisher equation: Nominal Rate ≈ Real Rate + Expected Inflation.

How do I calculate the return on a discount bond when there is no coupon?

• Because there is no coupon payment

• the bond's return comes entirely from the change in its price

• Return = Capital Gain Rate

How do I calculate current yield?

• Step 1: Annual Coupon = Face Value × Coupon Rate.

• Step 2: Current Yield = Annual Coupon / Current Price.

What does i represent?

The nominal interest rate.

What does πᵉ represent?

The expected inflation rate.

What is the approximate YTM formula?

YTM ≈ [C + (F − P)/n] ÷ [(F + P)/2]

How do you compare the YTM of two bonds?

Calculate each bond’s YTM, then compare the percentages. Higher YTM = higher yield.

Why does delaying loan payments lower YTM?

Payments farther in the future have lower present value, so the YTM must be lower.

How does maturity affect bond price sensitivity?

Longer-maturity bonds are more sensitive to YTM changes.

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The bond sells at face value: P = F.

If YTM < coupon rate, how does the bond sell?

At a premium: price > face value. If YTM > coupon rate, it sells at a discount.

Why is YTM useful for comparing financial instruments?

YTM

• converts different cash-flow structures into a common annual interest-rate measure

• making instruments easier to compare

When does current yield approximate YTM?

When the bond's price is close to its face value, so there is little capital gain or loss remaining.

What is the exact Fisher equation?

1 + i = (1 + r)(1 + πᵉ), where i = nominal rate, r = real rate, and πᵉ = expected inflation.

What is the Fisher adjustment term?

The adjustment term is rπᵉ. It appears when the exact Fisher equation is expanded: i = r + πᵉ + rπᵉ.

What happens when actual inflation is higher than expected?

Borrowers benefit relative to expectations because they repay with dollars worth less; lenders receive a lower real return than expected.

What happens when actual inflation is lower than expected?

• Lenders benefit relative to expectations because they receive a higher real return

• borrowers face a higher real cost than expected

What happens to a discount bond's price as maturity approaches?

• Its price moves upward toward face value

• assuming no default and no change in required yield

What happens to a premium bond's price as maturity approaches?

• Its price moves downward toward face value

• assuming no default and no change in required yield