4 Flashcards
what are interest rates
the proportion of a loan that is charged as interest to the borrower
Why do we need to value and compare payments made at different times?
To understand the true value of money received in the future versus today, taking into account the time value of money.
Why is a £1 paid in the future less valuable than a £1 paid today?
Due to the time value of money — money today can be invested to earn interest, making future payments worth less in today’s terms.
what is present value
Present value is the value today of a payment that is promised in the future, discounted to account for the time value of money.
What concept rewards time in financial payments?
The concept of interest rewards time by adding value to payments received in the future.
why do we need present (discounted) value
- needed to compare payments made at different times.
- It helps determine the current value of a future payment, taking into account the time value of money—the idea that money today is worth more than the same amount in the future.
formula for future cash flow
𝐶𝐹 = 𝑃𝑉 (1 + 𝑖)^𝑛
- 𝑃𝑉 ≡ today’s (present) value
- 𝐶𝐹 ≡ future cash flow (payment)
- 𝑖 ≡ the interest rate
- 𝑛 ≡ the number of years to maturity
What is the formula for future cash flow (CF) given present value (PV)?
CF = PV × (1 + i)ⁿ
How do you calculate the present value (PV) of a future cash flow?
PV = CF / (1 + i)ⁿ
credit market instructions (4)
- simple loan
- fixed payment loan
- coupon bond
- discount bond (aka zero coupon bond)
what is a simple loan
A simple loan involves borrowing a principal amount and paying back that principal along with interest at a set time in the future (usually at maturity).
what is a fixed payment loan
A fixed payment loan requires the borrower to make equal payments (comprising both principal and interest) over a fixed period. e.g. mortgage
what is a coupon bond
A coupon bond pays the bondholder a fixed interest payment (coupon) at regular intervals (e.g., annually or semi-annually) until the bond matures, at which point the face value (principal) is repaid.
what is a discount bond
A discount bond is purchased at a price below its face value, and the full face value is repaid at maturity. There are no periodic interest payments—just a return when the bond matures.
key differences between the instruments
Simple Loans: Repay principal and interest at maturity.
Fixed Payment Loans: Repay through equal periodic payments of principal and interest.
Coupon Bonds: Receive regular interest payments until maturity, and face value is paid at the end.
Discount Bonds: Buy below face value and receive the full face value at maturity with no regular interest payments.
what is yield to maturity
YTM is the “interest rate” that equates the present value of cash flow payments from a debt instrument with its value today.
What does the Yield to Maturity (YTM) reflect in terms of investment?
YTM reflects the return an investor will earn if the debt instrument is held until maturity, based on the current market price, interest payments, and the time to maturity.
What is another name for Yield to Maturity (YTM)
Internal Rate of Return (IRR).
What factors affect the calculation of Yield to Maturity (YTM)?
The calculation of YTM depends on the structure and the timing of payments and repayments.
key features of yield to maturity (3)
Equates Present Value to Current Price
Depends on Structure & Timing of Payments
Used for Comparing Bonds
YTM for simple loans formula
one single repayment at maturity
𝑃𝑉 = 𝐶𝐹 / (1 + i)ⁿ
YTM for fixed payment loans formula
The same cash flow payment every period throughout the life of the loan.
𝐿𝑉 = 𝐹𝑃 / (1 + 𝑖) + 𝐹𝑃 / (1 + 𝑖)² + ⋯ + 𝐹𝑃 / (1 + 𝑖)ⁿ
What is the general structure of a coupon bond?
A coupon bond pays fixed interest every year until maturity, plus a specified final amount (the face value)
What is the coupon rate formula for a bond?
The coupon rate is calculated as:
Coupon rate = 𝐶 / 𝐹
What is the formula for the price of a coupon bond (𝑃)?
𝑃 = 𝐶 / (1 + 𝑖) + 𝐶 / (1 + 𝑖)² + ⋯ + 𝐶 / (1 + 𝑖)ⁿ + 𝐹 / (1 + 𝑖)ⁿ
What happens when a coupon bond is priced at its face value?
When a coupon bond is priced at its face value, the yield to maturity (YTM) equals the coupon rate.
What is the relationship between the price of a coupon bond and the yield to maturity (YTM)?
The price of a coupon bond and the yield to maturity (YTM) are negatively related.
What happens to the yield to maturity (YTM) when the price of a bond is below its face value?
The yield to maturity (YTM) is greater than the coupon rate when the price of the bond is below its face value.
What is a consol or perpetuity?
A consol or perpetuity is a coupon bond with no face value and no maturity. It does not repay principal but pays a fixed coupon periodically, forever.
What does 𝑃𝑐 represent in the context of a consol?
𝑃𝑐 represents the price of a consol.
What does 𝐶 represent in the context of a consol?
𝐶 represents the yearly payment or coupon paid by the consol.
What is the formula to calculate the price of a consol (𝑃𝑐)?
The formula to calculate the price of a consol is:
𝑃𝑐 = 𝐶 / 𝑖𝑐
What is the formula to calculate the interest rate (𝑖𝑐) of a consol?
The formula to calculate the interest rate (𝑖𝑐) of a consol is:
𝑖𝑐 = 𝐶 / 𝑃𝑐
What is the relationship between coupon bonds and current yield?
For coupon bonds, the equation 𝑃𝑐 = 𝐶 / 𝑖𝑐 provides the current yield, which is an easy-to-calculate approximation of the yield to maturity.
What is a discount bond?
A discount bond is bought at a price below its face value, and the face value is repaid at maturity.
What is the formula to calculate the yield to maturity (𝑖) for a discount bond?
𝑖 = (𝐹 − 𝑃) / 𝑃
How is the yield to maturity (YTM) for a 1-year discount bond calculated?
The yield to maturity (YTM) for a 1-year discount bond equals the increase in price over the year divided by the initial price.
What is the relationship between the yield to maturity (YTM) and the current bond price?
The yield to maturity (YTM) is negatively related to the current bond price
YTM discount bond formula
𝑃 = 𝐹 / (1 + 𝑖)ⁿ
What is the Rate of Return?
The Rate of Return is the payments to the owner plus the change in value, expressed as a fraction of the purchase price.
How does the Rate of Return differ from yield to maturity?
The Rate of Return is not the same as yield to maturity. Yield to maturity is the “interest rate” that equates the present value of cash flow payments received from a debt instrument with its value today.
What does 𝑅 represent?
𝑅 represents the return from holding the bond from time 𝑡 to time 𝑡 + 1.
What does 𝑃𝑡 represent in the context of a coupon bond?
𝑃𝑡 represents the price of the bond at time 𝑡.
What does 𝑃𝑡+1 represent in the context of a coupon bond?
𝑃𝑡+1 represents the price of the bond at time 𝑡 + 1.
What does 𝐶 represent in the context of a coupon bond?
𝐶 represents the coupon payment of the bond.
What is the formula for the Rate of Return (𝑅) for a coupon bond?
𝑅 = (𝐶 + 𝑃𝑡+1 − 𝑃𝑡) / 𝑃𝑡
How can the Rate of Return (𝑅) be split into two terms for a coupon bond?
The Rate of Return (𝑅) can be split into:
𝑅 = (𝐶 / 𝑃𝑡) + ((𝑃𝑡+1 − 𝑃𝑡) / 𝑃𝑡)
This is equal to:
𝑅 = 𝑖𝑐 + (𝑃𝑡+1 − 𝑃𝑡) / 𝑃𝑡
What is the formula for Rate of Return (𝑅), using the components 𝑖𝑐 and 𝑔?
𝑅 = 𝑖𝑐 + 𝑔
𝑖𝑐 is the current yield
𝑔 is the rate of capital gain
How does the maturity of a bond affect its price change associated with an interest-rate change?
The more distant a bond’s maturity, the greater the size of the percentage price change associated with an interest-rate change.
How does the maturity of a bond affect its rate of return when interest rates increase?
The more distant a bond’s maturity, the lower the rate of return when the interest rate increases.
Can a bond with a high initial interest rate still have a negative return?
Yes, a bond with a high initial interest rate can still have a negative return if interest rates rise.
What is interest-rate risk?
Interest-rate risk is the riskiness of an asset’s return resulting from interest rate changes.
How does maturity affect the volatility of a bond’s price and return?
Prices and returns for long-term bonds are more volatile than those for shorter-term bonds because long-term bonds have higher interest rate risk.
Is there any interest-rate risk for bonds with a maturity equal to the holding period?
yes as interest rates may still rise
What is the nominal interest rate?
The nominal interest rate makes no allowance for inflation.
What is the real interest rate?
The real interest rate is adjusted for expected changes in the price level, making it a more accurate measure.
What does the real interest rate reflect?
The real interest rate reflects the cost of borrowing.
What is the difference between ex-ante and ex-post real interest rates?
The ex-ante real interest rate is adjusted for expected changes in the price level.
The ex-post real interest rate is adjusted for actual changes in the price level.
What does the Fisher Equation represent?
i=r+π
i = nominal interest rate
r = real interest rate
𝜋𝑒 = expected inflation
What does the nominal interest rate (i) in the Fisher equation represent?
The nominal interest rate (i) is the interest rate that is observed in the market, not adjusted for inflation.
What does the real interest rate (r) in the Fisher equation represent?
The real interest rate (r) reflects the true cost of borrowing or the true return on lending, adjusted for expected inflation.
What does expected inflation (𝜋𝑒) represent in the Fisher equation?
Expected inflation (𝜋𝑒) represents the anticipated rate of inflation over the period of the loan or investment
What happens when the real interest rate is low?
When the real interest rate is low, there are greater incentives to borrow and fewer incentives to lend.
Why is the real interest rate a better indicator for borrowing and lending incentives?
The real interest rate is a better indicator of the incentives to borrow and lend because it accounts for expected inflation, reflecting the true cost of borrowing and the true return on lending.
