L8 - Financial Maths

Question 1

What is the equation for Discounted Present value of a bond?

Answer 1

Question 2

What is the equation for Discounted Present Value of a bond with Semi-Annual Payments?

Answer 2

Question 3

What is Annuity and how is it calculated?

Answer 3

• An annuity is a financial product that pays out a fixed stream of payments to an individual. These financial products are primarily used as an income stream for retirees. Annuities are created and sold by financial institutions, which accept and invest funds from individuals. Upon annuitization, the holding institution will issue a stream of payments at a later point in time.
• The period of time when an annuity is being funded and before payouts begin is referred to as the accumulation phase. Once payments commence, the contract is in the annuitization phase.

Question 4

How is Annuities and Perpetuities linked?

Answer 4

Question 5

How do you calculate Market Value for different types of bonds?

Answer 5

Question 6

How do you calculate Yield to Maturity of different bonds?

Answer 6

Yield(y) = Yield is the market rate that when used for discounting values the bond at its ‘fair value’ price

• The issue with the calculation of the coupon bond is that is it a ccrude result adn does not allow an approximation of YTM when a bond sells at a premium

Question 7

What is a better approximation of YTM?

Answer 7

• works when bond is trading at a premium!
• want our answers in annual yield

Question 8

How does nominal and real interest rate effect the present value?

Answer 8

nominal - if you are expecting to receive £1000 in nominal terms in 3 years time means you expect a £1000 cash flow to occur in 3 years

real - in 3 years time you expect the equivalent of £1000 today to occur no just the cash flow of the future £1000

(1+r_nominal) = (1+r_real)(1+π)

( in calculations when using real include inflation to the power of the same time period)

Question 9

What are the two types of risk that affect bond prices?

Answer 9

Changes in bond prices imply changes in the interest rate received by bond holders

– Specific risk or default risk is associated with sales of bonds to a specific purchaser

– Interest rate risk is associated with returns to bond holders and varies with the maturity of bonds in the investor’s portfolio

Question 10

How are bonds affected by maturity?

Answer 10

• The longer the maturity of a debt instrument, the greater the price change associated with any given change in interest rates.

– For a given bond, the lower the coupon, the greater the price change associated with a given change in interest rates.

Question 11

How do you calculate Macaulay Duration?

Answer 11

Question 12

What is the difference between maturity and duration?

Answer 12

• Duration is a measure of the sensitivity of the price of a bond or other debt instrument to a change in interest rates - its a measure of the lie of a bond, and its importance lies in its use as a measure of interest rate risk
• Maturity is the date on which the life of a transaction or financial instrument ends, after which it must either be renewed, or it will cease to exist.

Question 13

What are the properties of Duration?

Answer 13

Duration is always less than or equal to maturity (as you multiple by time)

– For all bonds, duration increases with time to maturity

– Duration increases as coupon and yield decrease (gives greater time preference value to coupons that are received sooner rather than later)

– For pure discount bonds duration is equal to maturity

Question 14

Why is duration equal to maturity for pure discount bonds?

Answer 14

• as there is no interim cashflows
• duration - weight of average time for payments –> pure discount bonds there is no waiting time as there is no coupon payment –> thus Duration = Time to Maturity

Question 15

What is Duration a useful measurement of?

Answer 15

Duration is a measure of ‘the life of a bond’ and its importance lies in its use as a measure of ​interest rate risk

Question 16

How is Duration linked to Interest Rate Risk?

Answer 16

• Similar to Elasticity equation
• – The LHS of equation 10 is the elasticity of the bond with respect to (one plus) the yield to maturity.
• The RHS is (the negative of) duration. In other words, duration measures the interest rate elasticity of the bond’s price. It is therefore a measure of interest rate risk. The lower the duration, the less responsive is the bond’s value to interest rate fluctuations

Question 17

What is Modifed Duration?

Answer 17

• Uk market participants also use a concept known as Modifed Duration (sometimes known as Volatility)
• modified duration measures the price sensitivity of a bond when there is a change in the yield to maturity –> accounts for different yield of maturity
• Modified duration is measured in years as:

MD=D/1+r

D - Duration (more specifically this refers to Macaulay Duration)

Question 18

How do you calculate Bond Price Changes based on Modified Duration?

Answer 18

Question 19

How is Duration related to Volatility?

Answer 19

• In general the volatility of a bond is proportional to its duration.
• In particular, the greater the duration of a bond, the greater its price sensitivity or volatility for any given change in yields.

Question 20

How can the relationship between bond prices and yields be related?

Answer 20

• We know that the percentage increase in price when yields decline is greater than the percentage decrease when yields rise.
• This implies a convex relationship between bond prices and yields.

Question 21

How is Convexity related to Interest Rate Risk?

Answer 21

• Duration is a first order measure of interest rate risk.
• Convexity, on the other hand, can be regarded as a second order measure of risk.
• Convexity therefore describes the way in which duration changes as yield to maturity changes.

Question 22

What is the Formula for Convexity?

Answer 22

Question 23

What is an Alternate way of measuring Convexity?

Answer 23

Question 24

What is Immunisation?

Answer 24

• Managers often hold a portfolio of assets to meet a long term liability
• – To ensure the value of the portfolio at maturity is sufficient to meet the future liability, a portfolio must be immunised –> against changes in yield to maturity - and thus the price of the bond
• Immunisation –> the point where the value of a portfolio at maturity is sufficient to meet future liabilities
• – A portfolio cannot be immunised without knowledge of duration

Question 25

What is the Simpliest for of Immunisation?

Answer 25

• Immunisation implies ‘cash matching’ of assets and liabilities.
• – Suppose a fund manager has a liability of £1,000,000 maturing in ten years
• – The portfolio is fully immunised if the fund manager buys a pure discount bond with a duration of ten years and a maturity value of £1,000,000

Work but not very cost effective

Question 26

Why may immunisation by cash matching not be possible?

Answer 26

• Cash matching with pure discount bonds works well, but only if a suitable pure discount bond is available
• If this is not the case, immunisation is more difficult because: -
• – It might be necessary to take account of price risk
• – It will be necessary to take account of reinvestment risk

Question 27

What does Immunisation look like on a graph?

Answer 27

• In Figure 1, the desired holding period is t₀ to t₁ . •
  
  • Note that whatever happens to interest rates, the value of the portfolio is always the same at time t₁ .
• If interest rates do not change throughout this period the value of the portfolio (including reinvestment income) is shown by the solid line.
• • If interest rates rise at time t₀ the value of the portfolio will fall and its value is shown by dotted line.
• • If interest rates fall at time t₀ the value of the portfolio will rise and its value is shown by dashed line.
  
  • Being Immunisated means that at a future time period t₁ no matter the yield to maturity, the value of the portfolio in the future will be a specific value (the Macauley Duration) that would be equilivalent future liabilities

Question 28

How do you immunise a portfolio with a Single Payment Liability?

Answer 28

• Consider a funding manager that has a single payment liability of £1,931,000 maturing in 10 y ears time.
• – Assume that the yield to maturity is 10 per cent
• . – The PV of the liability is £745,000.
• – Since we are considering a single payment liability, the Macauley Duration of the liability equals its maturity (10 years).

If this was an annuity will multiple interim cashflow payments, you would need to find the PV of each payment, and invest in bonds with a duration of 10 years

Question 29

How does a change to the Yield to Maturity effect the ability of an immunised portfolio to meet its future liabilities?

Answer 29

• No matter if it goes up or down you will generate a profit, however if it stays the same you will just break even

Question 30

How do you immunise a Portfolio with multiple Liabilities and Bonds?

Answer 30

• if you have multiple liabilities unlikely to invest in just a single bond - would be hard to cash match based of the maturity of the debt obligation

1. Find PV of Liability
2. PV of Each of the Bonds
3. Duration of Each of the Bonds - for pure discount Duration = Time to Maturity
4. Assign Weight in the portfolio - which will be the sum of all duration multiplied by the proportion of the fund to be invested in that bond –> which should equal the duration of the liability
   
   1. sum of all proportion = 1
5. solve the via simultanous equations (the assigned weight in the portfolio and the sum of all proportion equations)
6. Based on the proportion calculate how much you will invest in each bond –> weight x PV of liability

Question 31

What are the assumption that apply when we have calculated immunised portfolios?

Answer 31

• Fund manager is risk adverse
• No defaults
• When the pure discount bond matures, another suitable bond is available to replace it in the portfolio - maintain full immunisation for the lifetime of the debt obligation
• Level term structure
• Re-balancing