Long term interest rates are more stable because government policy which affects short-term rates.Ie Long term rates are less volatile.

However long term bond prices are more volatile than short term rates as Durartion (price sensitivity increases with maturity generally).

A primary measure of risk for a bond. % change in bonds price for every 1% change in interest rates. 

* Macualy duration = (Sum of PVCF x T)/(Sum PVCF)

then MD = Duration / (1+r)

Allows issuer to call back bond at pre-set call price. Bond owners recieve a penalty for the issuers calling the bond.

If interest rates fall the issuer can call the bond and issue new bonds at lower interest rates meaning lower coupon payments

Gives the buyer of the bond the right to sell the bond back to the issuer at the pre-set put price.

If interest rates rise the buyer can 'put' the bond, ie sell back to the issuer and buy new bonds at a higher interest rate, ie. higher coupon payments.

MD* = (P- - P+) / (2Po x change in r)

Where

P- = the the price of the bond with r - change in r

P+ = the price of the bond with r + change in r

-Faster for bonds with many CF's

- Can be used for any securities where rates change (ie bonds with embedded options) 

Spot rate is an interest rate at t=0.

Forward rate is any interest rate in the future.

1. measure of interest rate risk (sensitivity), can compre interest rate risk with other bonds/portfolios
2. To estimate a bonds price when if interest rates changed.
3. Hedging and speculating

• Flat yield curve, this is false/ very rare (bootstrapping to construct yield curve)
• Parallel shifts of the curve, this is unlikely as short term rates are more volatile than long term rates.

Convexity adjusts for non linearity in price/ yield curve (because duration is linear), if there is a small change in r the difference is negligable, however for large change in r need to adjust for convexity.

Formula 

CM = [(P^- + P^{+) -} 2P₀] / (P₀ x change in r²⁾

using only duration 

P₁ = [1 - (MD x change in r)]

using duration and convexity 

P₁ = [1 - (MD x change in r) + 1/2(CM + change in r²)]

#NB. P₁ is approx. equal.

• T-Bills
• Coupon stock 
• Inflation Indexed bonds
• Kiwi Bonds

• Bank Bills
• Commercial Bills
• Corporate Bonds
• Kauri Bonds

• 90 day bank bill futures/options
• OCR Futures
• 3yr Government stock futures/options
• 10yr Government stock futures/options

Draw yield curves; 

• Upward sloping (normal) 
• Downward sloping (common in NZ)
• flat 

What are the 3 main theories explaining the yield curve?

1. Expectations
2. Liquidity Premium 
3. Market Segmentation 

2 Theories under Liquidity Premium and brief descriptions.

1. Liquidity Premium.

- Nothing to do with Liquidity at all, reall Maturity premium

- Forward rate = expected future spot + LP

- LP > 0 always, and increasing with maturity.

- always below yield curve since always +.

LP compensates for uncertainty, as longer you hold the more things that can change

2. Preffered Habitat.

- Similar to LP, exept LP can be <,=,> 0, and not strictly increasing with time.

- Borrowers and lenders have preffered maturities depending on goals.

- borrowers and lenders can change maturity if better rates at different maturities.

-Like Preffered habitat in that borrows/ lenders have a desired maturity, but different in that they are restricted to a specific maturity because of retirement and regulations.

-This theory is to strict, a large quantity of investors do have the flexibilty to take advantage of different interest rates at different maturites.

Interest rate of uncertainty. 

- Price uncertainty is bigger for longer maturities

- Interest rate uncertainty is bigger for shorter maturites.

- Disproves Expectations theories but allows LP.

•  constant standard deviation
• tree recombines
• No arbitrage trees, ie. price at end gives you P₀

Work backwards from the end of binomial tree.

Formula 

P^H = [(P^HH + P^HL) + CF] / (1 + r^H)

What happens to Duration as;

• coupon increases
• YTM increases
• Maturity increases

• duration decreases as its is price sensitivty to interest, if coupon increases greater weight of payment earlier in bonds life.
• Duration decreases, same reason???
• Duration increases (generally increases with maturity) as long term bonds have greater price sensitivity to changes in interest rates.

Level Loan (table loan).

- it is annuity with fixed payments, each time you make a payment you pay some principle so the balance you pay interest on decreases and amount of principle in each payment increases.

- Most are 20-25 year maturity fixed for approx. 5 years (largest fixed in NZ 20years and rare)

30year fixed loans.

- No penalty for prepayment, essentially free option to borrower.

-Homogenous contracts, easy for banks to package up and sell.

- Loans packaged up and sold in slices

- Banks do this to get loans of balance sheet so they can get more cash to lend out again.

- Equal participation for investors, ie. if own 20% recieve 20% of each CF.

Because no penalty for early payment mortguage holders have option to pay back loan and borrow at lower rate if interest rates fall, bad for investors as large reinvestment risk (have to reinvest at lower rates).

Like an SPV in that loans are packaged up and sold, however CMO's prioritise claims so that class 1 will get paid back first then 2 etc.

The main point of a CMO is certainty of maturity.

For bank take advantage of market segmentation and get people in each class to pay a little more (arbitrage)

Problem if there is not enough collateral to go around.

- when interest rates decrease (pay lower coupons)

- to eliminate a restrictive covenant

1. American option - can be called at any time
2. Bermudan option - can be called on specific dates, usually at coupon payment date.
3. European options -can only be called at one point in time.

Price of a callable bond is approx = to price of non-callable bond - call option 

-from an investors POV a call option lowers value

- Use binomial tree to find price of call option.

• Call price - the higher the call price the less likely a bond is to be called
• Time to maturity - the longer the maturity the more chance the price will increase to call price and bond will be called
• Shape of yield curve - upward decreases price of option, downward increases price of option
• Volatility of interest rates - the more volatile interest rates the more chance price increase to call price an bond called.

• Bond buyers have the right/option to sell back bond at preset price
• If interest rates increase, the cac sell back and buy bew bond with higher coupon payments. Or If their is a change in credit rating or a takeover (both affect price of bond)
• Price of a putable bond approx. = price of a non putable bond + price of option (adds value for investor)

Used to analyse any risky bond

-amount added to spot curve so that PV of CF = market price ie.

P₀ = CF₁/(1 +₀r₁ + z) +... CF_t/(1 + ₀r_t + z)

-no uncertainty about interest rates (hardcore pur expectatations)

- makes no adjustments for adjustments in CF's ie. if bond put/called because no uncertainty about interest rates.

OAS includes 

-Credit risk

- Liquidity risk 

- Mispricing

-used for evaluating price differences between similar bonds with different embedded options, Large OAS implies greater return for greater risk (always choose highest OAS if other factors constanint)

OAS = Static - option cost

P^H = [1/2(P^HH + P^HL) + CF]/ (1 + r^H + OAS)

Futures = agreement where price set today to exchange assets at a later date.

- standardized and done through market, liquid.

Forwards - hard to trade except with party you created with, illiquid.

Pricing Futures.

P = notional value (facevalue) / [1 + ([100 - F]/100)^x/365]

Where F = 100 - YTM

- Good faith deposit

-deposit to your broker who deposits at exchange.

When you trade a futures you trade with another party, exchange becomes buyer/ seller.

- Margin stops you from abondoning obligation.

- margin is not buying futures (pay nothing for it)

-intitutions can put T-bills and earn interest on it it, individuals must use $.

1. Arbitrage, if you calcualte future price to be different to market price buy/shortsell stock, and buy /shortsell futures.
2. Hedging, use to get certainty of return 
3. Speculation - if you think price going to increase/ decrease.

[(MD_t - MD_p)/MD_F] x (MV_p/FV) = N_F

IF N_F '+' buy, IF N_F '-' sell.

Where 

FV (FaceValue) = Futures price x multiplier (1000)

Futures price = 100 - YTM

NB- common target MD = 0

Swap is like future (agreement to exchange assets in the future) however have repeated exchanges, where futures only one.

1. FX (interest payments in different currencies)
2. Total return swaps (exchange total return on asset)
3. Credit default swaps (like insurance for credit event)
4. Interest rate swaps (fixed fro floating)

NZ government uses swaps to manages interest rate position.

- contract prices on futures change daily, so day to day gains/losses are exchanged.

- this reduces the chance of fleeing without paying.

NCF = (F - f) x Notional Value 

where

Fixed rate is known

Floating rate is payed in arears, ie. first payment is always known f₁ = ₀r₁, f₂ = ₁r₂.

• Hedging - protect against changes in interest rates
• Speculating - on changes in interest rates
• Arbitrage.

-Liquid (to open a swap, have to negotiate with other party to end early)

- Low cost, no commission

N_s = [(MD_T - MD_p)/MD_s] x MV_p

where 

MD_s = MD_F - MD_f - 1/2(reset period)

we only deal with swaps done annually so reset period=1

Futures Profit = (P_sell - P_Buy) x multiplier x N_F

if you short sell (ie you want to sell for higher than buy.

If bought futuers would be P_buy - P_sell

HPR = (P₁ - P₀ + CF) / P₀

In a futures you do not but $ for the principle, only the margin, therefore do not include marhin in investment (P₀) for return.

• an effective hedge gives certainty to return. If you make more than you planned still ineffective hedge because not certain return.
• The reason YTM does not normally equal return on a hedge is because the rates on the future and the bond change by different amounts over the time period.

1. Bankruptcy - either failure to pay or breach of covenant
2. Rating downgrades - bad from investors point (if owned bond)
3. Takeover - bonds may get lower priority
4. Restructuring - change of terms in contract (lower interest, longer maturity, principle reduction etc).
5. Repudation - deny having debt
6. Cross default 
7. cross acceleration

- Find bonds whose rating about to change, buy/sell before they do. (generally a delay between trouble and rating change)

-best to fond changes from BBB to BB (vise versa) because change from investment to junk grade bonds, many institutions have covenants restrcting what they can buy. 

- 'Fallen angels', good company in bad times, may have CF problems.

• Exchange the total return on assets 
• Banks/ Owners are not required to report
• Use to transfer economic owenership while maintaning ownership/control (may want less risky return but still control in company).

• A buyer buys protection, paying fixed quarterly payments to the protection seller. The protection sellers payment to protection buyer is conditional on a credit event.l
• Usually cash settlement (100 - auction price)

Used like insurance for bond holder, can speculate if you expect/ don't expect credit event.

1. Empirical 
2. Reduced form
3. Structural

- Logistics regression, ie statistical

- Altmans Z score, predict bankruptcy

<1.81 - default

1.81<z<2.99 - grey area

>3.00 - likely to survive

-Probability of bankruptcy is known

Timing is unknown,

-default is at the end of a branch(binomial tree)

-Bankruptcy is random (exogenous)

P(default) = e^-*t

P(No default) =1 - e^-*t

combinations of options can create positions that are the same as holdong the stock itself.

call + cash = stock

put + shortsell = stock

P(0,t) = Ke^*-rt - P(A, K, std.dev, rf, t)

from bondholders perspective.

• A = Assets, increase is good, more likely  to be paid off.
• K = FaceValue of debt, increase is bad, harder to make payments
• std. dev of debt, increase is bad, more uncertainty, lowers the value of the debt
• rf = ?
• t = time to maturity, increase is good, more time, more chance of using put.

P = Ke^-rt x N(-d2) - AN(-d1)

where K= FaceValue of debt

A = Assets

d1 = [ln(A/K) + (rf + [std.dev²/2])T]/ std.dev x T^1/2

d2 = [ln(A/K) + (rf - [std.dev²/2])T]/ std.dev x T^1/2

or d2 = d1 - (std.dev x T^1/2)

use normal distribution tables to N(-d1), N(-d2)

Value of risky debt = (e^-*t(1 - e-*t)S)r_f

ie. =(prob of not default x R_f) + (prob of default x R_f x recovery rate)

where S = to recovery rate

prob of deault in real world = N(-DD)

where DD = Distance to default

DD = [ln(A/K) + (m - (std.dev²/2))T]/ std.dev x T^1/2

where m = growth rate of assets,

NB (m - (std.dev²/2)) = arithmatic return - correction factor. = geometric return.

y = - [(ln(D/K))/T]

yield spread = y - rf

1. Target - towards specific CF's
2. Benchmark - try to beat the market

MARIOUS

M = Manageable - evaluate/ calculate over time

A = Appropriate, ie if managing a short term maturity portfolio should not benchmark against long term debt

R = Reflective - of current market views

I = Investable - should be able to invest in the benchmark

O = Owned - willing to trade/ build against it

U = Unambiguos - clear what the benchmark is

S = Set in advance - must determine what the benchmark is before you try to beat it, not after.

- replicate the index 

- "can't beat'em join'em"

- Tend to uderperform because costs; commission, fees, trading costs etc. (Passive portfolio minimised.)

-Bonds are hard to index, have to buy in proper proportions.

If can buy in proper proprtions and model index, called full replication (almost impossible to do)

1. Stratified Sampling - choose a subset of bonds based on various characteristics
2. Optimization - keep return on portfolio as close to return on benchmark as close as possible subject to a set of possible restraints

1. Duration / maturity 
2. Quality / rating
3. currency 
4. optionality
5. sector / industry

1. diversification level
2. Duration 
3. quality,

...