1.01.2005

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» Q&A: Which is the Z-spread for...
» Q&A: What's the 6-month forward rate...
» Q&A: What is the total cash flows received...
» Q&A: What is the reinvestment income if...
» Q&A: If an investor holds a 14.00%, 17.50 year bond bought at $950.00 to maturity...
» Q&A: Is there an arbitrage opportunity...
» Q&A: List at least 2 reasons why Yield-To-Maturity...
» Q&A: Under what circumstance should an American call...
» Q&A: Under what types of market structure would...
» Q&A: What is the duration/convexity of...

Q&A: Which is the Z-spread for...

...a 3.5-year, 9.60%, $100 par non-Treasury bond selling for 110.2950: 100 basis points, 143bp, or 165bp?

Period Spot Rate
1 0.03
2 0.033
3 0.035053
4 0.039164
5 0.044376
6 0.04752
7 0.049622

Answer : PV the cash flows (discounted at the appropriate spot rate + spread) and compare to the purchase price. The 143 Z-spread yields a purchase price of 4.70 + 4.58 + 4.46 + 4.32 + 4.15 + 4.00 + 84.08 = 110.2950.

  • Coupon = 9.6/2 = 4.8.
  • Coupon(i) PV = 4.8/(1+( Spot(i) + Z-spread)/2)^i
  • Period 1: 4.8/(1+(0.03+0.0143)/2) = 4.6960
  • Period 2: 4.8/(1+(0.033+0.0143)/2)^2 = 4.5808
  • Period 3: 4.8/(1+(0.035053+0.0143)/2)^3 = 4.4615
...
  • Period 7: (100+4.8)/(1+(0.049622+0.0143)/2)^7 = 84.0850

Category: C++ Quant > Finance > Debt > Valuation

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Q&A: What's the 6-month forward rate...

... 2 years from now given the following spot rates?

Period Years to Maturity Spot Rate
1 .5 4.250%
2 1 4.750%
3 1.5 5.050%
4 2.0 7.000%
5 2.5 8.750%
6 3.0 9.250%

Answer : 4th period forward rate can be derived from spot rates of 5th period and4th period. Forward rate = [(1 + 5th_spot_rate / 2)^5 / (1 + 4th_spot_rate / 2)^4] -1 = [(1 + 0.0875/2)5/(1 + 0.0700/2)4] -1 = 7.95%. Annualized the rate one gets 15.9%.

Bonus Points

  • forward rates assess the future interest rate for some period in the future. Useful when one can't decide how long to hold a bond, such as 4 periods or 5 periods.
  • The notation (1)f(4): the subscript before "f" is the length of time that the rate applies (ie. 1 period=6 months). The subscript after "f" refers to the period when the forward rate begins (ie. 4 periods = 2 years into the future).

Category: C++ Quant > Finance > Debt > Valuation

Your Turn!

 

Q&A: What is the total cash flows received...

... if an investor purchases a 3-year, 8% coupon bond that has a 10% YTM, assuming the semi-annual bond is held to maturity?

Answer : Total cash flow = FV of total Interest Payments + Capital gain = $272.08 + $50.76 = $322.84

  • FV of total Interest Payments: using Excel's Rate function with Nper = 3*2, Rate = 10%/2, PMT = 1000*(8/2)%, one gets FV = $272.08
  • Capital gain = Par - PV = 1000 - $949.24 = $50.76
    • using Excel's PV function with Nper = 3*2, Rate = 10%/2, PMT = 1000*(8/2)%, FV =1000, one gets PV = $949.24

Category: C++ Quant > Finance > Debt > Valuation

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Q&A: What is the reinvestment income if...

...an investor holds a 14.00%, 17.50 year bond bought at $950.00 to maturity, with the market interest rate at 15.00%?

Answer

  • FV of total cash flows = PV * (1+r)^N = $950.00 * (1+15/2/100)^(17.50*2) = $11940.43
  • Coupon Interest Payments = 14% * $1000 * 17.50 = $2,450.00
  • Capital gain/loss = $1000 - 950.00 = $50.00
  • Reinvested interest = total cash flow return - Coupon Interest Payments - Capital gain/loss = FV of total cash flows - Purchase price - Coupon Interest Payments - Capital gain/loss = $11940.43 - $950 - $2,450.00 - $50.00 = $8490.43

Bonus Points

  • In additon to reinvestment income, two other sources of income from holding a bond to maturity are coupon interest and capital gain/loss (when matures/called/sold)
  • Reinvested interest is also known as "interest on interest".
  • Two factors that affect the degree of reinvestment risk:
    • maturity: the longer the maturity, the higher the reinvestment risk
    • coupon rate: the higher the coupon rate, the higher the reinvestment risk. This implies that premium bond will be more dependent on reinvestment income than a bond selling at par.

Category: C++ Quant > Finance > Debt > Valuation

Your Turn!

 

Q&A: If an investor holds a 14.00%, 17.50 year bond bought at $950.00 to maturity...

...what is the reinvestment income, assuming rates remain constant?

Answer

  • Required Rate of Return = current yield/2 = Annual interest payment / Bond price / 2= 14/950 / 2= 0.7%
  • FV of total Interest Payments = Annuity * ( ((1 + r)^N - 1) / r )= $70.00 * ( ((1 + 0.7%)^35 - 1) / 0.7% ) = $2765.31
  • Coupon Interest Payments = 14% * $1000 * 17.50 = $2,450.00
  • Reinvestment income = FV of total Interest Payments - Coupon Interest Payments = $2765.31 - 2450 = $315.31

Category: C++ Quant > Finance > Debt > Valuation

Your Turn!

 

Q&A: Is there an arbitrage opportunity...

...given the following market data for $1000 par bonds?

Year .5 1.0 1.5 2.0
Market Price $900 $900 $900 $900
Coupon 10% 10% 10% 10%
Spot Rate 15% 16% 15% 14%

Answer : One should buy the 2 year Treasury, strip the coupons and principal, and sell at the spot rates for a risk free profit of $30.67. The conclusion is based on discounting each cash flow at the prevailing spot rate.

  • Period 1 = 50 / (1+15%/2) = 46.51
  • Period 2 = 50 / (1+16%/2)^2 = 42.87
  • Period 3 = 50 / (1+15%/2)^3 = 40.25
  • Period 4 = 1050 / (1+14%/2)^4 = 801.04
  • Total cash flow PV = Period 1 + Period 2 + Period 3 + Period 4 = $930.67 > $900

Bonus Points

  • Reconstitution of strips also occurs when the strips in the market are priced less than the bond (will continue until the bond price does not vary materially from the arbitrage-free value.)

Category: C++ Quant > Finance > Debt > Valuation

Your Turn!

 

Q&A: List at least 2 reasons why Yield-To-Maturity...

...of a bond may not be realized.

Answer

  • Reinvestment risk: future interest rates may be less than the YTM at the time the bond is purchased, leading to less reinvestment income (ie. the interest income generated by reinvesting coupon interest payments and any principal repayments from the time of receipt to the bond's maturity.)
  • Interest rate risk: cannot be held to maturity, and may have to be sold for less than the purchase price because the interest rate required by the market is higher than the YTM.

Category: C++ Quant > Finance > Debt > Valuation

Your Turn!

 

Q&A: Under what circumstance should an American call...

...be exercised early?

Answer : American call options are never exercised early unless the underlying makes cash payments. Stocks pay dividends, bonds pay interest, foreign currencies pay interest, and commodities have carrying costs. Whether to exercise early depends on the tradeoff between the time value of the call and the cash payments from the underlying.

All American puts may be exercised early, regardless of whether the underlying makes cash payments or not.

Category: C++ Quant > Finance > Derivatives > Options

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Q&A: Under what types of market structure would...

...it be possible to have two separate but simultaneous transactions where bond X was traded at unit prices of $105 and $95 respectively?

Answer : A broker gets involve when buyers and sellers of securities find themselves unable to identify each other without high costs (mostly due to heavy trading volume). The broker may not be able to execute the order instantaneously, so it is essential that transactions away from the best possible price must not be too costly.

The secondary market for the U.S. Treasury is organized as a dealer market. Dealers take outright positions (ie. inventory) in the market and is exposed to market risk. Therefore the bid-offer prices will vary from dealer to dealer. Their inventory save time spent in searching for buyers and sellers.

Bond X is more likely to have been traded in an illiquid dealer market.

Category: C++ Quant > Debt

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Q&A: What is the duration/convexity of...

...a 12%, 12-year, option-free bond selling at $125?

Answer : Duration = (V- - V+) / (2 * V0 * yield_change)

  • yield: with Excel Rate function, Nper = 24, PV=-125, PMT = 12/2, FV=100, which leads to rate= 4.31%. Annualize it one gets 8%.
  • Let yield change be 0.5% (50 bps)
    • for yield = 8.5%: Nper = 24, PMT = 12/2, Rate = (8.5/2)%, FV=100, PV = 126.01 = V+
    • for yield = 7.5%: Nper = 24, PMT = 12/2, Rate = (7.5/2)%, FV=100, PV = 135.20 = V-
  • Duration = (135.20 - 126.01)/(2 * 125 * 0.005) = 7.35

Convexity=(V+ + V- - 2*V0) / (2 * V0 * yield_change^2).

  • Convexity = (135.20 + 126.01 - 2*125) / (2*125*0.005^2) = 17.93

Bonus Points

  • It tells us that for 1% change in the required yield, the bond price will change by approximately 7.35%.
  • For a 10 basis point increase in yield, the approximate percentage price change is -7.35 x 0.001 x 100 = -0.74%
    • Approximate percentage price change = - duration x change in yield x 100
    • the negative sign due to the inverse relationship between price change and yield change - when yields increase, bond prices fall
  • Convexity adjustment takes into account the curvature of the price/yield relationship: convexity x yield_change^2 x 100 = 17.93*0.005^2*100 = 0.04%

Category: C++ Quant > Finance > Debt > Valuation

Your Turn!