Immunization Strategies

Immunization Strategies + Term Structure of Interest RatesDR. HIMANSHU JOSHI

Basic Simple Immunization Model

Consider the following situation: A firm has a known future obligation, Q. (An insurance firm, which knows that it has to make a payment in future.) The discounted value of this obligation is

V0 = Q/(1+r)N

Where r is the appropriate discounting rate.

Suppose that this future obligation is hedged by a bond held by the firm. That is, the firm currently holds a bond whose value VB is equal to the discounted value of future obligation V0.

If P1, P2, and …..PM is the stream of anticipated payments made by the bond, then the bond’

S present value s given by:

VB = ∑Mt=1 = Pt / (1+ r)t

A numerical Example Suppose you are trying to immunize a year-10 obligation whose present value is $1,000;

that is at the current interest rate of 6%, its future value is $1000*(1.06)10 = $1790.85.

You intend to immunize the obligation by purchasing $1000 worth of a bond or a combination of bonds.

You consider three bonds:

1. Bond 1 has 10 years remaining until maturity, a coupon rate of 6.7 percent, and a face value of $1000.

2. Bond 2 has 15 years until maturity, a coupon rate of 6.988 percent, and a bond face value of $1000.

3. Bond 3 has 30 years until, maturity, a coupon rate of 5.9 percent and a face value of $1000.

Simon Benninga “Financial Modeling”, The MIT Press, Cambridge, Massachusetts, USA, III Ed., pp-695-698.

Prices of Bonds at 6% Interest RateBond Prices at 6% interest rate 10 15 30Interest Rates 0.06Particulars Bond 1 Bond 2 Bond 3Settlement Date 1/1/2000 1/1/2000 1/1/2000Maturity Date 1/1/2010 1/1/2015 1/1/2030Coupon 0.067 0.06988 0.059Yield 0.06 0.06 0.06Redemption 100 100 100Frequency 1 1 1Basis 0 0 0Price 105.15206 109.595702 98.62351688Price of Bond in (,000) 1051.5206 1095.95702 986.2351688Face Value of the Bond 1000 1000 1000Face value equal to $1000 of Market Value 951.00371 912.4445409 1013.956946Duration 7.665498 10.00000886 14.6360603MDurarion 7.2316019 9.433970626 13.80760406% of Face Value bought 0.9510037 0.912444541 1.013956946

Immunization Problem Illustrated for 30 Year Bond.

Year 10: Future Obligation of

$1790.850

30

Buy $1014 face value of 30 year

bond Reinvest Coupon

from bond during 1-10

years

Sell bond for PV of

remaining coupons and

redemption in year 30

Immunization Problem Illustrated for 30 Year Bond.

When the Interest Rate Increases:

When the Interest Rate Decreases:

Value of Reinvested Coupon Increases Value of bond in year 10 decreases

Value of Reinvested Coupon Increases Value of bond in year 10 decreases

Basic Immunization Example: 3 BondsFuture Value Calculation of Three Bonds on 10 yearsNew YTM 0.06Particulars Bond 1 Bond 2 Bond 3Remaining Life 0 5 20Settlement Date 1/1/2000 1/1/2000 1/1/2000Maturity Date 1/1/2000 1/1/2005 1/1/2020Coupon 0.067 0.06988 0.059YTM 0.06 0.06 0.06Redemption 100 100 100Freq 1 1 1Basis 0 0 0Price after 10 years 100 104.1618154 98.85300788Price for $1000 value 1000 1041.618154 988.5300788Calculation of Reinvested CouponsValue of Coupons in $ 67 69.88 59Value of Coupons at 10th Year $883.11 $921.07 $777.67Total $1,883.11 $1,962.69 $1,766.20Multiply by Percent of face value bought $1,790.85 $1,790.85 $1,790.85

Immunization Data Table Showing FV of Bonds at different YTM..

Immunization Data TableBond 1 Bond 2 Bond 3YTM FV YTM FV YTM FV

$1,790.85 $1,790.85 $1,790.850.01 1617.6271 0.01 1844.71 0.01 2536.4160.02 1648.6898 0.02 1825.139 0.02 2315.61350.03 1681.4506 0.03 1810.0478 0.03 2137.23430.04 1715.9998 0.04 1799.3472 0.04 1994.02390.05 1752.4324 0.05 1792.9655 0.05 1880.13530.06 1790.8477 0.06 1790.8477 0.06 1790.84770.07 1831.3498 0.07 1792.9548 0.07 1722.3440.08 1874.0476 0.08 1799.2629 0.08 1671.53450.09 1919.0554 0.09 1809.7627 0.09 1635.916

0.1 1966.4926 0.1 1824.4591 0.1 1613.461

Min 1617.6271 Min 1790.8477 Min 1613.461Max 1966.4926 Max 1844.71 Max 2536.416STD 117.3268 17.664644 313.0389

Term Structure Modeling Description of Data:

Yahoo’s Bond Screener (http://screen.yahoo.com/bonds.html)

Original Source: ValuBond

http://screen.yahoo.com/bonds.html

CurrentYield(%)

Zero 15-Aug-2016 99.39 0 15-Aug-16 0.362 0 AAA No

Zero U S TREAS SEC STRIPPED INT PMT 15-Aug-2016 99.36 0 15-Aug-16 0.374 0 AAA No

Zero U S TREAS BD STRIPPED PRIN PMT 15-Nov-2016 99.1 0 15-Nov-16 0.463 0 AAA No

Zero U S TREAS SEC STRIPPED INT PMT 15-Nov-2016 99.11 0 15-Nov-16 0.457 0 AAA No

Zero U S TREAS NT STRIPPED PRIN PMT 15-Feb-2017 98.79 0 15-Feb-17 0.551 0 AAA No

Zero U S TREAS SEC STRIPPED INT PMT 15-Feb-2017 98.72 0 15-Feb-17 0.584 0 AAA No

Zero U S TREAS BD STRIPPED PRIN PMT 15-May-2017 98.4 0 15-May-17 0.657 0 AAA No

Zero U S TREAS NT STRIPPED PRIN PMT 15-May-2017 98.46 0 15-May-17 0.634 0 AAA No

Zero U S TREAS SEC STRIPPED INT PMT 15-May-2017 98.36 0 15-May-17 0.675 0 AAA No

Zero U S TREAS BD STRIPPED PRIN PMT 15-Aug-2017 97.98 0 15-Aug-17 0.756 0 AAA No

Zero U S TREAS SEC STRIPPED INT PMT 15-Aug-2017 97.91 0 15-Aug-17 0.78 0 AAA No

Zero U S TREAS NT STRIPPED PRIN PMT 15-Nov-2017 97.44 0 15-Nov-17 0.88 0 AAA No

Zero U S TREAS SEC STRIPPED INT PMT 15-Nov-2017 97.39 0 15-Nov-17 0.897 0 AAA No

Zero U S TREAS NT STRIPPED PRIN PMT 15-Feb-2018 96.93 0 15-Feb-18 0.974 0 AAA No

Zero U S TREAS SEC STRIPPED INT PMT 15-Feb-2018 96.89 0 15-Feb-18 0.987 0 AAA No






Zero U S TREAS BD STRIPPED PRIN PMT 15-Feb-2019 94.71 0 15-Feb-19 1.296 0 AAA No










ero U S TREAS BD STRIPPED PRIN PMT 15-Nov-2022 85.25 0 15-Nov-22 2.016 0 AAA No















Fitch Ratings

CallableType Issue Price Coupon(%) Maturity YTM(%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 450

0.5

1

1.5

2

2.5

3

Chart Title

Information on expected future short term rates can be implied from the yield curve

The yield curve is a graph that displays the relationship between yield and maturity

Three major theories are proposed to explain the observed yield curve

Overview of Term Structure

Figure 15.1 Treasury Yield Curves

Bond Pricing If yields on different maturity bonds are not all equal how should we value coupon bonds that makes payments at many different maturities?

Maturity YTM (%) Price

1 5% $952.38

2 6% $890.00

3 7% $816.00

4 8% $735.03

STRIPS A 1 year maturity T-bond paying semi-annual coupons can be split into:

A six month maturity zero (by selling the first coupon payment as stand alone security)

And a 12 month zero (corresponding to final payment)

If each security can be sold off as a separate security, then value of the whole bond should be the sum of the value of its cash flows bought by piece in STRIPS market.

STRIPS If it is not? Arbitrage opportunity: violation of law of one price.

Then there would be easier profits to be made.

If Bond Selling Price < Sum of its Parts in Strip Market (Bond Stripping)

If Bond Selling Price > Sum of its Parts in Strips Market (Bond Reconstitution)

Bond as portfolio of STRIPPED Zeros

The Pure Yield Curve: curve for stripped or zero coupon treasuries.

The On the Run Yield Curve: plot of yields as a function of maturity of recently issued coupon bonds selling at or near par value.

And there may be a significant difference between the two.

Yield Curve Under Certainty Why is curve some time upward slopping and other times downward slopping?

How do expectations for the evolution of interest rate affects the shape of today’s yield curve?

Certainty, here means one in which all investors already know the path of future interest rates.

Yield Curve Under Certainty

An upward sloping yield curve is evidence that short-term rates are going to be higher next year

When next year’s short rate is greater than this year’s short rate, the average of the two rates is higher than today’s rate

22 1 2

1

22 1 2

(1 ) (1 ) (1 )

1 (1 ) (1 )

y r x r

y r x r

Yield Curve Under Certainty Q. If the interest rates are certain what do we make of the fact that yield on 2 year zero coupon bond in previous example is higher than on the 1 year zero coupon?

Q. Can it be that one bond is expected to provide a higher rate of return than the other? (Given the market is competitive)

Yield Curve Under Certainty Instead the upward slopping yield curve is evidence, that short term rates are going to be higher next year than they are now.

Why?

Consider a two year holding strategies:

Figure 15.2 Two 2-Year Investment Programs

Yield Curve Under Certainty Remember both strategies must provide equal returns as neither entails any risk:

Buy and Hold two year zero = Rollover 1 year Bond

$890 (1.06)2 = $890 (1.05) (1+r2)

Yield Curve Under Certainty So, while the 1 year bond offers a lower Yield to Maturity than 2 year bond (5%vs 6%), we see it has a corresponding advantage. It allows you to rollover your funds into another short term bond next year when rates are higher.

Next year’s rate is higher than today’s by just enough to make rolling over 1 year bond equally attractive.

Figure 15.3 Short Rates versus Spot Rates

Yield Curve Under Certainty Not surprisingly, the two year spot rate is an average of today’s short rate r1 and next year’s short rate r2. but because of compounding, that average is a geometric one.

(1+y2)2 = (1+r1) (1+r2)

1+y2 = [(1+r1) (1+r2)]1/2 -----------------(1)

When r2> r1 then y2> r1 (upward slopping)

When r2<r1 then y2<r1 (downward slopping)

11)1(

)1()1(

n

n

nn

n y

yf

fn = one-year forward rate for period n

yn = yield for a security with a maturity of n

)1()1()1( 11 nn

nn

n fyy

Forward Rates from Observed Rates

Example 15.4 Forward Rates

4 yr = 8.00% 3yr = 7.00% fn = ?

(1.08)4 = (1.07)3 (1+fn)

(1.3605) / (1.2250) = (1+fn)

fn = .1106 or 11.06%

Holding period Returns Multiyear cumulative returns on all our competitive bonds ought to be equal.

What about holding period returns over shorter periods such as one year?

In world of certainty all bonds must give identical returns, or investors will flock to higher return securities, bidding up their prices, and reducing their returns.

Bond PricingMaturity YTM (%) Price

1 5% $952.38

2 6% $890.00

3 7% $816.00

4 8% $735.03

Holding period Returns 1 year bond can be bought today for $1000/1.05 = $952.38. rate of return = ($1000-$952.38)/$952.38 = .05

The two year bond can be bought at $1000/1.062 =$890.

Return after one year => 890 (1.06)2 =890 (1.05) (1+r) so r = 7.01%.

The next year price = $1000/1.071 = $934.49

One year holding period return = $934.49 - $890/$890 = 0.05

Interest Rate Uncertainty and Forward Rates

We have so far that argued that, in a certain world, different investment strategies with common terminal dates must provide equal rates of return. Therefore under certainty:

(1+r1) (1+r2) = (1+y2)2 -----------(1)

What can we say when r2 is not known today?

Suppose today’s rate is r1= 5%, and that expected short rate for the following year is E(r2) =6%.

If investors cared only about the expected value of interest rate, then the YTM on 2 year zero would be determined by using the expected short rate in eq. (1)


(1+y2)2 = (1+r1) [1+ E(r2)] =1.05*1.06

Price of two year zero would be $1000/1.05*1.06 = $898.47.

But now consider a short term investor who wishes to invest for 1 year only.

She can purchase one zero at $1000/1.05 =$952.38, and lock in a risk free rate of 5%.

She can also purchase the 2 year zero. Its expected return is also 5%.


Next year: bond will have 1 year to maturity, we expect that the 1 year interest rate will be 6%, Price =$1000/1.06 =$943.40

Holding period return = = ($943.40 - $898.47)/898.47 = 5%.

But the rate of return on the 2 year bond is risky.

If next year’s interest rates turn out to be higher than expected rate i.e., >6% bond price will be below $943.40.

And if next year’s interest rates <6%. Bond price will exceed $943.40..

Why should short term investor would buy two year bond offering same return (5%) as risk free one year bond?

Only when two year bond offer higher expected rate of return.

means, when two year bond sell at a price lower than the $898.47 value we derive when we ignore risk.

Bond prices and Forward Rates with Interest Rate Risk

Suppose that most investors have short term horizons and therefore are willing to hold 2 year bond only if price falls to $881.83.

Return at this price is ($943.40- $881.83)/$881.83 = 7%.

The risk premium of the two year bond is therefore 2%. It offers an expected rate of return of 7% versus 5%.

When bond prices reflect a risk premium, however , the forward rate, f2, no longer equals the expected short rate, E(r2) =6%.

YTM on two year zeros selling at $881.83 is 6.49%.

1+f2 = (1+y2)2 / (1+y1) = 1.06492 /1.05 = 8%.


Result of the above example is that the forward rate exceed short rate.

We defined the forward rate as the interest rate that would need to prevail in the second year to make the long term and short term investment equally attractive, ignoring risk.

When we account for risk, it is clear that short term investors will shy away from the long term bonds unless it offers an expected return greater than that of 1 year bond.

Lower the expectation of r2 greater the anticipated return on the long term bond.


Therefore, if most individuals are short term investors, bonds must have prices that makes f2 greater than E(r2).

The forward rate will embody a premium compared with the expected future short-interest rate.

This is called liquidity premium.

Which compensates short term investors for uncertainty about the prices at which they will be able to sell their long term bond at the end of one year.

Expectations

Liquidity Preference

Market Segmentation Theory

Preferred Habitat Theory

Theories of Term Structure

Figure 15.4 Yield Curves

Figure 15.4 Yield Curves (Concluded)

Interpreting the Term Structure

If the yield curve is to rise as one moves to longer maturities◦ A longer maturity results in the inclusion of a new forward

rate that is higher than the average of the previously observed rates

◦ Reason:◦ Higher expectations for forward rates or◦ Liquidity premium

Empirical Evidences from the bond Markets..

Interest rates on bonds of different maturities move together over time.

When short term interest rates are low, long run interest rates tend to be high, such that yield curves are upward sloped and vice versa.

Yield curves are generally almost always upward slopped.

Empirical Evidence of Term Structure..

Recession 1969 Quarter 4-1970 Quarter 4, Total GDP declines is .1%.

Term structure begins inversion in 1968 quarter 3 correctly. (predicts the recession four Quarter in advance.)

Recession 1973 Quarter 4-1975Quarter 1, total GDP decline is 4.2%.

Term structure begins inversion in 1973 Quarter 2 Correctly. (predict the recession with a two quarter lead time.)

Empirical Evidence.. Recession 1980 Quarter 1 -1980 Quarter 3, total GDP decline is 2.6%.

Term structure begins inversion in 1978 Quarter 4 correctly. (predicts the downturn with a five quarter lead.)

Recession 1990 Quarter 3-1991 Quarter 1, total GDP decline is 1.8%.

Term structure shows inversion in 1989 Quarter 2.

Expectation Theory.. The expectation theory asserts that the expected future spot rate is equal in magnitude to the forward rate. thus there is a world of certainty.

We can characterize this approach by one of the following propositions:

a) The return on holding a long term bond to maturity is equal to the expected return on repeated investment in a series of the short term bonds.

b) OR, we can say that the expected rate of return over the next holding period is the same for bonds of all maturities.

Expectation Theory… In this theory shape of the yield curve is based on the market participants expectations of future interest rates.

Ex. One year Spot rate 7%

Two year spot rate 8%.

Investment =$1.

Expected Short rate = 10% or 6%.

Expectation Theory..

So we can say that expectation theory explains the fact 1 and fact 2 stated in empirical evidences.

1. Interest rates on bonds of different maturities move together over time.

2. When short term interest rates are low, long run interest rates tend to be high, such that yield curves are upward sloped and vice versa.

Liquidity Preference Theory Liquidity preference theory asserts that risk aversion will cause forward rates to be systematically greater than the expected spot rates.

It starts with the notion that investors are primarily interested in purchasing short term securities. That is, even though some investors may have longer holding periods, there is a tendency for them to prefer short term securities.

These investors realize that they may need funds earlier than anticipated and they recognize that they face less “Price Risk” if they invest in short term securities.

The Market Segmentation Theory..

According to this theory, individuals have strong maturity preferences, thus bonds of different maturities trade in separate markets.

This means that market for bonds of different maturities are completely separated and segmented and can not substitutable.

Investors have to decide whether they need short term or long term instruments.

In this situation, we know that investor prefer their portfolio to be liquid. Thus they will prefer short term instruments to long term instruments.

This higher demand to the short term instruments will cause higher prices and lower yield and an upward sloping yield curve.

The Preferred Habitat Theory Preferred Habitat Theory is the combination of the market segmentation theory and expectations theory, because investors care both expected return and maturity.

Additionally, investors have different investment horizons and to buy bonds outside their habitat, they need a meaningful premium.

Thus this theory allows market participants to trade outside of their preferred maturity if adequately compensated for the additional risk.

Preferred Habitat Theory.. If the yield curve slopes slightly upward, investors predict interest rates to stay about the same.

If the yield curve slopes sharply upward, short term rates are predicted to rise.

If the yield curve slopes flat, short term rates are predicted to fall slightly.

If the yield curve is slopes downward sharply, the investors predict sharp decline in interest rates.

Interpreting the Term Structure If the yield curve reflects expectations of future short rates, then it offers a potentially powerful tool for fixed income investors.

If we can use the term structure to infer the expectation of other investors in the economy, we can use those expectations as benchmarks for our own analysis.

If we are relatively more optimistic than other investors, that int. rate will fall, we will be more willing to extend our portfolios into longer term bonds.

Interpreting the Term Structure Unfortunately while yield curve does reflect expectations of future interest rates it also reflect other factors such as liquidity premiums.

Forecast of interest rates may have different implications depending upon whether these changes are driven by :

Changes in expected inflation

Or the real rate.

Interpreting the Term Structure First we ask what factors can account for a rising yield curve?

Under certainty:

(1+yn) = [(1+r1)(1+r2)…….. (1+rn)]1/n

Under Uncertainty:

(1+yn) = [(1+r1)(1+f1)………(1+fn)]1/n

Thus there is a direct relationship between yields on various maturity bonds and forward interest rate.

Interpreting the Term StructureWhat factors can account for rising yield curve?

Mathematically, if the yield curve is rising

fn+1 > Yn

In words, the yield curve is slopping upwrds at any maturity date, n, for which the forward rate for the coming period is greater than the yield at that maturity .

Example: forward rates and the slope of yield curve..

If the yield to maturity on 3 year zero-coupon bond is 7%, then yield on a 4 year bond will satisfy this equation:

(1+y4)4 = (1.07)3 (1+f4)

If f4 = 7%, then y4= 7%.

If f4 > 7% say f4=8%

Then (1+y4)4 = (1.07)3 (1.08)

Y4= 7.25%

Interpreting the Term Structure Given that an upward sloping yield curve is always associated with a forward rate higher than the spot rate or current YTM.

We ask next what account for the higher forward rates?

Unfortunately there are always two possible answers:

Fn = E(rn) + Liquidity Premium

Figure 15.5 Price Volatility of Long-Term Treasury Bonds

Interpreting the Term Structure 1+ Nominal Rate = (1+Real Rate) (1+Inflation Premium)

Or

Approx. Nominal Rate= Real Rate + Inflation Rate.

Real Rate [ rapidly growing economy, high government budget deficits and tight monetary policy]

Inflation rate [either the same, or rapid expansion of money supply, or supply side shocks]

Figure 15.6 Term Spread: Yields on 10-Year Versus 90-Day Treasury Securities

Forward Rates as Forward Contracts

In general, forward rates will not equal the eventually realized short rate

◦ Still an important consideration when trying to make decisions :

◦ Locking in loan rates

Words from Wall StreetGrading Bonds on Inverted Curve

The bond market is having relationship issues that are getting harder to ignore. Normally yield on long term government securities are higher than yields on short term government securities. The relationship has been upside down since July, 2006.

This is known as inverted yield curve- has some economist wondering whether the bond market is signaling that the economy itself could turn upside down.

Yield inversions are indicators of hard times.

Words from Wall Street When bond investors see recession coming. They tend to buy long term treasury securities for two reasons.

1. they are safer than stocks.

2. they are appealing when inflation is low, and recession tends to beat down inflation.

The buying that comes with recession fears drives down a long term bond’s yield, some times lower than the prevailing yield on short term treasuries.

Seven time between 1965 to 2005, yields on 10 year notes have dropped below those on the three month treasury bill for an extended period.

In six of these instances, the US. Economy went into recession soon after.

CASE

They are discussing the FLAT YIELD CURVE observed towards the end of the previous day. Will it continue like this today? It was the subject matter of discussion.

Jan 15, 2001, 9:45. Trading Room of bank.

FLAT YIELD CURVE

They get a news that RBI is uncomfortable with FLAT YIELD CURVE and it is expected that it would like to have a normal positively sloped curve.

Jan 15, 2001, 10:00 Trading Room of bank.

FLAT YIELD CURVE

Jan 15, 2001, 10:15 Trading Room of bank.

FLAT YIELD CURVE

Now, it is final! For the day, start selling long term securities. Is that

O.K.???

Why this decision ?

Does the bank make money at the end of the day!!!!!?????????

The yield curve we had in the evening was……..

This end was lowered!!!!

Consequence!!!!!!!!

CONNECTING THEORY WITH PRACTICE…

His observation is … ‘Whenever a new bond issue comes to the bond market, the players have a tendency to become crazy about new issues and forgets about old issued bonds.

Now, meet another man who is working in an investment company …

He is Michael Hopkins.

He read the observation given by Mr. Harry Jones.

He himself watched the market and also has same understanding about it.

And, finally come out with a great idea………………………………..

The idea made …

Billions of dollars for the investment company !!!!!!!!!!!

Everyone was very happy in the company.

After some time …

The company became bankrupt!!!!!!

The reason…

LIQUIDITY RISK ……….

Please never forget about“Peso Problem”

Immunization Strategies

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