ACTSC 231 Final ReviewAugust 10, 2011

Introduction• Jeffrey Baer• 4A Actuarial Science• Work terms at Manulife and Towers Watson• Waterloo SOS President, May 2009 – Aug 2010

Outline1. Growth of money2. Equations of value and fund performance3. Annuities 4. Loan amortization and sinking funds5. Bonds6. Spot/forward rates7. Duration/price sensitivity/immunization

Growth of Money Accumulation Functions:

a(0) = 1 and a(t) ≥ a(0) for all t ≥ 0 Future Value (FV/AV at time t) = a(t) * Present Value (PV) AK(t) is the FV at time t of $K investment made at time 0 = K

* a(t)

a(t) means that our accumulation starts at time 0! Only works for money invested at (or discounted to) t0

i.e. money invested at t1 cannot be accumulated to t5 using a(4)

Example: Let a(t) = t2 + 1, t>=0. What is the accumulated value at time 2 of deposits of $1 at time 0 and $2 at time 1?

Compound interest• a(t) = (1 + i)t , t ≥ 0• Pays interest on balance earned so far• a(t)*a(s) = a(t + s)

Example: Edward invests $100 for 2n years at 8% compound interest per year, and then reinvests the proceeds for another n years at x% compound interest. Jacob invests $100 for 2n years at x% compound interest per year, and then reinvests the proceeds for another n years at 10% compound interest. Calculate x (x>0) if Edward and Jacob have the same amount of money after 3n years.

Effective Rate of Interest

• The amount of interest payable over a period as a proportion of the balance at the beginning of the period

• i[n-1, n] = in =

• Compound: i

Keith deposits $4,300 into an account on March 1, 1998. The bank guarantees that the annual effective rate for a balance under $5,000 is 3.5% and for a balance over $5,000 is 5%.

Suppose that there are no other deposits or withdrawals except for a withdrawal of $1,000 on March 1, 2003 and a deposit of $500 on March 1, 2004. Find his account balance on March 1, 2006.

Effective Rate of Interest Example

Discount Rate Effective rate of discount:

d =

Compound discount: a(t) = (1 – d)-t

(1+i)t = (1-d)-t

d= i/(1 + i) i = d/(1 – d)

Present Value Discount Functions:

v(t) = 1/a(t) PV = FV * v(t) Compound: v(t) = (1+i)-t =

v = v(1) = 1/(1 + i) = 1 – d

We can use either discount functions or accumulation functions to get PVs or AVs!

Nominal Rates Not effective interest rates

Cannot be used directly for PV/AV calculations! Convertible/compounded mthly: must divide

the nominal rate by m to get an effective mthly rate: (1 + i) = (1 + i(m)/m)m

(1 – d) = (1 – d(m)/m)m

Force of Interest Force of Interest

δt = or a(t) = e ; v(t) = e-

Used for continuous compounding Compound interest: δ = ln(1+i) => constant

a(t) = eδt ; v(t) = e-δt

Example: If the monthly nominal discount rate d(12) is 5%, calculate δ, d(4) and i(1/2).

dsSt

0 dsSt

0

Interest Rates ExampleThe accumulated value of $1 at time t (0<=t<=1) is given by a second degree polynomial in t.

You are given that the nominal rate of interest convertible semi-annually for the first half of the year is 5% per annum, and the effective rate of interest for the year is 4% per annum.

Calculate δ3/4.

Equations of Value

• PV of cash inflow = PV of cash outflowExample: A single payment of $800 is made to replace 3 payments: $100 in 2 years, $200 in 3 years, and $500 in 8 years.

When should the payment of $800 be made, given the annual effective rate of 5%?

Reinvestment rates

• Often times, two or more interest rates involved in a transaction– Consider all cash flows to get total yield on investment

Example: Yafang makes a one time investment of $200 in an account earning an annual effective rate of 5%. If the annual interest payments from the account are reinvested in another account earning 10%, what is Yafang’s annual effective yield at the end of 5 years?

Fund Performance

• Exact (usually involves using quadratic formula) :

• Simple interest approximation (DWRR):

• Basic midpoint approximation:

• Time weighted rate of return (from s to t):

Yield Rates Example

Irene’s investment account has a balance of $1000 at the beginning of the year. On Feb. 1, the account had a balance of $1050 and Irene deposited an additional $200. On Jul. 1, the balance was $1400 and Irene deposited an additional $X. On Aug. 1, the balance was $1800 and Irene withdrew $700. At the end of the year, Irene had $900 in her investment account.

If Irene’s dollar-weighted rate of return was 0.096, find her time-weighted rate of return.

Level Annuities An annuity is a regular series of payments Annuity immediate: payments are made at the end

of the year

Annuity due: payments are made at the beginning of the year

Three components: amount of payment, interest/yield rate (i), length (n)

Level Annuities PV of $S/year for n years at ann. eff. interest rate i:

Immediate: PV = S*an¯| = Sv + Sv2 + Sv3 + … + Svn

= Sv(1 + v + v2 + … + vn-1) = Sv(1-vn)/(1-v)

Since v/(1-v) = i : = S(1-vn)/i Due: PV = S*än¯| = S(1-vn)/d

AV of $S/year for n years, int rate i, at time n: Immediate: AV = S*sn¯| = S(1+i)n-1 + S(1+i)n-2 + … + S(1+i) + S

= S(1 + (1+i) + … + (1+i)n-1) = S(1 – (1+i)n)/(1 – (1+i)) = S[(1+i)n – 1]/i

Due: FV = S* n¯| = S[(1+i)n – 1]/d

Level Annuities Relationship between due and immediate :

än¯| = S(1-vn)/[i/(1+i)] = (1+i)S(1-vn)/i= (1+i)an¯| (same with s)

än¯| = 1 + an-1¯| sn¯| = 1 + n - 1¯|

Usage of the accumulation function: If the last payment was just made, the AV should be calculated using an

immediate annuity If there are n deposits at annual intervals, with the last deposit just

made, the AV = sn¯|, regardless of how the time diagram is labelled i.e. if deposits of $1 are made at the beginning of the year for five years

starting two years from now, what annuity symbol represents the AV right after the last deposit?

Use the TVM functions on your financial calculator to calculate unknown interest rates, if necessary Make sure you reset your calculator!

Deferred Annuities Deferred Annuities:

A deferred annuity begins payments at some time t ≠ 0 or 1 PV of deferred annuity starting at time t = vt-1(an¯|) or vt(än¯|)

Example:At time t = 0, Batman deposits P into a fund crediting interest at an annual effective rate of 8%. At the end of each year in years 6 through 20, Batman withdraws an amount sufficient to purchase an annuity due of 100 per month for 10 years at a nominal interest rate of 12% compounded monthly. Immediately after the withdrawal at the end of year 20, the fund value is zero. Calculate P.

Other Annuities Perpetuities:

A perpetuity is an annuity with payments lasting forever AV of a perpetuity is infinite! PVperp. immediate

= a∞¯| = 1/i PVperp. due = ä∞¯| = 1/d

Continuous Annuities: ān¯| = PV of $1 paid continuously throughout the year for n years

=

= (1-vn)/ δ = (i/ δ)* an¯|

an¯| < ān¯| < än¯| <=> d < d(m) < δ < i(m) < i

sbarn¯| = [(1+i)n – 1]/ δ PV of continuous annuity at rate f(t) per year:

n

tdtv0

^

dttvtfn

)()(0

1. Payments are made into an account continuously at a rate of 8Y +tY per year, for 0≤ t≤10. At time T = 10, the account is worth $20,000. Find Y if the account earns interest according to a force of interest δt = 1/(8 + t) at time t, for 0 ≤ t ≤ 10.

Continuous Annuity Examples

Increasing Annuities Arithmetic Progression:

i.e. payment of 5 at t1 , 8 at t2 , 11 at t3 , etc. In general, for annuity immediate with first payment P and

constant increase of Q each period: PV = P(an¯|) + Q(an¯| - nvn)/i AV = P(sn¯| ) + Q(sn¯| - n)/i When P = Q = k:

PV = k(Ian¯|) = k(än¯| - nvn)/i AV = (Isn¯|) = k( n¯| - n)/i PV increasing perpetuity = k(Ia∞¯|) = k/(id)

In general, for annuity due with first payment P and constant increase of Q each period:

PV = P(än¯|) + Q(an¯| - nvn)/d AV = P( n¯| ) + Q(sn¯| - n)/d When P = Q = k:

PV = (Iän¯|) = k(än¯| - nvn)/d AV = (I n¯|) = k( n¯| - n)/d PV perpetuity = k(Iän¯|) = k/d2

Increasing Annuities Geometric Progression:

Consider an annuity immediate with payments increasing by a constant factor of 1+k (i.e. 1 at t1, 2 at t2, 4 at t3, etc. if k=1)

For interest rate i, initial payment of X: PV = X[1 – [(1+k)/(1+i)]n]/(i-k)

If i = k : PV = nXv If i ≠ k : PV = (Xv)än ¯|j , where j = (i-k)/(1+k)

AV = PV(1+i)n PV or AV due = PV or AV immediate*(1+i)

Example:Tiger Woods purchases an increasing annuity immediate for $80,630 that makes annual payments for 20 years to his 20 mistresses as follows:(i) For the first 10 years, the first payment is P and each subsequent payment is P more than the previous one (i.e. P, 2P. . . 10P for t = 1, 2, . . . , 10;) and(ii) For the remaining 10 years, the first payment is 10P(1.05) and each subsequent payment is 5% larger than the previous one (i.e. 10P(1.05), 10P(1.05)2, . . . ,10P(1.05)10 for t = 11, 12, . . . , 20).If the annual effective rate of interest is 7%, then determine the value of P.

Increasing Annuities Example

Different Int. and Pmt. Periods

• Use:a(m)

n¯| = i/i(m) * an¯|ä(m)

n¯| = d/d(m) * än¯|s(m)

n¯| = [(1+i)n-1]/i(m) (m)

n¯| = [(1+i)n-1]/d(m)

Or… just convert interest rates

Santa takes out a 25-year $200,000 mortgage on his workshop with monthly payments at 5% compounded semi-annually.

If Santa had taken out a 25-year $200,000 mortgage with weekly payments at 5% compounded semi-annually instead, how much less would he have paid after two years? (Assume payments can be non-integral.)

Different Int. and Pmt. PeriodsExample

Loan Amortization Loan Terminology

Principal (L): Balance of the loan at t0

Outstanding Loan Balance (Bt): Remaining balance of the loan / principal not yet paid at time t

Loan Payments and Outstanding Loan Balance Loan payments are an annuity! (Principal = Payment * an¯|) Each payment Xt = interest paid (It) + principal repaid (Pt)

It = i*Bt-1

Pt = Bt-1 – Bt

Bt can be calculated in two ways: Retrospective (backward): Bt = L(1+i)t – Xst¯|

Prospective (forward): Bt = X an-t¯|

• Example: Cyntha takes out a 20-year $25,000 amortized loan with payments made at the end of each year at an annual effective rate of 8%. Calculate the regular payment, the outstanding balance after the 10th payment, and the principal paid on the 11th payment.

What would be the regular payment and OLB after the 5th payment if the interest rate is 8% for the first 10 years, and 7% thereafter?

Loan Amortization Example

Sinking Funds Sinking Funds:

Instead of payments consisting of part principal, part interest: Interest payments are paid on the initial loan principal, which remains constant

—hence interest payments remain constant Separate deposits made into a “sinking fund” (SF) eventually accumulate to the

initial loan value Interest rate for interest pmts (i) may differ from SF interest rate (j)

Total payment per period = interest pmt + SF pmt

= i(L) + L / sn¯|j OLB for SF method = L – SF Balance

Pt = SF balancet – SF balancet-1

= interest earned on SF in periodt + deposit to SF at time t Interest payments are being made, but interest is also being accumulated

in the sinking fund It = Net interest paid at time t = i(L) – j(st-1¯|j )

Sinking Fund ExampleA yacht owner pays back a loan of $20,000 using a sinking fund. He makes interest payments at the end of each year for 10 years, using an annual effective interest rate of 7%. He deposits $2,000 annually for the first 5 years and $1,000 annually for the next 4 years into the sinking fund account (all payments are made at the end of each year). If the sinking fund account earns an annual effective interest rate of 3%, what is his payment at the end of the 10th year to pay off the loan?

Show the first two years of the amortization schedule under the sinking fund and amortization (assuming end of year payments at 7%) methods.

Amortization Schedules

Time t Interest Pmt Principal Repmt Total Pmt OLB t

0 20,000.00

1 1,400.00 1,447.55 2,847.55 18,552.45

2 1,298.67 1,548.88 2,847.55 17,003.57

Amortization Method:

Sinking Fund Method:

Time t Interest Pmt SF Pmt Total Pmt OLB t

0 20,000.00

1 1,400.00 2,000.00 3,400.00 18,000.00

2 1,400.00 2,000.00 3,400.00 15,940.00

Bonds Bond Terminology:

Face/Par Value (F): used to calculate coupon payments Coupon rate (r): % of F given as a coupon each period

Nominal coupon rates are generally provided, so we must convert to the effective coupon rate per period

Coupon (Fr): generally fixed sum of money paid regularly to the bondholder Number of Periods (n): number of coupons paid Yield rate (i): effective rate/period at which CFs are discounted

Nominal yield rates are generally provided—again, convert Redemption Value (C): future value of the bond at expiry

Unless otherwise stated, C = F (bond matures/redeemable at par: “Par Value”)

i.e. a $1,000 bond with 8% semi-annual coupons maturing in 10 yearsat par

Bonds Bond Pricing at Issue

Price = PV(Coupons) + PV(Redemption Value)

= (Fr) an¯|i + Cvin , where vi is determined using effective

yield rate per period

Premiums and Discounts If Price of bond > Redemption Value: Premium = P - C

Define g as Fr/C: then if g > i, Premium = Price – C = (Cg-Ci) an¯|i If Price of bond < Redemption Value: Discount = C - P

Define g as Fr/C: then if g < i, Discount = C – Price = (Ci – Cg) an¯|i If g = i, Price of bond = Redemption Value (“Sold at Par”)

A $1,500 4% 12-year bond is sold to yield 5% convertible semi-annually. The discount on the bond is $100. Find the redemption amount of the bond.

Bond Pricing Example

Valuing Bonds at Coupon Dates Book Value (directly after coupon payment)

Same concept as prospective Outstanding Loan Balance Bt = Fr an-t¯|i + Cvi

n-t

B0 = Price Bn = C n-t = number of remaining coupon payments

Valuing Bonds between Coupon Dates

Book Value between coupon payments To get dirty price, use Bt, book value of bond on

coupon date prior to redemption date Dirty Price: sale price of a bond between coupon

payments Dt+k = Bt(1+i)k, where k is a fraction of a period, 0<k<1 i.e. if semi-annual coupon date is July 23rd and sale is on

November 3rd, k = (8+31+30+31+3)/(8+31+30+31+30+31+23) = 103/184

(Semipractical) Clean Price: quoted (in a newspaper) price of a bond between coupon payments Ct+k = Bt(1+i)k – k(Fr) = Dt+k – k(Fr)

Bond Valuation Example

A $1000 bond redeemable at $1050 has 7.5% semi-annual coupons and matures on July 1, 2017.

Find the actual selling price of this bond on November 15, 2013, and the price that would be quoted in a financial newspaper on the same date, based on a nominal annual yield of 5.80% compounded semiannually.

Financial Analysis Spot Rate (rt):

Annual effective yield rate for an investment of length t years made at t0 (e.g. zero-coupon bond) i.e. if 3-year spot rate is 6%, can invest $1 at time 0 at 6% effective/year and

receive 1.063 at time 3

Forward Rate (ft,t+n) Annual effective yield rate earned on an investment made t years

from now for the period t -> t+n ft, t+n =

i.e. if 2-year forward rate from time 4 (f4,6) is 6%, we can make an investment 4 years from now that will earn 6% annual effective from time 4 to time 6

Forward Rate Example

If the 1-year spot rate is 2% and the price of a 4-year $100 zero coupon bond is $84.69, calculate the 3-year forward rate from time 1.

Duration Macaulay Duration

Represents average timing of an asset’s cash flows weighted by the PV of each cash flow

-P’(δ)/P(δ) or -d/dδ[ln P(δ)] , where P(δ) = asset price = [Σ(Ct*vt*t)]/[Σ(Ct*vt)]

= [Σ(Ct*vt*t)]/(Price of Asset) Single Cash Flow at time n: MacD = n Level annuity immediate: MacD = (Ia)n¯| / an¯|

Modified Duration -P’(i)/P(i) ModD = v(MacD)

Duration and Price Sensitivity Duration of a Portfolio

If a portfolio contains multiple assets with known durations and present values: Portfolio duration = Σ(Asset duration*PV asset)/(PV portfolio) Weighted average of individual asset durations, with weights = asset prices as a percentage of

the total portfolio price

Convexity (how sensitive duration is wrt δ or i) Macaulay: d2/dδ2(Price)); [ΣCt(vt)(t2)]/Price Modified: d2/di2(Price)); [ΣCt(vt+2) (t)(t+1)]/Price = v2[C(δ) +D(δ)]

Price Sensitivity Change in price of asset due to change in force of interest (Δ) New price= Original Price – (MacD)(Original Price)(Δ)

+0.5(Convexity)(Original Price)Δ2

Higher duration → More Price Sensitive

If interest rates increase, what happens to the price of our asset?

LeBron repays a loan with two payments. One year from today, he pays $1,000, and two years from today, he pays another $1,000. Assuming the annual effective rate of interest is 10%, find the following:(a) The Macaulay duration and convexity of the loan.(b) The new PV of his loan payments if the yield decreases to 9%.

Duration Example

Redington Immunization Redington Immunization:

Protects against small changes in interest rate Given a portfolio of assets and liabilities, immunized if: 1. PV Assets = PV Liabilities 2. Duration of Assets = Duration of Liabilities 3. Convexity of Assets > Convexity of Liabilities

Manulife needs to make three payments of $100 at times 2, 4, and 6. Manulife plans to meet these obligations with an investment plan that produces asset cash flows of X at time 1 and Y at time 5. The effective rate of interest is 10%.

Determine X and Y so that the investment plan has the same present value and duration as the liability cash flows.

Immunization Example

Questions?

Good Luck!jeffreybaer@gmail.com

Questions