FMP16 - Properties of Interest Rates

Question 1

Describe Treasury rates, Libor, Secured Overnight Financing Rate (SOFR), and repo rates and explain what is meant by the “risk free” rate

Answer 1

• Treasury rate: interest rate paid by government on its borrowings on its own currency
• Libor: London Interbank Overnight Rate, rate charged between banks for borrowing money overnight
• Repo rate: rate at which securities are sold between parties with the view of repurchasing at a later date for slightly more, each currency has its own index
• SOFR is the US index for repo rate for overnight cash requirements, used as US risk free rate

Question 2

Calculate the value of an investment using different compounding frequencies

Answer 2

Future value = A (1 + R/m) ^ (mT)
A = amount, R = rate, m = times per year compounded, T = years

If continuous compounding, F = Ae^RT

Question 3

Convert interest rates based on different compounding rates

Answer 3

Set future values equal to each other with only unknown being the rate compounded differently to what you have and solve for R2

Question 4

Calculate the theoretical price of a bond using spot rates

Answer 4

Discount each cashflow coming from the bond using that year’s spot rate and sum them

Question 5

Calculate the Macauley duration, modified duration, and dollar duration of a bond

Answer 5

Macauley duration = ((sum(t = 1 to n) t * C / (1 + y)t) + (n * A / (1 + y)^n)) / current bond price
C = coupon, n = duration of bond in periods, A = principal

Modified duration = Macauley Duration / (1 + y/m)
m = times paid per peiod, y is rate

Dollar duration D$ = - deltaB / (B * deltay)
B = price of bond, y = yield

Question 6

Evaluate limits of duration and explain how convexity addresses some of them

Answer 6

Duration limited when shifts in interest rates are non-parallel, convexity solves by allowing for non-linearity

Question 7

Calculate the change in a bond’s price given its duration, convexity, and change in interest rates

Answer 7

dB = - D * B * dy + 0.5 * C * B * (dy)^2

Question 8

Derive forward interest rates from a set of spot rates

Answer 8

Method:
1. calculate how much one dollar grows by at T1 (= V1)
2. calculate how much one dollar grows by at T2 (= V2)
3. Calculate what dollar at T1 is equivalent of at T2 (=V2 / V1)
4. solve 1 + F/m = V2 / V1 for F

Question 9

Derive the value of the cashflows from a forward rate agreement (FRA)

Answer 9

Agreement that a certain interest rate will apply to a principal for a period in the future

Payoff = (R - Rk) * T * A / (1 + R * T)
R is realised rate, Rk is realised rate